diff --git a/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/2301.02174v1.pdf.txt b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/2301.02174v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..146962bf794a9132300be168b9e746ceeaf05290 --- /dev/null +++ b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/2301.02174v1.pdf.txt @@ -0,0 +1,2425 @@ +arXiv:2301.02174v1 [math.PR] 5 Jan 2023 +Large time behavior of semilinear stochastic partial differential +equations perturbed by a mixture of Brownian and fractional +Brownian motions +Marco Dozzi∗ +Ekaterina T. Kolkovska† +Jos´e A. L´opez-Mimbela† +Rim Touibi‡ +Abstract +We study the trajectorywise blowup behavior of a semilinear partial differential equation that is +driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different +types of random perturbations. The linear operator is supposed to have an eigenfunction of constant +sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters +of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for +the probability of finite time blowup and of blowup before a given fixed time. Essential tools are +the mild and weak form of an associated random partial differential equation. +Keywords Stochastic reaction-diffusion equation; mixed fractional noise; finite-time blowup of +trajectories +AMS Mathematics Subject Classification 60H15 60G22 35R60 35B40 35B44 35K58 +1 +Introduction +In this paper we study existence, uniqueness and the blowup behavior of solutions to the fractional +stochastic partial differential equation of the form +du(x, t) += +�1 +2k2(t)Lu(x, t) + g(u(x, t)) +� +dt + u(x, t) dNt, +x ∈ D, +t > 0, +u(x, 0) += +ϕ(x) ≥ 0, +u(x, t) += +0, +x ∈ ∂D, +t ≥ 0, +(1.1) +where D ⊂ Rd is a bounded Lipschitz domain, L is the infinitesimal generator of a strongly continuous +semigroup of contractions which satisfies conditions (3.18), (3.19) below, and ϕ ∈ L∞(D), where L∞(D) +is the space of real-valued essentially bounded functions on D. Additionally, g is a nonnegative locally +Lipschitz function and N is a process given by +Nt = +� t +0 +a(s) dB(s) + +� t +0 +b(s) dBH(s), +t ≥ 0, +(1.2) +∗corresponding author, marco.dozzi@univ-lorraine.fr, UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, +France +†Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico. +‡UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France. +1 + +where B is Brownian motion and BH is fractional Brownian motion with Hurst parameter H > 1/2, +a is continuous and b is H¨older continuous of order α > 1 − H. Both, B and BH, are supposed to be +defined on a filtered probability space (Ω, F, (Ft, t ≧ 0), P) and adapted to the filtration (Ft, t ≧ 0). +Such models have recently been studied under the name of ‘mixed models’ in the context of stochastic +differential equations, see [19] and [20]. When N = 0, L = ∆, k = 1, g(u) = u1+β we obtain the +classical Fujita equation which was studied in [10]. In [7] and [1] there were considered the cases when +N is a Brownian motion, in [5] it was investigated the case when N is a fractional Brownian motion +with Hurst parameter H > 1/2 and D ⊂ Rd, and in [6] the case of H ≥ 1/2 and D = Rd. +The fractional Brownian motion (fBm) appears in many stochastic phenomena, where rough exter- +nal forces are present. The principal difference, compared to Brownian motion, is that fBm is not a +semimartingale nor a Markov process, hence classical theory of stochastic integration cannot be applied. +Since H > 1/2, the stochastic integral with respect to BH in (1.1) can be understood as a fractional +integral. Also the presence of both, Brownian and fractional Brownian motion in (1.1), due to their +different analytic and probabilistic properties, modelize different aspects of the random evolution in +time of the solution. The factor k2/2 in front of L affects dissipativity, which in several cases is in favor +of retarding or even preventing blowup. +We consider both, weak and mild solutions of (1.1), which we prove are equivalent and unique. +Beyond existence and uniqueness of weak and mild solutions we are interested in their qualitative +behaviour. In Theorem 3 below we obtain a random time τ ∗ which is an upper bound of the explosion +time τ. In Theorem 8 we obtain a lower bound τ∗ of τ so that a.s. +τ∗ ≤ τ ≤ τ ∗. +The random times τ∗ and τ ∗ are given by exponential functionals of the mixture of a Brownian and a +fractional Brownian motion. The laws of such kind of functionals presently are not known. In order to +study the distribution of τ ∗ we use the well-known representation of BH in the form +BH +t = +� t +0 +KH(t, s) dWs, +where the kernel KH is given in (3.23) and W is a Brownian motion defined in the same filtered +probability space as B. In general, W can be different from the Brownian motion B appearing in the +first integral of (1.2). We obtain estimates of the probability P(τ < ∞), and of the tail distribution +of τ ∗. To achieve this we make use of recent results of N.T. Dung [8, 9] from the Malliavin theory for +continuous isonormal Gaussian processes. +In Theorem 4 we obtain upper bounds for P(τ ∗ ≤ T) in the case when B = W, and in Theorem +5 when B is independent of W, and when B and W are general Brownian motions. In Theorem 6 +we obtain lower bounds for P(τ < ∞) when B = W. As a result in the case when W = B we get +specific configurations of the coefficients a, b and k under which the weak solution (hence also the mild +solution) of equation (1.1) exhibits finite time blow-up. To be concrete suppose that g(z) ≥ Cz1+β for +2 + +some constants C > 0, β > 0, BH +t = +� t +0 KH(t, s) dBs, and +� t +0 +a2(r) dr ∼ t2l, +� t +0 +b2(r) dr ∼ t2m, +� t +0 +k2(r) dr ∼ t2p +as +t → ∞ +for some nonnegative constants l, m and p. If β ∈ (0, 1/2) and max{p, l} > H + m − 1/2, or if β = 1/2 +and p > H +m−1/2, or if β > 1/2 and p > max{l, H +m−1/2}, then all nontrivial positive solutions +of (1.1) suffer finite-time blowup with positive probability. +Our approach here is to transform the equation (1.1) into a random partial differential equation +(RPDE) (2.5), whose solution blows up at the same random time τ as the solution of (1.1), and to +work with this equation. The blowup behavior of (2.5) is easier to determine because N appears as +a coefficient, and not as stochastic integrator as in (1.1). Such transformations are indeed known for +more general SPDEs than (1.1), including equations whose stochastic term does not depend linearly +on u, see [17]. But for the RPDE’s associated to more general SPDE’s it seems difficult to find explicit +expressions for upper and lower bounds for the blowup time, and this is an essential point in our study. +Another reason for having chosen the relatively simple form of (1.1) and (2.5) is that we consider the +blowup trajectorywise which is a relatively strong notion compared, e.g., to blowup of the moments of +the solution (see, e.g. [4]). The crucial ingredient in the proofs is the existence of a positive eigenvalue +and an eigenfunction with constant sign of the adjoint operator of L. Special attention is given to the +case H ∈ (3 +4, 1) because then the process N is equivalent to a Brownian motion [3]. This allows us to +apply a result by Dufresne and Yor [27] on the law of exponential functionals of the Brownian motion +to get in Theorem 7 an explicit lower bound for the probability of blowup in finite time. +We finish this section by introducing some notations and definitions we will need in the sequel. A +stopping time τ : Ω → (0, ∞) with respect to the filtration (Ft, t ≧ 0) is a blowup time of a solution u +of (1.1) if +lim sup +tրτ +sup +x∈D +|u(x, t)| = +∞ +P-a.s. +Let (P D +t , t ≧ 0) and ((P D)∗ +t , t ≧ 0) +be the strongly continuous semigroups corresponding to the +operator L and its adjoint L∗ : +� +D +f(x)P D +t g(x)dx = +� +D +g(x)(P D)∗ +tf(x)dx, +f, g ∈ L2(D). +(1.3) +As usual, Lf := lim +t→0 +1 +t (P D +t f − f) for all f ∈ L2(D) in the domain of L, denoted by Dom(L). Due to +the Hille-Yosida theorem, Dom(L) and Dom(L∗) are dense in L2(D). Let P D +t (x, Γ) and (P D)∗ +t (x, Γ) +denote the associated transition functions, where t > 0, x ∈ D, and Γ ∈ B(D), the Borel sets on D. +In the sequel we will assume that they admit densities, i.e. there exist families of continuous functions +3 + +(pD(t, ·, ·), t > 0) and ((pD)∗(t, ·, ·), t > 0) on D × D such that +P D +t g(x) += +� +D +g(y)P D +t (x, dy) = +� +D +g(y)pD(t, x, y)dy, +(P D)∗ +t f(x) += +� +D +f(y)(P D)∗ +t (x, dy) = +� +D +f(y)(pD)∗(t, x, y)dy. +Due to (1.3), +(pD)∗(t, x, y) = pD(t, y, x) +for all t > 0 and x, y ∈ D. +(1.4) +2 +The weak solution of the associated random partial differential +equation, equivalence with the mild solution +Let us consider the random partial differential equation +∂v +∂t (x, t) += +1 +2k2(t)Lv(x, t) − 1 +2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)), +(2.5) +v(x, 0) += +ϕ(x), x ∈ D, +v(x, t) += +0, t ≥ 0, x ∈ ∂D. +In this section we transform the weak form of (1.1) into the weak form of (2.5) using the transformation +v(x, t) = exp(−Nt)u(x, t), x ∈ D, t ≥ 0. Hence, if blowup takes place in finite time, it occurs of course +at the same time and at the same place x ∈ D for the solutions of both equations. +In the following we write ⟨·, ·⟩D for the scalar product in L2(D). +Definition 1. An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ∈ [0, T], x ∈ D) with values in +L2(D) is a weak solution of (2.5) if, for all t ∈ [0, T] and all f ∈ Dom(L∗), P-a.s. +⟨v(·, t), f⟩D += +⟨ϕ, f⟩D + +� t +0 +�1 +2k2(s) ⟨v(·, s), L∗f⟩D − 1 +2a2(s) ⟨v(·, s), f⟩D +� +ds ++ +� t +0 +exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds. +(2.6) +Since g is supposed to be locally Lipschitz, a blowup in finite time of v may occur, and the blowup +time τ depends in general on ω ∈ Ω. A weak solution of (2.5) up to τ is defined as an (Ft, t ≧ 0)- +adapted random field v that satisfies (2.6) for all t ∈ (0, T ∧ τ) P-a.s. If ω is such that v(ω, ·, ·) does +not blowup in finite time, we set τ(ω) = ∞. +Definition 2. An (Ft, t ≧ 0)-adapted random field u = (u(x, t), t ∈ [0, T], x ∈ D) with values in L2(D) +is a weak solution of (1.1) up to τ if, for all t ∈ (0, T ∧ τ) and all f ∈ Dom(L∗), P-a.s. +(i) +� t +0 +a2(s) +� +1 + ⟨u(·, s), f⟩2 +D +� +ds < ∞, +b(•) ⟨u(·, •), f⟩D ∈ Cβ[0, t] for some β > 1 − H, +4 + +(ii) +� t +0 +� +k2(s) |⟨u(·, s), L∗f⟩D| + |⟨g(u(·, s)), f⟩D| +� +ds < ∞, +and +⟨u(·, t), f⟩D = ⟨ϕ, f⟩D + +� t +0 +�1 +2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s), f⟩D +� +ds ++ +� t +0 +⟨u(·, s), f⟩D dNs. +(2.7) +Conditions (i) and (ii) in the above definition are sufficient for the Itˆo, the fractional and the Lebesgue +integrals in (2.7) to be well defined P-a.s. +We proceed now to the relation between (2.7) and (2.6). +Proposition 1. If u is a weak solution of (1.1) up to a random time τ, then v(x, t) = exp(−Nt)u(x, t) +is a weak solution of (2.5) up to τ, and viceversa. +Remark 1. We notice that ⟨v(·, s), f⟩D is absolutely continuous in s if v is a weak solution of (2.5). +With the choice u(x, t) := exp(Nt)v(x, t) condition (i) is satisfied. In fact, for t < T ∧ τ(ω), +� t +0 +⟨u(·, s), f⟩D a(s) dBs = +� t +0 +⟨v(·, s), f⟩D exp(Ns)a(s) dBs +is well defined since +� t +0( +� +D v(x, s)f(x) dx)2 exp(2Ns)a2(s) ds < ∞ P-a.s. +Recall that the fractional integral +� T +0 f(x)dg(x) is defined (in the sense of Z¨ahle [28]) in [18, Def. +2.1.1] for f, g belonging to fractional Sobolev spaces. If 0 < ε < H, f and g are H¨older continuous +of exponents α and H − ε respectively, and α + H − ε > 1, this fractional integral coincides with +the corresponding generalized Riemann-Stieltjes integral; see [18, Thm. 2.1.7]. Hence, the fractional +integral +� t +0 +⟨u(·, s), f⟩D b(s) dBH +s = +� t +0 +⟨v(·, s), f⟩D exp(Ns)b(s) dBH +s +(2.8) +is well defined for t < T ∧ τ(ω) because, on the one hand, N· = +� · +0(a(s) dBs + b(s) dBH +s ) is P-a.s. +H¨older continous of order 1/2 − ǫ for all ǫ > 0 by the theorem of Kolmogorov and [22, Proposition +4.1]. On the other hand b(·) is α-H¨older continuous (with α > 1 − H) and BH is H¨older continuous +with exponent H − ε for any ε > 0. Hence, choosing ε < min{H/2 − 1/4, α + H − 1} we get that +the integrand on the right side of (2.8) is H¨older continuous of order min{α, 1/2 − ε}, and therefore +H −ε+min{α, 1/2−ε} > 1 and the integral is well defined as a generalized Riemann-Stieltjes integral. +Proof. Let T > 0. It suffices to prove the assertion for t ∈ (0, T ∧ τ). We apply (a slight generalisation +5 + +of) the Itˆo formula in [18, page 184]. Let +Y 1 +t += +� t +0 +a(s) dBs , +Y 2 +t = +� t +0 +⟨u(·, s), f⟩D a(s) dBs, +Y 3 +t += +� t +0 +b(s) dBH +s , +Y 4 +t = +� t +0 +⟨u(·, s), f⟩D b(s) dBH +s , +Y 5 +t += +⟨ϕ, f⟩D + +� t +0 +�1 +2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D +� +ds, +and let F(y1, y2, y3, y4, y5) = exp(−y1 − y3)(y5 + y2 + y4). Then +F(Y 1 +t , . . . , Y 5 +t ) = exp(−Nt) ⟨u(·, t), f⟩D = ⟨v(·, t), f⟩D . +The above mentioned Itˆo formula then reads +F(Y 1 +t , . . . , Y 5 +t ) += +F(Y 1 +0 , . . . , Y 5 +0 ) + +5 +� +i=1 +� t +0 +∂F +∂yi +(Y 1 +s , . . . , Y 5 +s ) dY i +s + 1 +2 +2 +� +i,j=1 +� t +0 +∂2F +∂yi∂yj +(Y 1 +s , . . . , Y 5 +s ) d +� +Y i +s , Y j +s +� +. +Since u is a weak solution of (1.1), +⟨v(·, t), f⟩D += +⟨ϕ, f⟩D − +� t +0 +exp(−Ns) ⟨u(·, s), f⟩D +� +a(s) dBs + b(s) dBH +s +� ++ +� t +0 +exp(−Ns) ⟨u(·, s), f⟩D +� +a(s)dBs + b(s)dBH +s +� ++ +� t +0 +exp(−Ns) +�1 +2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D +� +ds +−1 +2 +� t +0 +exp(−Ns) ⟨u(·, s), f⟩D a2(s)ds += +⟨ϕ, f⟩D + +� t +0 +�1 +2k2(s) ⟨v(·, s), L∗f⟩D − 1 +2a2(s) ⟨v(·, s), f⟩D +� +ds ++ +� t +0 +exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds. +Therefore v is a weak solution of (2.5). Similarly we obtain the viceversa result. +In order to define the mild solutions of equations (1.1) and (2.5) we define first the evolution families +of contractions corresponding to the generator 1 +2k2(t)L. For 0 ≤ s < t let +K(t, s) = 1 +2 +� t +s +k2(r) dr, +A(t, s) = 1 +2 +� t +s +a2(r) dr, +K(t) = K(t, 0), +A(t) = A(t, 0), +(2.9) +6 + +and set pD(s, x; t, y) = pD(K(t, s), x, y), x, y ∈ D × D, 0 ≦ s < t. For f ∈ L2(D) the corresponding +evolution families of contractions on L2(D) are given by +U D(t, s)f(x) += +� +D +pD(s, x; t, y)f(y)dy = P D +K(t,s)f(x), +(U D)∗(t, s)f(x) += +� +D +pD(s, y; t, x)f(y)dy = (P D)∗ +K(t,s)f(x). +Definition 3. An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ≧ 0, x ∈ D) with values in L2(D) is +a mild solution of (2.5) on [0, T] if, for all t ∈ [0, T], P-a.s. +v(x, t) += +U D(t, 0)ϕ(x) − 1 +2 +� t +0 +a2(s)U D(t, s)v(x, s) ds ++ +� t +0 +exp(−Ns)U D(t, s) +� +g((exp Ns)v(x, s)) +� +ds. +Proposition 2. The mild form of (2.5) can be written as +v(x, t) += +exp(−A(t))U D(t, 0)ϕ(x) ++ +� t +0 +exp(−Ns − A(t, s))U D(t, s)g(exp(Ns)v(·, s))(x)ds, +(2.10) +where A(t, s) and A(t) are given in (2.9). +Remark 2. Since g and ϕ are supposed to be nonnegative, +v(x, t) ≥ exp(−A(t))U D(t, 0)ϕ(x) ≥ 0 for all x ∈ D and t ≥ 0. +Proof. Let w(x, t) = exp(A(t))v(x, t). For f ∈ L2(D), we get from the definition of the mild solution +d +dt⟨w(·, t), f⟩D += +1 +2a2(t) exp(A(t))⟨v(·, t), f⟩D + exp(A(t)) d +dt⟨v(·, t), f⟩D += +1 +2a2(t) exp(A(t))⟨v(·, t), f⟩D ++ exp(A(t)) +�1 +2k2(t) ⟨v(·, t), L∗f⟩D − 1 +2a2(t) ⟨v(·, t), f⟩D +� ++ exp(A(t)) exp(−Nt) ⟨g(exp(Nt)v(·, t)), f⟩D += +1 +2 exp(A(t))k2(t) ⟨v(·, t), L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt)v(·, t)), f⟩D += +1 +2k2(t) ⟨w(·, t), L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt − A(t))w(·, t)), f⟩D , +with boundary conditions w(x, 0) = ϕ(x) for x ∈ D and w(x, t) = 0 for x ∈ ∂D. Therefore w is a weak +solution of the RPDE formally given by +d +dtw(x, t) = 1 +2k2(t)Lw(x, t) + exp(A(t) − Nt)g(exp(Nt − A(t))w(x, t)). +7 + +By the definition of the mild solution +w(x, t) = U D(t, 0)ϕ(x) + +� t +0 +exp(A(s) − Ns)U D(t, s)g(exp(Ns − A(s))w(·, s))(x) ds. +Consequently, +v(x, t) += +exp(−A(t))w(x, t) += +exp(−A(t))U D(t, 0)ϕ(x) + +� t +0 +ds exp(−A(t, s) − Ns)U D(t, s)g(exp(Ns)v(·, s))(x). +Theorem 1. The equation (2.10) has a unique non-negative local mild solution, i.e. there exists t > 0 +such that (2.10) has a mild solution in L∞([0, t[×D). +Proof. Let T > 0 and denote ET = {v : [0, T] × D → L∞(D) : []v[] < ∞} , where +[]v[] := sup +0≤t≤T +∥v(t, ·)∥∞. +Let PT = {v ∈ ET : v ≥ 0} and for R > 0 let CR = {v ∈ ET : []v[] ≤ R}. Then ET is a Banach space +and PT and CR are closed subsets of ET . Let us now define +ψ(v)(t, x) = e−A(t)U D(t, 0)ϕ(x) + +� t +0 +e−A(t,s)−NsU D(t, s)g +� +eNsv(·, s) +� +(x) ds. +We will prove that for sufficiently big R and sufficiently small T, ψ is contraction on PT ∩ CR. Let +v1, v2 ∈ PT ∩ CR. Then +[]ψ(v1) − ψ(v2)[] ≤ +sup +0≤t≤T +� t +0 +��e−Ns � +g +� +eNsv1 +� +− g +� +eNsv2 +���� +∞ ds. +Let AT = sup0≤s≤T e|Ns| and GR = sup|x| 0 centered at 0. Then, +sup +0≤s≤T +��e−Nsvi(s, ·) +�� +∞ ≤ AT R, +i = 1, 2, +and +��e−Nsg +� +eNsv1(s, ·) +� +− e−Nsg +� +eNsv2(s, ·) +��� +∞ ≤ A2 +T KAT R ∥v1(s) − v2(s)∥∞ . +Therefore, +[]ψ(v1) − ψ(v2)[] ≤ +sup +0≤t≤T +� t +0 +A2 +T KAT R []v1 − v2[] ds = TA2 +T KAtR []v1 − v2[]. +8 + +We need +TA2 +T KAT R < 1. +(2.11) +In addition, we require that CR ∩ PT be mapped by ψ into itself. Let v ∈ CR ∩ PT . Using that for 0 ≤ +s ≤ T the operator U D(t, s) is a contraction, and that ∥eNsv(·, s)∥∞ ≤ AT R, we get ∥g(eNsv(·, s))∥∞ ≤ +GAT R. It follows that +[]ψ(v)[] ≤ ∥ϕ∥∞ + sup +0≤t≤T +� t +0 +���e−N(s)��� +∞ ds GAT R ≤ ∥ϕ∥∞ + TAT GAT R. +Hence, we need that +∥ϕ∥∞ + TAT GAT R < R. +(2.12) +Let R be such that R ≥ 2∥ϕ∥∞. Since limT→0 AT = 1, we choose ε1 > 0 so that AT < 2 if T < ε1, and +ε < +R +4G2R +∧ +1 +4K2R +∧ ε1. +Using that GA ≤ GB and KA ≤ KB if A ≤ B, we get for R > 2∥ϕ∥∞ and T < ε, +∥ϕ∥∞ + TAT GAT R ≤ ∥ϕ∥∞ + 2εG2R < R +2 + R +2 = R +and +TA2 +T KAT R < 4εK2R < 1. +We proceed to prove equivalence of weak and mild solutions of (2.5). The proof of this theorem +follows the method in [24, Theorem 9.15], where this equivalence is shown for SPDE’s with autonomous +differential operators and driven by L´evy noise. For a comparison of weak and mild solutions of SPDEs +driven by fractional Brownian motion we refer to [11]. +We state first the Kolmogorov backward and forward equations for U D. By the Kolmogorov back- +ward equation for P D, the transition density pD(u, x, y) satisfies, for any y fixed, +∂ +∂upD(u, x, y) = +LpD(u, x, y). Then (s, x) → pD(s, x; t, y) satisfies, for (t, y) fixed, the equation +− ∂ +∂spD(s, x; t, y) = − ∂ +∂spD(K(t, s), x, y) = − ∂ +∂upD(u, x, y) |u=K(t,s) +∂ +∂sK(t, s) += 1 +2k2(s)LpD(K(t, s), x, y) = 1 +2k2(s)LpD(s, x; t, y). +(2.13) +Similarly, by the Kolmogorov forward equation for P D, for any x fixed, pD(u, x, y) satisfies +∂ +∂upD(u, x, y) = L∗pD(u, x, y). +9 + +Then (t, y) → pD(s, x; t, y) satisfies, for (s, x) fixed, the equation +∂ +∂tpD(s, x; t, y) = ∂ +∂tpD(K(t, s), x, y) = ∂ +∂upD(u, x, y) |u=K(t,s) +∂ +∂tK(t, s) += 1 +2k2(t)L∗pD(K(t, s), x, y) = 1 +2k2(t)L∗pD(s, x; t, y). +(2.14) +Theorem 2. Consider the random partial differential equation (2.5). Then v is a weak solution of +(2.5) on [0, T] if and only if v is a mild solution of (2.5) on [0, T]. +Proof. Assume that v is a weak solution of (2.5). Let h ∈ C1([0, ∞), R), f ∈ Dom(L∗), and G(x, t) := +− 1 +2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)). The integration by parts formula is applicable since h ∈ +C1([0, ∞), R) (see [24] Proposition 9.16) and yields +⟨v(·, t), h(t)f(·)⟩D += +⟨v(·, 0), h(0)f(·)⟩D + +� t +0 +⟨v(·, s), h′(s)f(·)⟩D ds ++ +� t +0 +⟨v(·, s), 1 +2h(s)k2(s)L∗f(·)⟩D ds + +� t +0 +⟨G(·, s), h(s)f(·)⟩D ds. +Since the functions h · f are dense in C1([0, ∞), Dom(L∗)), for each z ∈ C1([0, ∞), Dom(L∗)) we have +⟨v(·, t), z(·, t)⟩D += +⟨v(·, 0), z(·, 0)⟩D + +� t +0 +⟨v(·, s), ∂ +∂sz(·, s)⟩D ds +(2.15) ++ +� t +0 +⟨v(·, s), 1 +2k2(s)L∗z(·, s)⟩D ds + +� t +0 +⟨G(·, s), z(·, s)⟩D ds. +For each f ∈ Dom(L∗) we define +ψ(x, s) := (U D)∗(t, s)f(x) = + + + +⟨pD∗(s, x; t, ·), f(·)⟩D +if s < t, +f(x) +if s = t, +hence ψ ∈ C1([0, ∞), Dom(L∗)). Taking z = ψ(x, s) in (2.15) we get, for any t ∈ [0, T] fixed, +⟨v(·, t), ψ(·, t)⟩D += +⟨v(·, 0), ψ(·, 0)⟩D + +� t +0 +� +v(·, s), d +dsψ(·, s) + 1 +2k2(s)L∗ψ(·, s) +� +D +ds ++ +� t +0 +⟨G(·, s), ψ(·, s)⟩D ds. +(2.16) +Now we evaluate the terms above: +⟨v(·, 0), ψ(·, 0)⟩D += +� +D +v(x, 0) +� +D +pD∗(0, x; t, y)f(y) dy dx += +� +D +f(y) +� +D +pD∗(0, x; t, y)v(x, 0) dx dy = +� +U D(t, 0)v(·, 0), f(·) +� +D . +10 + +By applying the Kolmogorov backward equation to (x, s) → (U D)∗(t, s)f(x) we get +− d +dsψ(x, s) += +− ∂ +∂s +� +(pD)∗(s, x; t, ·), f(·) +� +D += +1 +2k2(s)L∗ � +(pD)∗(s, x; t, ·), f(·) +� +D = 1 +2k2(s)L∗ψ(x, s). +Moreover, from Fubini’s theorem and (1.4) +⟨G(·, s), ψ(·, s)⟩D += +� +D +G(x, s) +� +D +pD∗(s, x; t, y)f(y) dy dx += +� +D +f(y) +� +D +pD(s, y; t, x)G(x, s) dx dy = +� +U D(t, s)G(·, s), f(·) +� +D . +Therefore, from (2.16), ⟨v(·, t), f(·)⟩D = +� +U D(t, 0)v(·, 0), f(·) +� +D + +� t +0⟨U D(t, s)G(·, s), f(·)⟩D ds for all +f ∈ Dom(L∗). Since Dom(L∗) is dense in L2(D) we obtain that v is a mild solution of (2.5) on [0, T]. +To prove the converse let v be a mild solution of (2.5) on [0, T]. For f ∈ Dom(L∗), +� t +0 +� +v(·, s), 1 +2k2(s)L∗f(·) +� +D +ds += +� t +0 +� +U D(s, 0)v(·, 0), 1 +2k2(s)L∗f(·) +� +D +ds ++ +� t +0 +�� s +0 +χ[0,s](r)U D(s, r)G(·, r) dr, 1 +2k2(s)L∗f(·) +� +D +ds += +� t +0 +� +v(·, 0), (U D)∗(s, 0)1 +2k2(s)L∗f(·) +� +D +ds ++ +� t +0 +� t +r +� +U D(s, r)G(·, r), 1 +2k2(s)L∗f(·) +� +D +ds dr. +(2.17) +By applying the Kolmogorov forward equation to (U D)∗ we get for the first integral on the right side +of (2.17): +(U D)∗(s, 0)(1 +2k2(s)L∗f)(x) = +� +D +pD∗(0, x; s, y)1 +2k2(s)L∗f(y) dy += +� +D +(1 +2k2(s)L)pD∗(0, x; s, y)f(y) dy = +� +D +∂ +∂spD∗(0, x; s, y)f(y) dy, +and therefore +� t +0 +� +v(·, 0), (U D)∗(s, 0)(1 +2k2(s)L∗)f(·) +� +D +ds += +� t +0 +� +v(·, 0), +� +D +∂ +∂spD∗(0, ·; s, y)f(y) dy +� +D +ds = +� +v(·, 0), +� +D +pD∗(0, ·; t, y)f(y)dy − f(·) +� +D += +� +v(·, 0), (U D)∗(t, 0)f(·) +� +D − ⟨v(·, 0), f(·)⟩D. +11 + +In the same way we get for the second integral on the right side of (2.17) +� +U D(s, r)G(·, r), 1 +2k2(s)L∗f(·)) +� +D += +� +G(·, r), (U D)∗(s, r)(1 +2k2(s)L∗f)(·) +� +D += +� +G(·, r), +� +D +∂ +∂spD∗(r, ·; s, y)f(y)dy +� +D +, +and therefore +� t +r +� +U D(s, r)G(·, r), 1 +2k2(s)L∗f(·) +� +D +ds = +� t +r +� +G(·, r), +� +D +∂ +∂spD∗(r, ·; s, y)f(y)dy +� +D +ds += +� +G(·, r), +� +D +pD∗(r, ·; t, y)f(y)dy − f(·) +� +D += +� +G(·, r), (U D)∗(t, r)f(·) − f(·) +� +D += +� +U D(t, r)G(·, r), f(·) +� +D − ⟨G(·, r), f(·)⟩D . +In this way we obtain +� t +0 +⟨v(·, s), 1 +2k2(s)L∗f(·)⟩D ds += +� +U D(t, 0)v(·, 0) + +� t +0 +U D(t, r)G(·, r)dr, f(·)⟩D − ⟨v(·, 0), f(·) +� +D +− +� t +0 +⟨G(·, r), f(·)⟩D dr += +⟨v(·, t), f(·)⟩D − ⟨v(·, 0), f(·)⟩ D − +� t +0 +⟨G(·, r), f(·)⟩D dr, +since v is a mild solution on [0, T]. It follows that v is a weak solution on [0, T]. +Corollary 1. The equations (1.1) and (2.5) possess unique weak solutions. +Proof. Theorem 2 and Proposition 1 show the existence and uniqueness of a local weak and mild +solution of (2.5), and Proposition 1 shows the uniqueness of a weak solution of (1.1). +Remark 3. We refer to [23] for an existence and uniqueness theorem of the variational solution of +an SPDE with a nonautonomous second order differential operator and driven by fractional Brownian +motion, and to [26] for the existence and uniqueness of the mild solution. In [20] the existence and +uniqueness of the mild solution is shown for equations with the same differential operator and driven +by mixed noise. +3 +An upper bound for the blowup time and probability estimates +3.1 +An upper bound for the blowup time +In the remaining part of the paper we will assume that L and L∗ admit strictly positive eigenfunctions: +there exists a positive eigenvalue λ0 and strictly positive eigenfunctions ψ0 ∈ Dom(L) for P D +t +and +12 + +ϕ0 ∈ Dom(L∗) for (P D)∗ +t with +� +D ψ0(x)dx = +� +D ϕ0(x)dx = 1 such that +(P D +t − e−λ0t)ψ0 = ((P D)∗ +t − e−λ0t)ϕ0 = 0, +(3.18) +hence +(L + λ0)ψ0 = (L∗ + λ0)φ0 = 0. +(3.19) +For generators of a general class of L´evy processes, properties (3.18) and (3.19) follow from [14, 2]. +Another example are the diffusion processes: for f ∈ C2 +0(D), the set of twice continously differentiable +functions with compact support in D, let us define the differential operator +Lf = +d +� +j,k=1 +∂ +∂xj +� +ajk +∂ +∂xk +f +� ++ +d +� +j=1 +bj +∂ +∂xj +f − cf, +where aj,k, bj, j, k = 1, ..., d are bounded smooth functions on D and c is bounded and continous. We +assume that the matrix (aj,k, j, k = 1, ..., d) is symmetric and uniformly elliptic. In this case properties +(3.18) and (3.19) follow from [12, Theorem 11, Chapter 2]. +Theorem 3. Assume (3.19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given +constants. Let us define +τ ∗ = inf +� +t > 0 : +� t +0 +exp [−β(λ0K(r) + A(r)) + βNr] dr ≥ +1 +Cβ ⟨ϕ, φ0⟩−β +D +� +, +(3.20) +where the functions K and A are defined in (2.9). Then, on the event {τ ∗ < ∞} the solution v of (2.5) +and the solution u of (1.1) blow up in finite time τ, and τ ≤ τ ∗ P-a.s. +Proof. Using the hypothesis on g and Jensen’s inequality we get for the terms in (2.6): +⟨v(·, s), L∗φ0⟩D += +−λ0⟨v(·, s), φ0⟩D, +exp(−Ns) ⟨g(exp(Ns)v(·, s)), φ0⟩D +≧ +C exp(βNs) +� +v1+β(·, s), φ0 +� +D , +≧ +C exp(βNs)⟨v(·, s), φ0⟩1+β +D +. +Applying these lower bounds to (⟨v(·, t + ε), φ0⟩D − ⟨v(·, t), φ0⟩D)/ε and letting ε → 0 we get +d +dt⟨v(·, t), φ0⟩D ≧ −1 +2(λ0k2(t) + a2(t))⟨v(·, t), φ0⟩D + C exp(βNt)⟨v(·, t), φ0⟩1+β +D +. +(3.21) +The corresponding differential equality reads +d +dtI(t) = −1 +2(λ0k2(t) + a2(t))I(t) + C exp(βNt)I(t)1+β, +and I(t) is a subsolution of (3.21), i.e. ⟨v(·, t), φ0⟩D ≧ I(t). Then +I(t) = exp[−(λ0K(t) + A(t))] +� +⟨ϕ, φ0⟩−β +D − βC +� t +0 +exp [−β(λ0K(s) + A(s)) + βNs] ds +�−1/β +13 + +for all t ∈ [0, τ ∗), where τ ∗ is given by (3.20). Therefore τ ∗ is an upper bound for the blowup time +of ⟨v(·, t), φ0⟩D, and the function t �→ ∥v(·, t)∥∞ = exp(−Nt)∥u(·, t)∥∞ can not stay finite on [0, τ ∗] if +τ ∗ < ∞. Therefore u and v blow up before τ ∗ if τ ∗ < ∞. +Remark 4. Notice that τ ∗ depends on L only by the positive eigenvalue λ0 and the associated eigen- +function φ0. Moreover, τ ∗ is a decreasing function of ϕ, φ0 and C, and an increasing function of λ0K. +Therefore small functions ϕ, φ0 and a small constant C, as well as high values of λ0K postpone the +blowup of I and have, in this sense, the tendency to postpone the blowup of v and u. +3.2 +A tail probability estimate for the upper bound of the blowup time +In the following theorem we apply a tail probability estimate for exponential functionals of fBm studied +by N.T. Dung [8] to estimate the probability that τ ∗ occurs before a fixed time T. Here we assume +that the process BH is given by the formula +BH +t = +� t +0 +KH(t, s) dBs, +(3.22) +where the kernel KH is given for H > 1/2 by +KH(t, s) = + + + +CHs1/2−H � t +s (σ − s)H−3/2σH−1/2dσ +if t > s, +0 +if t ≦ s, +(3.23) +where CH = [ +H(2H−1) +B(2−2H,H−1/2)] +1 +2 and B is the usual beta function (see Section 5.1.3 in [21] for a general +representation formula of fBm with H > 1/2). Notice that BH and B are dependent in this case. +Theorem 4. Under assumptions (3.19) and (3.22), let g(z) ≥ Cz1+β for all z > 0, where C > 0, +β > 0, are given constants, and let µ(T) = +� T +0 exp[−β(λ0K(t) + A(t))]E [exp(βNt)] dt. Then, for any +T > 0 such that +1 +Cβ⟨ϕ, φ0⟩−β +D > µ(T), +P {τ ∗ ≤ T} ≤ 2 exp + +− +ln2 � +Cβ⟨ϕ, φ0⟩β +D µ(T) +� +2M(T) + + , +where +M(T) = 2β2 +� T +0 +a2(r) dr + 4β2HT 2H−1 +� T +0 +b2(u) du. +Proof. For t ≥ 0, using (3.22), we have the following representation: +Xt +:= +−β(λ0K(t) + A(t)) + βNt +(3.24) += +−β(λ0K(t) + A(t)) + β +�� t +0 +a(s) dBs + +� t +0 +� t +s +b(r) ∂ +∂rKH(r, s) dr dBs +� +. +14 + +From [8, Theorem 3.1] it follows that for any T ≥ 0 and any x > µ(T), there holds +P +�� T +0 +eXtdt ≥ x +� +≤ 2 exp +� +−(ln x − ln µ(T))2 +2M(T) +� +, +(3.25) +where µ(T) = +� T +0 E +� +eXt� +dt and M(T) is such that +sup +t∈[0,T] +� T +0 +|DrXt|2 dr ≤ M(T) +P-a.s. +(3.26) +Here DrXt denotes the Malliavin derivative of Xt. In the following we will find an upper bound M(T) +such that (3.26) holds. For r < t we have, using the representation (3.25), +DrXt = β +� +a(r) + +� t +r +b(s) ∂ +∂sK(s, r) ds +� +. +Hence +� t +0 |DrXt|2 dr ≤ 2β2 � t +0 a2(r) dr + 2β2 � t +0( +� t +r b(s) ∂ +∂sK(s, r) ds)2 dr and +� t +0 +�� t +r +b(s) ∂ +∂sK(s, r) ds +�2 +dr += +� t +0 +�� t +r +b(s) ∂ +∂sK(s, r) ds +� �� t +r +b(s′) ∂ +∂s′ K(s′, r) ds′ +� +dr += +� t +0 +b(s) ds +� s +0 +∂ +∂sK(s, r) dr +� t +r +b(s′) ∂ +∂s′ K(s′, r) ds′ += +� t +0 +ds b(s) +� t +0 +dr1[0,s](r) ∂ +∂sK(s, r) +� t +r +b(s′) ∂ +∂s′ K(s′, r) ds′ += +� t +0 +ds b(s) +� t +0 +ds′b(s′) +� s′ +0 +1[0,s](r) ∂ +∂sK(s, r) ∂ +∂s′ K(s′, r) dr += +� t +0 +ds +� t +0 +ds′ b(s)b(s′) +� s∧s′ +0 +∂ +∂sK(s, r) ∂ +∂s′ K(s′, r) dr += +� t +0 +ds +� t +0 +ds′ b(s)b(s′)Φ(s, s′) += +� t +0 +ds +� s +0 +ds′ b(s)b(s′)Φ(s, s′) + +� t +0 +ds +� t +s +ds′ b(s)b(s′)Φ(s, s′) += +2 +� t +0 +ds +� s +0 +ds′ b(s)b(s′)Φ(s, s′), +where Φ(s, s′) = +� s∧s′ +0 +∂ +∂sK(s, r) ∂ +∂s′ K(s′, r) dr. Since +∂ +∂sK(s, r) = CHr1/2−H(s − r)H−3/2sH−1/2, using +(5.7) in [21] we obtain +Φ(s, s′) = C2 +H(ss′)H−1/2 +� s∧s′ +0 +r1−2H(s − r)H−3/2(s′ − r)H−3/2 dr = H(2H − 1)(s − s′)2H−2 +15 + +for s′ < s, hence +� t +0 +�� t +r +b(s) ∂ +∂sK(s, r) ds +�2 +dr +≤ 2H(2H − 1) +� t +0 +ds +� s +0 +|b(s)b(s′)|(s − s′)2H−2 ds′ +≤ H(2H − 1) +�� t +0 +b(s)2 +� s +0 +(s − s′)2H−2 ds′ ds + +� t +0 +� s +0 +b(s′)2(s − s′)2H−2 ds′ ds +� += H +� t +0 +b(s)2s2H−1 ds + H(2H − 1) +� t +0 +b(s′)2 +� t +s′ (s − s′)2H−2 ds ds′ += H +� t +0 +b(s)2(s2H−1 + (t − s)2H−1) ds +≤ 2Ht2H−1 +� t +0 +b(s)2 ds. +(3.27) +From the above inequalities we obtain +sup +t∈[0,T] +� T +0 +|DrXt|2dr ≤ 2β2 +� T +0 +a2(r)dr + 4β2HT 2H−1 +� T +0 +b2(u)du := M(T). +(3.28) +Now, from (3.20) +P(τ ∗ ≦ T) += +P +�� T +0 +exp[−β(λ0K(t) + A(t)) + βNt] dt ≧ +1 +Cβ ⟨ϕ, φ0⟩−β +D +� += +P +�� T +0 +exp[X(t)] dt ≥ x +� +, +(3.29) +where x = +1 +Cβ⟨ϕ, φ0⟩−β +D . The result follows from (3.25) and (3.28). +In the following theorem we obtain upper bounds for the tail of τ ∗ in the case when the Brownian +motion B and the fractional Brownian motion BH have general dependence structure. +Theorem 5. Assume (3.19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given +constants. +1. Assume that BH +t += +� t +0 KH(t, s) dWs, where W is a Brownian motion defined in the same proba- +bility space, and adapted to the same filtration as the Brownian motion B. Then +P(τ ∗ ≤ T) +≤ +Cβ⟨ϕ, φ0⟩β +D +� T +0 +� +e−βλ0 +� t +0 k2(s) ds+2β2 � t +0 a2(s) ds + e−β � t +0 a2(s) ds+4β2Ht2H−1 � t +0 b2(s) ds� +dt. +16 + +2. If B and BH are independent, then +P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β +D +� T +0 +e−βλ0K(t)+ β2−β +2 +� t +0 a2(s) ds+β2Ht2H−1 � t +0 b2(s) ds. +Proof. +1. Using H¨older’s and Chebishev’s inequalities we obtain +P(τ ∗ ≤ T) += +P +�� T +0 +e−βλ0K(t)+β � t +0 a(s) dBs−βA(t)+β � t +0 b(s) dBH +s dt ≥ +1 +Cβ⟨ϕ, φ0⟩−β +D +� +≤ +P + + +�� T +0 +e−2βλ0K(t)+2β � t +0 a(s) dBs dt +� 1 +2 +× +�� T +0 +e−2βA(t)+2β � t +0 b(s) dBH +s dt +� 1 +2 +≥ +1 +Cβ ⟨ϕ, φ0⟩−β +D + + +≤ +P +�� T +0 +e−2βλ0K(t)+2β � t +0 a(s) dBs dt ≥ +1 +Cβ ⟨ϕ, φ0⟩−β +D +� ++P +�� T +0 +e−2βA(t)+2β � t +0 b(s) dBH +s dt ≥ +1 +Cβ⟨ϕ, φ0⟩−β +D +� +≤ +E +�� T +0 e−2βλ0K(t)+2β � t +0 a(s) dBs dt +� ++ E +�� T +0 e−2βA(t)+2β � t +0 b(s) dBH +s dt +� +1 +Cβ⟨ϕ, φ0⟩−β +D +≤ +� T +0 +� +e−2βλ0K(t)+2β2 � t +0 a2(s) ds� +dt + +� T +0 e−2βA(t)E +� +e2β � t +0 b(s) dBH +s +� +dt +1 +Cβ⟨ϕ, φ0⟩−β +D +, +(3.30) +where we have used the fact that E +� +exp +�� t +0 f(s) dB(s) +�� += exp +� +1 +2 +� t +0 f 2(s) ds +� +to obtain the +last inequality. In addition, +E +� +e2β +� t +0 b(s) dBH +s +� += E +� +e2β +� t +0 +� t +s b(r) ∂ +∂r KH(r,s) dr dWs� += e2β2 � t +0[ +� t +s b(r) ∂ +∂r KH(r,s) dr] +2 ds, +where the last equality follows from [13, Theorem 4.12]. Therefore, using (3.27) we get +E +� +e2β +� t +0 b(s) dBH +s +� +≤ exp +� +4β2Ht2H−1 +� t +0 +b2(s) ds +� +. +(3.31) +Substituting (3.31) into (3.30) we obtain the desired bound. +17 + +2. Using Chebishev’s inequality, the independence of B and BH and the proof of (3.31), +P(τ ∗ ≤ T) += +P +�� T +0 +e−βλ0K(t)+β � t +0 a(s) dBs−βA(t)+β � t +0 b(s) dBH +s ) dt ≥ +1 +Cβ ⟨ϕ, φ0⟩−β +D +� +≤ +Cβ⟨ϕ, φ0⟩β +D +� T +0 +E +� +e−βλ0K(t)+β � t +0 a(s) dBs� +E +� +e−βA(t)+β � t +0 b(s) dBH +s +� +dt +≤ +Cβ⟨ϕ, φ0⟩β +D +� T +0 +exp +� +−βλ0K(t) + β2 − β +2 +� t +0 +a2(s) ds + β2Ht2H−1 +� t +0 +b2(s) ds +� +dt. +4 +Lower bounds for the blowup time and for the probability of finite +time blowup +4.1 +A lower bound for the probability of finite time blowup +In the following theorem we give a lower bound for the probability of finite time blow up of the weak +solution of (1.1). If f, g are nonnegative functions and c is a constant, we write f(t) ∼ cg(t) as t → ∞ +if limt→∞ f(t)/g(t) = c. +Theorem 6. Assume (3.19) and (3.22). Let g(z) ≥ Cz1+β and +� t +0 +a2(r) dr ∼ C1t2l, +� t +0 +b2(r) dr ∼ C2t2m, +� t +0 +k2(r) dr ∼ C3t2p +as t → ∞ for some nonnegative constants l, m, p and positive constants C, β, C1, C2 and C3. Suppose +additionally that +1. if β ∈ (0, 1/2), then max{p, l} > H + m − 1 +2, +2. if β = 1/2, then H+m − 1 +2 < p, +3. if β > 1/2, then p > max{l, H + m − 1 +2}. +Under these assumptions the solution of (1.1) blows up in finite time with positive probability. Moreover, +P(τ < ∞) ≧ P(τ ∗ < ∞) ≧ 1 − exp +� +−(mξ − 1)2 +2Lξ +� +, +(4.32) +where +ξ = +1 +Cβ ⟨ϕ, φ0⟩−β +D , +Lξ = sup +t≧0 +M(t) +(ln(ξ + 1) + f(t))2 , +(4.33) +18 + +with f(t) = tmax{H+m−1/2, l} and +mξ = E + +sup +t≧0 +ln +�� t +0 exp (−β(λ0K(s) + A(s)) + βNs) ds + 1 +� ++ f(t) +ln(ξ + 1) + f(t) + + . +(4.34) +Proof. From (3.29) it follows that P(τ ∗ < ∞) = P( +� ∞ +0 eXt dt ≥ ξ). In order to estimate P( +� ∞ +0 +eXt dt ≥ ξ) +we use [9, Theorem 3.1], with a = 0 and σ = 1 : +Proposition 3 ([9]). Assume that the stochastic process X is adapted and satisfies +a) +� ∞ +0 +EeXs ds < ∞, +b) For each t ≥ 0, Xt ∈ D1,2, +c) There exists a function f : R+ → R+ such that limt→∞ f(t) = ∞ and for each x > 0, +sup +t≧0 +sups∈[0,t] +� t +0 |DrXs|2dr +(ln(x + 1) + f(t))2 +≤ Lx < ∞ +a.s. +(4.35) +Then +P +�� ∞ +0 +eXt dt < x +� +≤ exp +� +−(mx − 1)2 +2Lx +� +, +where +mx = E +� +sup +t≥0 +ln( +� t +0 eXs ds + 1) + f(t) +ln(x + 1) + f(t) +� +. +We now verify that conditions a) - c) of the above proposition hold. +For condition a) we have from (3.25), +� ∞ +0 +E exp[Xt] dt += +� ∞ +0 +E exp +� +−βλ0 +2 +� t +0 +k2(s) ds − β +2 +� t +0 +a2(s) ds ++ β +�� t +0 +a(s) dBs + +� t +0 +� t +s +b(r) ∂ +∂rKH(r, s) dr dBs +�� +dt += +� ∞ +0 +E exp +� +−βλ0 +2 +� t +0 +k2(s) ds − β +2 +� t +0 +a2(s) ds + β +� t +0 +� +a(s) + +� t +s +b(r) ∂ +∂rKH(r, s) dr +� +dBs +� +dt += +� ∞ +0 +exp +� +−βλ0 +2 +� t +0 +k2(s) ds − β +2 +� t +0 +a2(s) ds + β2 +2 +� t +0 +� +a(s) + +� t +s +b(r) ∂ +∂rKH(r, s) dr +�2 +ds +� +dt, +where, again, we have used [13, Theorem 4.12] to obtain the last equality. Therefore, using (3.27), +� ∞ +0 +E exp[Xt] dt +≤ +� ∞ +0 +exp +� +−βλ0 +2 +� t +0 +k2(s) ds − β +2 +� t +0 +a2(s) ds + β2 +2 +� t +0 +2a2(s) ds ++ 2β2Ht2H−1 +� t +0 +b2(s) ds +� +dt. +(4.36) +19 + +The integral (4.36) is finite if and only if the leading power of t in the term +−βλ0 +2 +� t +0 +k2(s) ds + 2β2 − β +2 +� t +0 +a2(s) ds + 2β2Ht2H−1 +� t +0 +b2(s) ds +has negative coefficient, which follows from our assumptions. +Condition b) is a consequence of (3.28). +For condition c) we use the inequality (3.28), which implies that for any x > 0 and any fixed +function f, +sup +t≧0 +sups∈[0,t] +� t +0 |DrXs|2dr +(ln(x + 1) + f(t))2 +≤ sup +t≥0 +M(t) +(ln(x + 1) + f(t))2 . +(4.37) +Due to our assumptions, for big t, the leading power of t in the numerator is max{2l, 2H + 2m − 1}. +It follows that +lim +t→∞ +M(t) +� +ln(x + 1) + tmax{l,H+m−1/2}�2 < ∞, +and therefore the supremum in (4.37) is finite. The result follows from Proposition 3. +The cases when a = 0 (presence only of fractional Brownian motion) or b = 0 (presence only of +Brownian motion), are simpler: +Corollary 2. Under the assumptions in Theorem 6, +1. When a(t) ≡ 0 and p > H + m − 1/2 the solution of (1.1) explodes in finite time with positive +probability for all β > 0. +2. If a(t) ≡ 0 and p = H + m − 1/2, the solution of (1.1) explodes in finite time with positive +probability for all β > 0 satisfying β < C3λ0 +4C2H . +3. When b(t) ≡ 0 and 0 < β ≤ 1 +2 the solution of (1.1) exhibits explosion in finite time with positive +probability for all values of p and l. +4. If b(t) ≡ 0 and β > 1/2, the solution of (1.1) exhibits explosion in finite time with positive +probability if p > l or if p = l and C3λ0 > C1(2β − 1). +Notice that mξ given in (4.34) satisfies mξ > 1 due to Theorem 3.1 in [9]. The formula for mξ +shows interactions between ϕ and K that have an influence on the lower bound in (4.32). Increasing +values of K decrease the lower bound in (4.32). In this sense high values of K are in favour of absence +of finite time blowup. +20 + +4.2 +The case H > 3/4 and independent B and BH +In order to find more explicit lower bounds for P(τ < +∞), we consider in this subsection the case +H ∈ (3/4, 1) and suppose that B and BH are independent and b(s) = ca(s) for all s ≧ 0, where c +is a constant. Then Nt = +� t +0 a(s)dMs with Ms = Bs + cBH +s . By [3] M is equivalent to a Brownian +motion �B, and therefore Nt is equivalent to ˜Nt := +� t +0 a(s) d �Bs. Here equivalence means equality of the +laws of the processes on (C[0, T], B), the space of continous functions defined on [0, T] endowed with +the σ−algebra generated by the cylinder sets. Furthermore, ( ˜Nt)t≧0 is a continous martingale and +therefore a time-changed Brownian motion: ˜Nt = �B2A(t). +Theorem 7. Assume (3.19). Let H ∈ (3/4, 1), B and BH be independent and b(s) = ca(s) for all +s ≧ 0, where c is a constant.We assume also that g(z) ≥ Cz1+β, that the functions k and a are positive +continuous on R+ and that there exist constants η ∈ (0, +∞] and c1 > 0 such that +1 +a2(t) exp(−βλ0K(t)) ≥ c1 exp +� +−2β A(t) +η +� +, +t ∈ R+. +(4.38) +Then +P(τ < +∞) ≥ P(Zµ ≤ θ), +(4.39) +where τ is the blowup time of (1.1), Zµ is a gamma-distributed random variable with parameter µ := +2 +β( 1 +η + 1 +2), θ := 2c1 +β2ξ and ξ := +1 +Cβ⟨ϕ, φ0⟩−β +D . +Proof. From Theorem 3, +P(τ ∗ = +∞) += +P +�� t +0 +dr exp +� +−β(λ0K(r) + A(r)) + β ˜Nr +� +< ξ for all t > 0 +� += +P +�� ∞ +0 +dr exp +� +−β(λ0K(r) + A(r)) + β ˜Nr +� +≤ ξ +� +. +By the change of variable q = 2A(r) we get +P(τ ∗ = +∞) = P +�� ∞ +0 +dr exp +� +−β(λ0K(r) + A(r)) + β ˜B2A(r) +� +≤ ξ +� += P +�� ∞ +0 +dq +a2(A−1(q/2)) exp +� +−β(λ0K(A−1(q/2)) + 1 +2q) + β ˜Bq +� +≤ ξ +� +. +Applying (4.38) to t = A−1(q/2) yields +1 +a2(A−1(q/2)) exp +� +−β(λ0K(A−1(q/2)) +� +≥ c1 exp +� +−β +η q +� +, +q ∈ R+. +21 + +Therefore +P(τ ∗ = +∞) +≤ +P +� +c1 +� ∞ +0 +dq exp +� +−βq +�1 +η + 1 +2 +� ++ β ˜Bq +� +≤ ξ +� += +P +�� ∞ +0 +dq exp +� +β( ˜Bq − ˜µq) +� +≤ ξ +c1 +� +, +where ˜µ := 1 +η + 1 +2. A second change of variable q = 4s +β2 yields +P(τ ∗ = +∞) ≤ P +�� ∞ +0 +ds exp +� +2( ˜Bs − µs) +� +≤ β2ξ +4c1 +� +, +where µ := ˜µ 2 +β. Due to [27, Corollary 1.2, page 95], +� ∞ +0 +e2( ˜ +Bs−µs) ds L= +1 +2Zµ +, +where Zµ is a gamma-distributed random variable with parameter µ. Therefore +P(τ = +∞) ≤ P(τ ∗ = +∞) ≤ P +� 1 +2Zµ +≤ β2ξ +4c1 +� += P +� +Zµ ≥ 2c1 +β2ξ +� +. +This implies the statement of the theorem. +Remark 5. If k, a and b are constants, a more explicit lower bound for P(τ < +∞) is available +without the assumption (4.38). Indeed, starting with (3.20), a straightforward calculation gives a lower +bound in terms of a gamma-distributed random variable Z again, but this time with parameter �µ := +(λ0k2 + a2)/(a2β). More precisely, +P(τ < ∞) ≧ P(τ ∗ < ∞) = P +� +Z�µ ≦ 2C +a2β ⟨ϕ, φ0⟩β +D +� +. +4.3 +A lower bound for the blowup time +Our next goal is to obtain a lower bound for the blowup time τ. Since the proofs of the following results +are close to those in [1] (where b = 0), we omit them here. +Theorem 8. Let the function g be such that g(0) = 0, z → g(z)/z is increasing, and g(z) ≤ Λz1+β for +some positive constant Λ. Then τ ≥ τ∗, where +τ∗ = inf +� +t > 0 : +� t +0 +exp(β(Nr − A(r))) +��U D(r, 0)ϕ +��β +∞ dr ≧ 1 +Λβ +� +. +(4.40) +Let us define for 0 ≦ t < τ∗, +J(t) = +� +1 − Λβ +� t +0 +exp(β(Nr − A(r))) +��U D(r, 0)ϕ +��β +∞ dr +�−1/β +. +22 + +Then the solution u of (1.1) satisfies, for x ∈ D, 0 ≦ t < τ∗, P-a.s. +0 ≦ u(x, t) ≦ J(t) exp(Nt − A(t))U D(t, 0)ϕ(x). +(4.41) +Remark 6. More precisely, the proof of this theorem shows that the mild solution v of (2.5) satisfies +(4.41) without the factor exp(Nt). By Theorem 2, v is also the weak solution of (2.5), hence the weak +solution u(·, t) = exp(Nt)v(·, t) of (1.1) satisfies (4.41). +Corollary 3. Assume that +Λβ +� ∞ +0 +exp[β(Nr − A(r))] +��U D(r, 0)ϕ +��β +∞ dr < 1. +Then the solution u of (1.1) satisfies (4.41) P-a.s. for all t. +Remark 7. For the special choice of ϕ = pψ0, p > 0, the integrals appearing in (3.20) and (4.40) are +the same exponential functionals of N. In fact, U D(r, 0)ψ0 = exp(−λ0K(r))ψ0, and τ∗ becomes +τ∗ = inf +� +t > 0 : +� t +0 +exp +� +β(Nr − λ0K(r) − A(r)) +� +dr ≧ p−β +Λβ ∥ψ0∥−β +∞ +� +, +(4.42) +whereas +τ ∗ = inf +� +t > 0 : +� t +0 +exp +� +β(Nr − λ0K(r) − A(r)) +� +dr ≥ p−β +Cβ ⟨ψ0, φ0⟩−β +D +� +. +(4.43) +In fact τ∗ ≦ τ ∗ if C ≦ Λ, since ⟨ψ0, φ0⟩D ≦ ∥ψ0∥∞ +� +D φ0(x)dx = ∥ψ0∥∞. In order to apply both bounds +simultaneously, we have to suppose Cz1+β ≦ g(z) ≦ Λz1+β, z > 0. It is therefore of interest to know +the law of the integral appearing in (4.42) and (4.43). This seems possible only for bH = 0, since, to +our best knowledge, the law of exponential functionals of fractional Brownian motion is still unknown. +For the moment it seems that only estimates of the type of those in Section 3.2 are available. See also +Theorem 7 for H > 3/4. +5 +A sufficient condition for finite time blowup +We consider now the mild form of (2.5) obtained in Proposition 2, and obtain a sufficient condition for +finite time blowup. +Theorem 9. Suppose that g(z) ≥ Cz1+β and that there exists w∗ > 0 such that +exp(βA(w∗)) ∥ U D(w∗, 0)ϕ ∥−β +∞ < βC +� w∗ +0 +exp(βNs) ds . +(5.44) +Then for the explosion time τ of (1.1) there holds τ ≤ w∗. +23 + +Remark 8. Inequality (5.44) is understood trajectorywise. Therefore w∗ is random. (5.44) is harder to +satisfy with a small initial condition ϕ and with a small value of C. Due to the different interpretations +of the integrals in N, the effects on blowup of B and BH are different. +If N = 0, (5.44) reads ∥ +U D(w∗, 0)ϕ ∥−β +∞ < βCw∗ and in this case w∗ is deterministic; if in addition ϕ = ψ0, (5.44) reads +exp(λ0βK(w∗)) ∥ ψ0 ∥−β +∞ < βCw∗. +Proof. We use the approach in [25, Lemma 15.6]; see also [15]. Suppose that v(x, t), x ∈ D, t ≥ 0, is +a global solution of (2.5), and let 0 < t < t′. Using the semigroup property of the evolution system +(U D(t, r))0≦r 0. Then +d +dtΨ(ψ(t)) = − +ψ′(t) +(ψ(t))1+β ≦ −C exp(βNt). +Hence +C +� t′ +0 +exp(βNs) ds ≦ Ψ(ψ(0)) − Ψ(ψ(t′)) = +� ψ(t′) +ψ(0) +dz/z1+β < +� ∞ +exp(−A(t′))UD(t′,0)ϕ(·)(x) +dz/z1+β +for all x ∈ D and all t′ > 0. Therefore βC +� t′ +0 exp(βNs) ds ≦ exp(βA(t′))∥U D(t′, 0)ϕ∥−β +∞ for all t′ > 0. +This contradicts (5.44). +Acknowledgement The authors are grateful to two anonymous referees for their valuable comments, +which greatly improved our paper. The second- and third-named authors acknowledge the hospitality +of Institut ´Elie Cartan de Lorraine, where part of this work was done. The research of the second- +named author was partially supported by CONACyT (Mexico), Grant No. 652255. The fourth-named +author would like to express her gratitude to the entire staff of the IECL for their hospitality and +strong support during the completion of her Ph.D. dissertation there. +References +[1] A. Alvarez, J.A. L´opez-Mimbela, N. Privault. Blowup estimates for a family of semilinear SPDEs +with time-dipendent coefficients. Differential Equations and Applications 2 (2015), 201-219. +[2] X. Chen, J. Wang. Intrinsic ultracontractivity for general L´evy processes on bounded open sets. +Illinois J. Math. 58 (2014), 1117-1144. +[3] P. Cheridito. Mixed fractional Brownian motion. Bernoulli 7 (2001), 913-934. +[4] P.L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean Lp-norm. J. +Diff. Equations 250 (2011), 2567-2580. +[5] M. Dozzi, E.T. Kolkovska, J.A. L´opez-Mimbela. Finite-time blowup and existence of global positive +solutions of a semi-linear SPDE with fractional noise. In: Modern Stochastics and Applications, +V. Korolyuk, N. Limnios, Y. 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Fields 111 (1998) 333-374. +27 + diff --git a/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f83ca1590333ef44ef8eeba9c6980d4653f44065 --- /dev/null +++ b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt @@ -0,0 +1,896 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf,len=895 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='02174v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='PR] 5 Jan 2023 Large time behavior of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions Marco Dozzi∗ Ekaterina T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Kolkovska† Jos´e A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L´opez-Mimbela† Rim Touibi‡ Abstract We study the trajectorywise blowup behavior of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different types of random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We give estimates for the probability of finite time blowup and of blowup before a given fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Essential tools are the mild and weak form of an associated random partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Keywords Stochastic reaction-diffusion equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' mixed fractional noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' finite-time blowup of trajectories AMS Mathematics Subject Classification 60H15 60G22 35R60 35B40 35B44 35K58 1 Introduction In this paper we study existence, uniqueness and the blowup behavior of solutions to the fractional stochastic partial differential equation of the form du(x, t) = �1 2k2(t)Lu(x, t) + g(u(x, t)) � dt + u(x, t) dNt, x ∈ D, t > 0, u(x, 0) = ϕ(x) ≥ 0, u(x, t) = 0, x ∈ ∂D, t ≥ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) where D ⊂ Rd is a bounded Lipschitz domain, L is the infinitesimal generator of a strongly continuous semigroup of contractions which satisfies conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) below, and ϕ ∈ L∞(D), where L∞(D) is the space of real-valued essentially bounded functions on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Additionally, g is a nonnegative locally Lipschitz function and N is a process given by Nt = � t 0 a(s) dB(s) + � t 0 b(s) dBH(s), t ≥ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2) ∗corresponding author, marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='dozzi@univ-lorraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='fr, UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France †Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ‡UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1 where B is Brownian motion and BH is fractional Brownian motion with Hurst parameter H > 1/2, a is continuous and b is H¨older continuous of order α > 1 − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Both, B and BH, are supposed to be defined on a filtered probability space (Ω, F, (Ft, t ≧ 0), P) and adapted to the filtration (Ft, t ≧ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Such models have recently been studied under the name of ‘mixed models’ in the context of stochastic differential equations, see [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When N = 0, L = ∆, k = 1, g(u) = u1+β we obtain the classical Fujita equation which was studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In [7] and [1] there were considered the cases when N is a Brownian motion, in [5] it was investigated the case when N is a fractional Brownian motion with Hurst parameter H > 1/2 and D ⊂ Rd, and in [6] the case of H ≥ 1/2 and D = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The fractional Brownian motion (fBm) appears in many stochastic phenomena, where rough exter- nal forces are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The principal difference, compared to Brownian motion, is that fBm is not a semimartingale nor a Markov process, hence classical theory of stochastic integration cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since H > 1/2, the stochastic integral with respect to BH in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) can be understood as a fractional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Also the presence of both, Brownian and fractional Brownian motion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), due to their different analytic and probabilistic properties, modelize different aspects of the random evolution in time of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The factor k2/2 in front of L affects dissipativity, which in several cases is in favor of retarding or even preventing blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We consider both, weak and mild solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), which we prove are equivalent and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Beyond existence and uniqueness of weak and mild solutions we are interested in their qualitative behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 3 below we obtain a random time τ ∗ which is an upper bound of the explosion time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 8 we obtain a lower bound τ∗ of τ so that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' τ∗ ≤ τ ≤ τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The random times τ∗ and τ ∗ are given by exponential functionals of the mixture of a Brownian and a fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The laws of such kind of functionals presently are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to study the distribution of τ ∗ we use the well-known representation of BH in the form BH t = � t 0 KH(t, s) dWs, where the kernel KH is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='23) and W is a Brownian motion defined in the same filtered probability space as B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In general, W can be different from the Brownian motion B appearing in the first integral of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We obtain estimates of the probability P(τ < ∞), and of the tail distribution of τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To achieve this we make use of recent results of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Dung [8, 9] from the Malliavin theory for continuous isonormal Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 4 we obtain upper bounds for P(τ ∗ ≤ T) in the case when B = W, and in Theorem 5 when B is independent of W, and when B and W are general Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 6 we obtain lower bounds for P(τ < ∞) when B = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' As a result in the case when W = B we get specific configurations of the coefficients a, b and k under which the weak solution (hence also the mild solution) of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits finite time blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To be concrete suppose that g(z) ≥ Cz1+β for 2 some constants C > 0, β > 0, BH t = � t 0 KH(t, s) dBs, and � t 0 a2(r) dr ∼ t2l, � t 0 b2(r) dr ∼ t2m, � t 0 k2(r) dr ∼ t2p as t → ∞ for some nonnegative constants l, m and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If β ∈ (0, 1/2) and max{p, l} > H + m − 1/2, or if β = 1/2 and p > H +m−1/2, or if β > 1/2 and p > max{l, H +m−1/2}, then all nontrivial positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) suffer finite-time blowup with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Our approach here is to transform the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) into a random partial differential equation (RPDE) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), whose solution blows up at the same random time τ as the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), and to work with this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The blowup behavior of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) is easier to determine because N appears as a coefficient, and not as stochastic integrator as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Such transformations are indeed known for more general SPDEs than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), including equations whose stochastic term does not depend linearly on u, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' But for the RPDE’s associated to more general SPDE’s it seems difficult to find explicit expressions for upper and lower bounds for the blowup time, and this is an essential point in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Another reason for having chosen the relatively simple form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) is that we consider the blowup trajectorywise which is a relatively strong notion compared, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', to blowup of the moments of the solution (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The crucial ingredient in the proofs is the existence of a positive eigenvalue and an eigenfunction with constant sign of the adjoint operator of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Special attention is given to the case H ∈ (3 4, 1) because then the process N is equivalent to a Brownian motion [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This allows us to apply a result by Dufresne and Yor [27] on the law of exponential functionals of the Brownian motion to get in Theorem 7 an explicit lower bound for the probability of blowup in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We finish this section by introducing some notations and definitions we will need in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A stopping time τ : Ω → (0, ∞) with respect to the filtration (Ft, t ≧ 0) is a blowup time of a solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) if lim sup tրτ sup x∈D |u(x, t)| = +∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let (P D t , t ≧ 0) and ((P D)∗ t , t ≧ 0) be the strongly continuous semigroups corresponding to the operator L and its adjoint L∗ : � D f(x)P D t g(x)dx = � D g(x)(P D)∗ tf(x)dx, f, g ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3) As usual, Lf := lim t→0 1 t (P D t f − f) for all f ∈ L2(D) in the domain of L, denoted by Dom(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to the Hille-Yosida theorem, Dom(L) and Dom(L∗) are dense in L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let P D t (x, Γ) and (P D)∗ t (x, Γ) denote the associated transition functions, where t > 0, x ∈ D, and Γ ∈ B(D), the Borel sets on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the sequel we will assume that they admit densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' there exist families of continuous functions 3 (pD(t, ·, ·), t > 0) and ((pD)∗(t, ·, ·), t > 0) on D × D such that P D t g(x) = � D g(y)P D t (x, dy) = � D g(y)pD(t, x, y)dy, (P D)∗ t f(x) = � D f(y)(P D)∗ t (x, dy) = � D f(y)(pD)∗(t, x, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3), (pD)∗(t, x, y) = pD(t, y, x) for all t > 0 and x, y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='4) 2 The weak solution of the associated random partial differential equation, equivalence with the mild solution Let us consider the random partial differential equation ∂v ∂t (x, t) = 1 2k2(t)Lv(x, t) − 1 2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) v(x, 0) = ϕ(x), x ∈ D, v(x, t) = 0, t ≥ 0, x ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this section we transform the weak form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) into the weak form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) using the transformation v(x, t) = exp(−Nt)u(x, t), x ∈ D, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, if blowup takes place in finite time, it occurs of course at the same time and at the same place x ∈ D for the solutions of both equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following we write ⟨·, ·⟩D for the scalar product in L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ∈ [0, T], x ∈ D) with values in L2(D) is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) if, for all t ∈ [0, T] and all f ∈ Dom(L∗), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ⟨v(·, t), f⟩D = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨v(·, s), L∗f⟩D − 1 2a2(s) ⟨v(·, s), f⟩D � ds + � t 0 exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6) Since g is supposed to be locally Lipschitz, a blowup in finite time of v may occur, and the blowup time τ depends in general on ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) up to τ is defined as an (Ft, t ≧ 0)- adapted random field v that satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6) for all t ∈ (0, T ∧ τ) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If ω is such that v(ω, ·, ·) does not blowup in finite time, we set τ(ω) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field u = (u(x, t), t ∈ [0, T], x ∈ D) with values in L2(D) is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) up to τ if, for all t ∈ (0, T ∧ τ) and all f ∈ Dom(L∗), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (i) � t 0 a2(s) � 1 + ⟨u(·, s), f⟩2 D � ds < ∞, b(•) ⟨u(·, •), f⟩D ∈ Cβ[0, t] for some β > 1 − H, 4 (ii) � t 0 � k2(s) |⟨u(·, s), L∗f⟩D| + |⟨g(u(·, s)), f⟩D| � ds < ∞, and ⟨u(·, t), f⟩D = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s), f⟩D � ds + � t 0 ⟨u(·, s), f⟩D dNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) Conditions (i) and (ii) in the above definition are sufficient for the Itˆo, the fractional and the Lebesgue integrals in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) to be well defined P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We proceed now to the relation between (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) up to a random time τ, then v(x, t) = exp(−Nt)u(x, t) is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) up to τ, and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We notice that ⟨v(·, s), f⟩D is absolutely continuous in s if v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' With the choice u(x, t) := exp(Nt)v(x, t) condition (i) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In fact, for t < T ∧ τ(ω), � t 0 ⟨u(·, s), f⟩D a(s) dBs = � t 0 ⟨v(·, s), f⟩D exp(Ns)a(s) dBs is well defined since � t 0( � D v(x, s)f(x) dx)2 exp(2Ns)a2(s) ds < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Recall that the fractional integral � T 0 f(x)dg(x) is defined (in the sense of Z¨ahle [28]) in [18, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1] for f, g belonging to fractional Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If 0 < ε < H, f and g are H¨older continuous of exponents α and H − ε respectively, and α + H − ε > 1, this fractional integral coincides with the corresponding generalized Riemann-Stieltjes integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' see [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, the fractional integral � t 0 ⟨u(·, s), f⟩D b(s) dBH s = � t 0 ⟨v(·, s), f⟩D exp(Ns)b(s) dBH s (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='8) is well defined for t < T ∧ τ(ω) because, on the one hand, N· = � · 0(a(s) dBs + b(s) dBH s ) is P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' H¨older continous of order 1/2 − ǫ for all ǫ > 0 by the theorem of Kolmogorov and [22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' On the other hand b(·) is α-H¨older continuous (with α > 1 − H) and BH is H¨older continuous with exponent H − ε for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, choosing ε < min{H/2 − 1/4, α + H − 1} we get that the integrand on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='8) is H¨older continuous of order min{α, 1/2 − ε}, and therefore H −ε+min{α, 1/2−ε} > 1 and the integral is well defined as a generalized Riemann-Stieltjes integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It suffices to prove the assertion for t ∈ (0, T ∧ τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We apply (a slight generalisation 5 of) the Itˆo formula in [18, page 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let Y 1 t = � t 0 a(s) dBs , Y 2 t = � t 0 ⟨u(·, s), f⟩D a(s) dBs, Y 3 t = � t 0 b(s) dBH s , Y 4 t = � t 0 ⟨u(·, s), f⟩D b(s) dBH s , Y 5 t = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D � ds, and let F(y1, y2, y3, y4, y5) = exp(−y1 − y3)(y5 + y2 + y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then F(Y 1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 t ) = exp(−Nt) ⟨u(·, t), f⟩D = ⟨v(·, t), f⟩D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The above mentioned Itˆo formula then reads F(Y 1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 t ) = F(Y 1 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 0 ) + 5 � i=1 � t 0 ∂F ∂yi (Y 1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 s ) dY i s + 1 2 2 � i,j=1 � t 0 ∂2F ∂yi∂yj (Y 1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 s ) d � Y i s , Y j s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), ⟨v(·, t), f⟩D = ⟨ϕ, f⟩D − � t 0 exp(−Ns) ⟨u(·, s), f⟩D � a(s) dBs + b(s) dBH s � + � t 0 exp(−Ns) ⟨u(·, s), f⟩D � a(s)dBs + b(s)dBH s � + � t 0 exp(−Ns) �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D � ds −1 2 � t 0 exp(−Ns) ⟨u(·, s), f⟩D a2(s)ds = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨v(·, s), L∗f⟩D − 1 2a2(s) ⟨v(·, s), f⟩D � ds + � t 0 exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Similarly we obtain the viceversa result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to define the mild solutions of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) we define first the evolution families of contractions corresponding to the generator 1 2k2(t)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For 0 ≤ s < t let K(t, s) = 1 2 � t s k2(r) dr, A(t, s) = 1 2 � t s a2(r) dr, K(t) = K(t, 0), A(t) = A(t, 0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9) 6 and set pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = pD(K(t, s), x, y), x, y ∈ D × D, 0 ≦ s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ L2(D) the corresponding evolution families of contractions on L2(D) are given by U D(t, s)f(x) = � D pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy = P D K(t,s)f(x), (U D)∗(t, s)f(x) = � D pD(s, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, x)f(y)dy = (P D)∗ K(t,s)f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ≧ 0, x ∈ D) with values in L2(D) is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T] if, for all t ∈ [0, T], P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' v(x, t) = U D(t, 0)ϕ(x) − 1 2 � t 0 a2(s)U D(t, s)v(x, s) ds + � t 0 exp(−Ns)U D(t, s) � g((exp Ns)v(x, s)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The mild form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) can be written as v(x, t) = exp(−A(t))U D(t, 0)ϕ(x) + � t 0 exp(−Ns − A(t, s))U D(t, s)g(exp(Ns)v(·, s))(x)ds, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) where A(t, s) and A(t) are given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since g and ϕ are supposed to be nonnegative, v(x, t) ≥ exp(−A(t))U D(t, 0)ϕ(x) ≥ 0 for all x ∈ D and t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let w(x, t) = exp(A(t))v(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ L2(D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' we get from the definition of the mild solution d dt⟨w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2a2(t) exp(A(t))⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D + exp(A(t)) d dt⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2a2(t) exp(A(t))⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D + exp(A(t)) �1 2k2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D − 1 2a2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D � + exp(A(t)) exp(−Nt) ⟨g(exp(Nt)v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2 exp(A(t))k2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt)v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2k2(t) ⟨w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt − A(t))w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' with boundary conditions w(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 0) = ϕ(x) for x ∈ D and w(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t) = 0 for x ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore w is a weak solution of the RPDE formally given by d dtw(x, t) = 1 2k2(t)Lw(x, t) + exp(A(t) − Nt)g(exp(Nt − A(t))w(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 7 By the definition of the mild solution w(x, t) = U D(t, 0)ϕ(x) + � t 0 exp(A(s) − Ns)U D(t, s)g(exp(Ns − A(s))w(·, s))(x) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Consequently, v(x, t) = exp(−A(t))w(x, t) = exp(−A(t))U D(t, 0)ϕ(x) + � t 0 ds exp(−A(t, s) − Ns)U D(t, s)g(exp(Ns)v(·, s))(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) has a unique non-negative local mild solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' there exists t > 0 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) has a mild solution in L∞([0, t[×D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let T > 0 and denote ET = {v : [0, T] × D → L∞(D) : []v[] < ∞} , where []v[] := sup 0≤t≤T ∥v(t, ·)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let PT = {v ∈ ET : v ≥ 0} and for R > 0 let CR = {v ∈ ET : []v[] ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then ET is a Banach space and PT and CR are closed subsets of ET .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let us now define ψ(v)(t, x) = e−A(t)U D(t, 0)ϕ(x) + � t 0 e−A(t,s)−NsU D(t, s)g � eNsv(·, s) � (x) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We will prove that for sufficiently big R and sufficiently small T, ψ is contraction on PT ∩ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let v1, v2 ∈ PT ∩ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then []ψ(v1) − ψ(v2)[] ≤ sup 0≤t≤T � t 0 ��e−Ns � g � eNsv1 � − g � eNsv2 ���� ∞ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let AT = sup0≤s≤T e|Ns| and GR = sup|x| 0 centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, sup 0≤s≤T ��e−Nsvi(s, ·) �� ∞ ≤ AT R, i = 1, 2, and ��e−Nsg � eNsv1(s, ·) � − e−Nsg � eNsv2(s, ·) ��� ∞ ≤ A2 T KAT R ∥v1(s) − v2(s)∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, []ψ(v1) − ψ(v2)[] ≤ sup 0≤t≤T � t 0 A2 T KAT R []v1 − v2[] ds = TA2 T KAtR []v1 − v2[].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 8 We need TA2 T KAT R < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='11) In addition, we require that CR ∩ PT be mapped by ψ into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let v ∈ CR ∩ PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using that for 0 ≤ s ≤ T the operator U D(t, s) is a contraction, and that ∥eNsv(·, s)∥∞ ≤ AT R, we get ∥g(eNsv(·, s))∥∞ ≤ GAT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that []ψ(v)[] ≤ ∥ϕ∥∞ + sup 0≤t≤T � t 0 ���e−N(s)��� ∞ ds GAT R ≤ ∥ϕ∥∞ + TAT GAT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, we need that ∥ϕ∥∞ + TAT GAT R < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12) Let R be such that R ≥ 2∥ϕ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since limT→0 AT = 1, we choose ε1 > 0 so that AT < 2 if T < ε1, and ε < R 4G2R ∧ 1 4K2R ∧ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using that GA ≤ GB and KA ≤ KB if A ≤ B, we get for R > 2∥ϕ∥∞ and T < ε, ∥ϕ∥∞ + TAT GAT R ≤ ∥ϕ∥∞ + 2εG2R < R 2 + R 2 = R and TA2 T KAT R < 4εK2R < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We proceed to prove equivalence of weak and mild solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The proof of this theorem follows the method in [24, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15], where this equivalence is shown for SPDE’s with autonomous differential operators and driven by L´evy noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For a comparison of weak and mild solutions of SPDEs driven by fractional Brownian motion we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We state first the Kolmogorov backward and forward equations for U D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By the Kolmogorov back- ward equation for P D, the transition density pD(u, x, y) satisfies, for any y fixed, ∂ ∂upD(u, x, y) = LpD(u, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then (s, x) → pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) satisfies, for (t, y) fixed, the equation − ∂ ∂spD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = − ∂ ∂spD(K(t, s), x, y) = − ∂ ∂upD(u, x, y) |u=K(t,s) ∂ ∂sK(t, s) = 1 2k2(s)LpD(K(t, s), x, y) = 1 2k2(s)LpD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='13) Similarly, by the Kolmogorov forward equation for P D, for any x fixed, pD(u, x, y) satisfies ∂ ∂upD(u, x, y) = L∗pD(u, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 9 Then (t, y) → pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) satisfies, for (s, x) fixed, the equation ∂ ∂tpD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = ∂ ∂tpD(K(t, s), x, y) = ∂ ∂upD(u, x, y) |u=K(t,s) ∂ ∂tK(t, s) = 1 2k2(t)L∗pD(K(t, s), x, y) = 1 2k2(t)L∗pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='14) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Consider the random partial differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T] if and only if v is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let h ∈ C1([0, ∞), R), f ∈ Dom(L∗), and G(x, t) := − 1 2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The integration by parts formula is applicable since h ∈ C1([0, ∞), R) (see [24] Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16) and yields ⟨v(·, t), h(t)f(·)⟩D = ⟨v(·, 0), h(0)f(·)⟩D + � t 0 ⟨v(·, s), h′(s)f(·)⟩D ds + � t 0 ⟨v(·, s), 1 2h(s)k2(s)L∗f(·)⟩D ds + � t 0 ⟨G(·, s), h(s)f(·)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since the functions h · f are dense in C1([0, ∞), Dom(L∗)), for each z ∈ C1([0, ∞), Dom(L∗)) we have ⟨v(·, t), z(·, t)⟩D = ⟨v(·, 0), z(·, 0)⟩D + � t 0 ⟨v(·, s), ∂ ∂sz(·, s)⟩D ds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15) + � t 0 ⟨v(·, s), 1 2k2(s)L∗z(·, s)⟩D ds + � t 0 ⟨G(·, s), z(·, s)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For each f ∈ Dom(L∗) we define ψ(x, s) := (U D)∗(t, s)f(x) = \uf8f1 \uf8f2 \uf8f3 ⟨pD∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·)⟩D if s < t, f(x) if s = t, hence ψ ∈ C1([0, ∞), Dom(L∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Taking z = ψ(x, s) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15) we get, for any t ∈ [0, T] fixed, ⟨v(·, t), ψ(·, t)⟩D = ⟨v(·, 0), ψ(·, 0)⟩D + � t 0 � v(·, s), d dsψ(·, s) + 1 2k2(s)L∗ψ(·, s) � D ds + � t 0 ⟨G(·, s), ψ(·, s)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16) Now we evaluate the terms above: ⟨v(·, 0), ψ(·, 0)⟩D = � D v(x, 0) � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y) dy dx = � D f(y) � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)v(x, 0) dx dy = � U D(t, 0)v(·, 0), f(·) � D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 10 By applying the Kolmogorov backward equation to (x, s) → (U D)∗(t, s)f(x) we get − d dsψ(x, s) = − ∂ ∂s � (pD)∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·) � D = 1 2k2(s)L∗ � (pD)∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·) � D = 1 2k2(s)L∗ψ(x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, from Fubini’s theorem and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='4) ⟨G(·, s), ψ(·, s)⟩D = � D G(x, s) � D pD∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y) dy dx = � D f(y) � D pD(s, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, x)G(x, s) dx dy = � U D(t, s)G(·, s), f(·) � D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16), ⟨v(·, t), f(·)⟩D = � U D(t, 0)v(·, 0), f(·) � D + � t 0⟨U D(t, s)G(·, s), f(·)⟩D ds for all f ∈ Dom(L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since Dom(L∗) is dense in L2(D) we obtain that v is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To prove the converse let v be a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ Dom(L∗), � t 0 � v(·, s), 1 2k2(s)L∗f(·) � D ds = � t 0 � U D(s, 0)v(·, 0), 1 2k2(s)L∗f(·) � D ds + � t 0 �� s 0 χ[0,s](r)U D(s, r)G(·, r) dr, 1 2k2(s)L∗f(·) � D ds = � t 0 � v(·, 0), (U D)∗(s, 0)1 2k2(s)L∗f(·) � D ds + � t 0 � t r � U D(s, r)G(·, r), 1 2k2(s)L∗f(·) � D ds dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17) By applying the Kolmogorov forward equation to (U D)∗ we get for the first integral on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17): (U D)∗(s, 0)(1 2k2(s)L∗f)(x) = � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)1 2k2(s)L∗f(y) dy = � D (1 2k2(s)L)pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy = � D ∂ ∂spD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy, and therefore � t 0 � v(·, 0), (U D)∗(s, 0)(1 2k2(s)L∗)f(·) � D ds = � t 0 � v(·, 0), � D ∂ ∂spD∗(0, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy � D ds = � v(·, 0), � D pD∗(0, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy − f(·) � D = � v(·, 0), (U D)∗(t, 0)f(·) � D − ⟨v(·, 0), f(·)⟩D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 11 In the same way we get for the second integral on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17) � U D(s, r)G(·, r), 1 2k2(s)L∗f(·)) � D = � G(·, r), (U D)∗(s, r)(1 2k2(s)L∗f)(·) � D = � G(·, r), � D ∂ ∂spD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y)dy � D , and therefore � t r � U D(s, r)G(·, r), 1 2k2(s)L∗f(·) � D ds = � t r � G(·, r), � D ∂ ∂spD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y)dy � D ds = � G(·, r), � D pD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy − f(·) � D = � G(·, r), (U D)∗(t, r)f(·) − f(·) � D = � U D(t, r)G(·, r), f(·) � D − ⟨G(·, r), f(·)⟩D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this way we obtain � t 0 ⟨v(·, s), 1 2k2(s)L∗f(·)⟩D ds = � U D(t, 0)v(·, 0) + � t 0 U D(t, r)G(·, r)dr, f(·)⟩D − ⟨v(·, 0), f(·) � D − � t 0 ⟨G(·, r), f(·)⟩D dr = ⟨v(·, t), f(·)⟩D − ⟨v(·, 0), f(·)⟩ D − � t 0 ⟨G(·, r), f(·)⟩D dr, since v is a mild solution on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that v is a weak solution on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) possess unique weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 2 and Proposition 1 show the existence and uniqueness of a local weak and mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), and Proposition 1 shows the uniqueness of a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We refer to [23] for an existence and uniqueness theorem of the variational solution of an SPDE with a nonautonomous second order differential operator and driven by fractional Brownian motion, and to [26] for the existence and uniqueness of the mild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In [20] the existence and uniqueness of the mild solution is shown for equations with the same differential operator and driven by mixed noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3 An upper bound for the blowup time and probability estimates 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 An upper bound for the blowup time In the remaining part of the paper we will assume that L and L∗ admit strictly positive eigenfunctions: there exists a positive eigenvalue λ0 and strictly positive eigenfunctions ψ0 ∈ Dom(L) for P D t and 12 ϕ0 ∈ Dom(L∗) for (P D)∗ t with � D ψ0(x)dx = � D ϕ0(x)dx = 1 such that (P D t − e−λ0t)ψ0 = ((P D)∗ t − e−λ0t)ϕ0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) hence (L + λ0)ψ0 = (L∗ + λ0)φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) For generators of a general class of L´evy processes, properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) follow from [14, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Another example are the diffusion processes: for f ∈ C2 0(D), the set of twice continously differentiable functions with compact support in D, let us define the differential operator Lf = d � j,k=1 ∂ ∂xj � ajk ∂ ∂xk f � + d � j=1 bj ∂ ∂xj f − cf, where aj,k, bj, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', d are bounded smooth functions on D and c is bounded and continous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We assume that the matrix (aj,k, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', d) is symmetric and uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this case properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) follow from [12, Theorem 11, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let us define τ ∗ = inf � t > 0 : � t 0 exp [−β(λ0K(r) + A(r)) + βNr] dr ≥ 1 Cβ ⟨ϕ, φ0⟩−β D � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) where the functions K and A are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, on the event {τ ∗ < ∞} the solution v of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) and the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) blow up in finite time τ, and τ ≤ τ ∗ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using the hypothesis on g and Jensen’s inequality we get for the terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6): ⟨v(·, s), L∗φ0⟩D = −λ0⟨v(·, s), φ0⟩D, exp(−Ns) ⟨g(exp(Ns)v(·, s)), φ0⟩D ≧ C exp(βNs) � v1+β(·, s), φ0 � D , ≧ C exp(βNs)⟨v(·, s), φ0⟩1+β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Applying these lower bounds to (⟨v(·, t + ε), φ0⟩D − ⟨v(·, t), φ0⟩D)/ε and letting ε → 0 we get d dt⟨v(·, t), φ0⟩D ≧ −1 2(λ0k2(t) + a2(t))⟨v(·, t), φ0⟩D + C exp(βNt)⟨v(·, t), φ0⟩1+β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='21) The corresponding differential equality reads d dtI(t) = −1 2(λ0k2(t) + a2(t))I(t) + C exp(βNt)I(t)1+β, and I(t) is a subsolution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ⟨v(·, t), φ0⟩D ≧ I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then I(t) = exp[−(λ0K(t) + A(t))] � ⟨ϕ, φ0⟩−β D − βC � t 0 exp [−β(λ0K(s) + A(s)) + βNs] ds �−1/β 13 for all t ∈ [0, τ ∗), where τ ∗ is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore τ ∗ is an upper bound for the blowup time of ⟨v(·, t), φ0⟩D, and the function t �→ ∥v(·, t)∥∞ = exp(−Nt)∥u(·, t)∥∞ can not stay finite on [0, τ ∗] if τ ∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore u and v blow up before τ ∗ if τ ∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that τ ∗ depends on L only by the positive eigenvalue λ0 and the associated eigen- function φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, τ ∗ is a decreasing function of ϕ, φ0 and C, and an increasing function of λ0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore small functions ϕ, φ0 and a small constant C, as well as high values of λ0K postpone the blowup of I and have, in this sense, the tendency to postpone the blowup of v and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 A tail probability estimate for the upper bound of the blowup time In the following theorem we apply a tail probability estimate for exponential functionals of fBm studied by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Dung [8] to estimate the probability that τ ∗ occurs before a fixed time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Here we assume that the process BH is given by the formula BH t = � t 0 KH(t, s) dBs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22) where the kernel KH is given for H > 1/2 by KH(t, s) = \uf8f1 \uf8f2 \uf8f3 CHs1/2−H � t s (σ − s)H−3/2σH−1/2dσ if t > s, 0 if t ≦ s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='23) where CH = [ H(2H−1) B(2−2H,H−1/2)] 1 2 and B is the usual beta function (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3 in [21] for a general representation formula of fBm with H > 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that BH and B are dependent in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under assumptions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22), let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants, and let µ(T) = � T 0 exp[−β(λ0K(t) + A(t))]E [exp(βNt)] dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, for any T > 0 such that 1 Cβ⟨ϕ, φ0⟩−β D > µ(T), P {τ ∗ ≤ T} ≤ 2 exp \uf8eb \uf8ed− ln2 � Cβ⟨ϕ, φ0⟩β D µ(T) � 2M(T) \uf8f6 \uf8f8 , where M(T) = 2β2 � T 0 a2(r) dr + 4β2HT 2H−1 � T 0 b2(u) du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For t ≥ 0, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22), we have the following representation: Xt := −β(λ0K(t) + A(t)) + βNt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='24) = −β(λ0K(t) + A(t)) + β �� t 0 a(s) dBs + � t 0 � t s b(r) ∂ ∂rKH(r, s) dr dBs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 14 From [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1] it follows that for any T ≥ 0 and any x > µ(T), there holds P �� T 0 eXtdt ≥ x � ≤ 2 exp � −(ln x − ln µ(T))2 2M(T) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25) where µ(T) = � T 0 E � eXt� dt and M(T) is such that sup t∈[0,T] � T 0 |DrXt|2 dr ≤ M(T) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='26) Here DrXt denotes the Malliavin derivative of Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following we will find an upper bound M(T) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For r < t we have, using the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25), DrXt = β � a(r) + � t r b(s) ∂ ∂sK(s, r) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence � t 0 |DrXt|2 dr ≤ 2β2 � t 0 a2(r) dr + 2β2 � t 0( � t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds)2 dr and � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds �2 dr = � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds � �� t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ � dr = � t 0 b(s) ds � s 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr � t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ = � t 0 ds b(s) � t 0 dr1[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s](r) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) � t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ = � t 0 ds b(s) � t 0 ds′b(s′) � s′ 0 1[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s](r) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr = � t 0 ds � t 0 ds′ b(s)b(s′) � s∧s′ 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr = � t 0 ds � t 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = � t 0 ds � s 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) + � t 0 ds � t s ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = 2 � t 0 ds � s 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' where Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = � s∧s′ 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since ∂ ∂sK(s, r) = CHr1/2−H(s − r)H−3/2sH−1/2, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) in [21] we obtain Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = C2 H(ss′)H−1/2 � s∧s′ 0 r1−2H(s − r)H−3/2(s′ − r)H−3/2 dr = H(2H − 1)(s − s′)2H−2 15 for s′ < s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' hence � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds �2 dr ≤ 2H(2H − 1) � t 0 ds � s 0 |b(s)b(s′)|(s − s′)2H−2 ds′ ≤ H(2H − 1) �� t 0 b(s)2 � s 0 (s − s′)2H−2 ds′ ds + � t 0 � s 0 b(s′)2(s − s′)2H−2 ds′ ds � = H � t 0 b(s)2s2H−1 ds + H(2H − 1) � t 0 b(s′)2 � t s′ (s − s′)2H−2 ds ds′ = H � t 0 b(s)2(s2H−1 + (t − s)2H−1) ds ≤ 2Ht2H−1 � t 0 b(s)2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27) From the above inequalities we obtain sup t∈[0,T] � T 0 |DrXt|2dr ≤ 2β2 � T 0 a2(r)dr + 4β2HT 2H−1 � T 0 b2(u)du := M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28) Now, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) P(τ ∗ ≦ T) = P �� T 0 exp[−β(λ0K(t) + A(t)) + βNt] dt ≧ 1 Cβ ⟨ϕ, φ0⟩−β D � = P �� T 0 exp[X(t)] dt ≥ x � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='29) where x = 1 Cβ⟨ϕ, φ0⟩−β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The result follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following theorem we obtain upper bounds for the tail of τ ∗ in the case when the Brownian motion B and the fractional Brownian motion BH have general dependence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that BH t = � t 0 KH(t, s) dWs, where W is a Brownian motion defined in the same proba- bility space, and adapted to the same filtration as the Brownian motion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β D � T 0 � e−βλ0 � t 0 k2(s) ds+2β2 � t 0 a2(s) ds + e−β � t 0 a2(s) ds+4β2Ht2H−1 � t 0 b2(s) ds� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If B and BH are independent, then P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β D � T 0 e−βλ0K(t)+ β2−β 2 � t 0 a2(s) ds+β2Ht2H−1 � t 0 b2(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using H¨older’s and Chebishev’s inequalities we obtain P(τ ∗ ≤ T) = P �� T 0 e−βλ0K(t)+β � t 0 a(s) dBs−βA(t)+β � t 0 b(s) dBH s dt ≥ 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � ≤ P \uf8ee \uf8f0 �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt � 1 2 × �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt � 1 2 ≥ 1 Cβ ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D \uf8f9 \uf8fb ≤ P �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt ≥ 1 Cβ ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � +P �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt ≥ 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � ≤ E �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt � + E �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt � 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D ≤ � T 0 � e−2βλ0K(t)+2β2 � t 0 a2(s) ds� dt + � T 0 e−2βA(t)E � e2β � t 0 b(s) dBH s � dt 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='30) where we have used the fact that E � exp �� t 0 f(s) dB(s) �� = exp � 1 2 � t 0 f 2(s) ds � to obtain the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In addition, E � e2β � t 0 b(s) dBH s � = E � e2β � t 0 � t s b(r) ∂ ∂r KH(r,s) dr dWs� = e2β2 � t 0[ � t s b(r) ∂ ∂r KH(r,s) dr] 2 ds, where the last equality follows from [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27) we get E � e2β � t 0 b(s) dBH s � ≤ exp � 4β2Ht2H−1 � t 0 b2(s) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='30) we obtain the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using Chebishev’s inequality, the independence of B and BH and the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31), P(τ ∗ ≤ T) = P �� T 0 e−βλ0K(t)+β � t 0 a(s) dBs−βA(t)+β � t 0 b(s) dBH s ) dt ≥ 1 Cβ ⟨ϕ, φ0⟩−β D � ≤ Cβ⟨ϕ, φ0⟩β D � T 0 E � e−βλ0K(t)+β � t 0 a(s) dBs� E � e−βA(t)+β � t 0 b(s) dBH s � dt ≤ Cβ⟨ϕ, φ0⟩β D � T 0 exp � −βλ0K(t) + β2 − β 2 � t 0 a2(s) ds + β2Ht2H−1 � t 0 b2(s) ds � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4 Lower bounds for the blowup time and for the probability of finite time blowup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 A lower bound for the probability of finite time blowup In the following theorem we give a lower bound for the probability of finite time blow up of the weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If f, g are nonnegative functions and c is a constant, we write f(t) ∼ cg(t) as t → ∞ if limt→∞ f(t)/g(t) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let g(z) ≥ Cz1+β and � t 0 a2(r) dr ∼ C1t2l, � t 0 b2(r) dr ∼ C2t2m, � t 0 k2(r) dr ∼ C3t2p as t → ∞ for some nonnegative constants l, m, p and positive constants C, β, C1, C2 and C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose additionally that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β ∈ (0, 1/2), then max{p, l} > H + m − 1 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β = 1/2, then H+m − 1 2 < p, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β > 1/2, then p > max{l, H + m − 1 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under these assumptions the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) blows up in finite time with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, P(τ < ∞) ≧ P(τ ∗ < ∞) ≧ 1 − exp � −(mξ − 1)2 2Lξ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32) where ξ = 1 Cβ ⟨ϕ, φ0⟩−β D , Lξ = sup t≧0 M(t) (ln(ξ + 1) + f(t))2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='33) 18 with f(t) = tmax{H+m−1/2, l} and mξ = E \uf8ee \uf8f0sup t≧0 ln �� t 0 exp (−β(λ0K(s) + A(s)) + βNs) ds + 1 � + f(t) ln(ξ + 1) + f(t) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='29) it follows that P(τ ∗ < ∞) = P( � ∞ 0 eXt dt ≥ ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to estimate P( � ∞ 0 eXt dt ≥ ξ) we use [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1], with a = 0 and σ = 1 : Proposition 3 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that the stochastic process X is adapted and satisfies a) � ∞ 0 EeXs ds < ∞, b) For each t ≥ 0, Xt ∈ D1,2, c) There exists a function f : R+ → R+ such that limt→∞ f(t) = ∞ and for each x > 0, sup t≧0 sups∈[0,t] � t 0 |DrXs|2dr (ln(x + 1) + f(t))2 ≤ Lx < ∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='35) Then P �� ∞ 0 eXt dt < x � ≤ exp � −(mx − 1)2 2Lx � , where mx = E � sup t≥0 ln( � t 0 eXs ds + 1) + f(t) ln(x + 1) + f(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We now verify that conditions a) - c) of the above proposition hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For condition a) we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' � ∞ 0 E exp[Xt] dt = � ∞ 0 E exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β �� t 0 a(s) dBs + � t 0 � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr dBs �� dt = � ∞ 0 E exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β � t 0 � a(s) + � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr � dBs � dt = � ∞ 0 exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β2 2 � t 0 � a(s) + � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr �2 ds � dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' we have used [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12] to obtain the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27), � ∞ 0 E exp[Xt] dt ≤ � ∞ 0 exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β2 2 � t 0 2a2(s) ds + 2β2Ht2H−1 � t 0 b2(s) ds � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='36) 19 The integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='36) is finite if and only if the leading power of t in the term −βλ0 2 � t 0 k2(s) ds + 2β2 − β 2 � t 0 a2(s) ds + 2β2Ht2H−1 � t 0 b2(s) ds has negative coefficient, which follows from our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Condition b) is a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For condition c) we use the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28), which implies that for any x > 0 and any fixed function f, sup t≧0 sups∈[0,t] � t 0 |DrXs|2dr (ln(x + 1) + f(t))2 ≤ sup t≥0 M(t) (ln(x + 1) + f(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='37) Due to our assumptions, for big t, the leading power of t in the numerator is max{2l, 2H + 2m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that lim t→∞ M(t) � ln(x + 1) + tmax{l,H+m−1/2}�2 < ∞, and therefore the supremum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='37) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The result follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The cases when a = 0 (presence only of fractional Brownian motion) or b = 0 (presence only of Brownian motion), are simpler: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under the assumptions in Theorem 6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When a(t) ≡ 0 and p > H + m − 1/2 the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) explodes in finite time with positive probability for all β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If a(t) ≡ 0 and p = H + m − 1/2, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) explodes in finite time with positive probability for all β > 0 satisfying β < C3λ0 4C2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When b(t) ≡ 0 and 0 < β ≤ 1 2 the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits explosion in finite time with positive probability for all values of p and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If b(t) ≡ 0 and β > 1/2, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits explosion in finite time with positive probability if p > l or if p = l and C3λ0 > C1(2β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that mξ given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='34) satisfies mξ > 1 due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The formula for mξ shows interactions between ϕ and K that have an influence on the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Increasing values of K decrease the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this sense high values of K are in favour of absence of finite time blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 The case H > 3/4 and independent B and BH In order to find more explicit lower bounds for P(τ < +∞), we consider in this subsection the case H ∈ (3/4, 1) and suppose that B and BH are independent and b(s) = ca(s) for all s ≧ 0, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then Nt = � t 0 a(s)dMs with Ms = Bs + cBH s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By [3] M is equivalent to a Brownian motion �B, and therefore Nt is equivalent to ˜Nt := � t 0 a(s) d �Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Here equivalence means equality of the laws of the processes on (C[0, T], B), the space of continous functions defined on [0, T] endowed with the σ−algebra generated by the cylinder sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Furthermore, ( ˜Nt)t≧0 is a continous martingale and therefore a time-changed Brownian motion: ˜Nt = �B2A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let H ∈ (3/4, 1), B and BH be independent and b(s) = ca(s) for all s ≧ 0, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='We assume also that g(z) ≥ Cz1+β, that the functions k and a are positive continuous on R+ and that there exist constants η ∈ (0, +∞] and c1 > 0 such that 1 a2(t) exp(−βλ0K(t)) ≥ c1 exp � −2β A(t) η � , t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38) Then P(τ < +∞) ≥ P(Zµ ≤ θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='39) where τ is the blowup time of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), Zµ is a gamma-distributed random variable with parameter µ := 2 β( 1 η + 1 2), θ := 2c1 β2ξ and ξ := 1 Cβ⟨ϕ, φ0⟩−β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' From Theorem 3, P(τ ∗ = +∞) = P �� t 0 dr exp � −β(λ0K(r) + A(r)) + β ˜Nr � < ξ for all t > 0 � = P �� ∞ 0 dr exp � −β(λ0K(r) + A(r)) + β ˜Nr � ≤ ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By the change of variable q = 2A(r) we get P(τ ∗ = +∞) = P �� ∞ 0 dr exp � −β(λ0K(r) + A(r)) + β ˜B2A(r) � ≤ ξ � = P �� ∞ 0 dq a2(A−1(q/2)) exp � −β(λ0K(A−1(q/2)) + 1 2q) + β ˜Bq � ≤ ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38) to t = A−1(q/2) yields 1 a2(A−1(q/2)) exp � −β(λ0K(A−1(q/2)) � ≥ c1 exp � −β η q � , q ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 21 Therefore P(τ ∗ = +∞) ≤ P � c1 � ∞ 0 dq exp � −βq �1 η + 1 2 � + β ˜Bq � ≤ ξ � = P �� ∞ 0 dq exp � β( ˜Bq − ˜µq) � ≤ ξ c1 � , where ˜µ := 1 η + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A second change of variable q = 4s β2 yields P(τ ∗ = +∞) ≤ P �� ∞ 0 ds exp � 2( ˜Bs − µs) � ≤ β2ξ 4c1 � , where µ := ˜µ 2 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to [27, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2, page 95], � ∞ 0 e2( ˜ Bs−µs) ds L= 1 2Zµ , where Zµ is a gamma-distributed random variable with parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore P(τ = +∞) ≤ P(τ ∗ = +∞) ≤ P � 1 2Zµ ≤ β2ξ 4c1 � = P � Zµ ≥ 2c1 β2ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This implies the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If k, a and b are constants, a more explicit lower bound for P(τ < +∞) is available without the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Indeed, starting with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20), a straightforward calculation gives a lower bound in terms of a gamma-distributed random variable Z again, but this time with parameter �µ := (λ0k2 + a2)/(a2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' More precisely, P(τ < ∞) ≧ P(τ ∗ < ∞) = P � Z�µ ≦ 2C a2β ⟨ϕ, φ0⟩β D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3 A lower bound for the blowup time Our next goal is to obtain a lower bound for the blowup time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since the proofs of the following results are close to those in [1] (where b = 0), we omit them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let the function g be such that g(0) = 0, z → g(z)/z is increasing, and g(z) ≤ Λz1+β for some positive constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then τ ≥ τ∗, where τ∗ = inf � t > 0 : � t 0 exp(β(Nr − A(r))) ��U D(r, 0)ϕ ��β ∞ dr ≧ 1 Λβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='40) Let us define for 0 ≦ t < τ∗, J(t) = � 1 − Λβ � t 0 exp(β(Nr − A(r))) ��U D(r, 0)ϕ ��β ∞ dr �−1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 22 Then the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies, for x ∈ D, 0 ≦ t < τ∗, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 0 ≦ u(x, t) ≦ J(t) exp(Nt − A(t))U D(t, 0)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' More precisely, the proof of this theorem shows that the mild solution v of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) without the factor exp(Nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By Theorem 2, v is also the weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), hence the weak solution u(·, t) = exp(Nt)v(·, t) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that Λβ � ∞ 0 exp[β(Nr − A(r))] ��U D(r, 0)ϕ ��β ∞ dr < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For the special choice of ϕ = pψ0, p > 0, the integrals appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='40) are the same exponential functionals of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In fact, U D(r, 0)ψ0 = exp(−λ0K(r))ψ0, and τ∗ becomes τ∗ = inf � t > 0 : � t 0 exp � β(Nr − λ0K(r) − A(r)) � dr ≧ p−β Λβ ∥ψ0∥−β ∞ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='42) whereas τ ∗ = inf � t > 0 : � t 0 exp � β(Nr − λ0K(r) − A(r)) � dr ≥ p−β Cβ ⟨ψ0, φ0⟩−β D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='43) In fact τ∗ ≦ τ ∗ if C ≦ Λ, since ⟨ψ0, φ0⟩D ≦ ∥ψ0∥∞ � D φ0(x)dx = ∥ψ0∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to apply both bounds simultaneously, we have to suppose Cz1+β ≦ g(z) ≦ Λz1+β, z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It is therefore of interest to know the law of the integral appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='42) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This seems possible only for bH = 0, since, to our best knowledge, the law of exponential functionals of fractional Brownian motion is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For the moment it seems that only estimates of the type of those in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' See also Theorem 7 for H > 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 5 A sufficient condition for finite time blowup We consider now the mild form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) obtained in Proposition 2, and obtain a sufficient condition for finite time blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose that g(z) ≥ Cz1+β and that there exists w∗ > 0 such that exp(βA(w∗)) ∥ U D(w∗, 0)ϕ ∥−β ∞ < βC � w∗ 0 exp(βNs) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) Then for the explosion time τ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) there holds τ ≤ w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 23 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) is understood trajectorywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore w∗ is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) is harder to satisfy with a small initial condition ϕ and with a small value of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to the different interpretations of the integrals in N, the effects on blowup of B and BH are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If N = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) reads ∥ U D(w∗, 0)ϕ ∥−β ∞ < βCw∗ and in this case w∗ is deterministic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if in addition ϕ = ψ0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) reads exp(λ0βK(w∗)) ∥ ψ0 ∥−β ∞ < βCw∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We use the approach in [25, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' see also [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose that v(x, t), x ∈ D, t ≥ 0, is a global solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), and let 0 < t < t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using the semigroup property of the evolution system (U D(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r))0≦r 2. To do this, we can construct our polynomials by enforcing a fixed sparsity +on each of the matrix powers; for simplicity we chose the sparsity of Aff. This is particularly suited to convection +operators given that the fill-in should be small. For example, if we consider a third-order polynomial (i.e., m = 4), and +denoting the sparsity pattern of Aff as S ⊂ {(i, j) | (Aff)i, j � 0}, we enforce that +( ˜A2 +ff)i,j = (AffAff)i,j, +( ˜A3 +ff)i, j = ( ˜A2 +ffAff)i, j +(i, j) ∈ S, +(29) +and for (i, j) not in S , the entries are zero. This is simply computing A2 +ff with no fill-in, using this approximation when +computing subsequent matrix-powers and again enforcing no fill-in on the result. Our fixed-sparsity approximation to +A−1 +ff with m = 4 would then be given by +ˆA +−1 +ff = α0 + α1Aff + α2 ˜A2 +ff + α3 ˜A3 +ff ≈ q3(Aff). +(30) +Fixing the sparsity of the matrix powers reduces the memory consumption of our hierarchy and also allows us to +optimise the construction of our polynomial. For example, with m = 4 it costs two matrix-matrix additions and two +matmatmults to explicitly construct our polynomial, but given the shared sparsity, the additions can be performed +quickly and the matmatmults with m > 2 can share the same row data required when computing A2 +ff. In parallel this +means we only require the communication of off-processor row data once with m > 2. Computing low-order GMRES +polynomials with fixed sparsity with the Krylov basis therefore only requires the communication associated with m +matvecs, a single all-reduce and if m > 2 the matmatmult which produces A2 +ff, regardless of the polynomial order. +This has the potential to scale well, in contrast to some of the other methods which approximate A−1 +ff described +above. We cannot use nAIR with truncated Neumann series due to the lack of lower triangular structure in our spatial +discretisation, while with lAIR we found we require greater than distance 2 neighbours for scalability (we exam- +ine this in Section 7), but the communication required for this becomes prohibitive. Using ILU factorisations make +parallelisation difficult given the sequential nature of the underlying Gaussian elimination; approximate ILU factori- +sations have more parallelism [69, 70] but they still require triangle solves (which could then also be approximated +with truncated Neumann series for better performance in parallel, for example). The SAIs used by [46] should scale +well, given that if the fixed sparsity of Aff is used, then the formation of an approximate inverse only requires the +same communication as computing A2 +ff; we must however also consider the local cost of computing a SAI and the +effectiveness of its approximation; we examine this in Section 7. +Typically with AIR the approximations ˆA +−1 +ff on each level are thrown away once the grid-transfer operators have +been built and different F-point smoothers are used; as mentioned we save them to use as smoothers instead. The GM- +RES polynomials are very strong smoothers (which don’t require the calculation of any extra dampening parameters) +but storing them explicitly takes extra memory. We examine the total memory consumption of the AIRG hierarchy +in Section 7, but note that given the fixed sparsity of ˆA +−1 +ff , our experiments with pure streaming show it has approxi- +mately 65% as many non-zeros as R. We could instead throw away ˆA +−1 +ff after our grid transfer operators are computed +on each level and perform F-point smoothing by applying qm−1(Aff) matrix-free. With m ≤ 2 the number of matvecs +required is the same and hence ˆA +−1 +ff could be discarded. With m > 2 however, applying qm−1(Aff) matrix-free would +require more matvecs in order to apply the matrix-powers. As such we believe the (constant sized) extra memory +required is easily justified. We also investigated using Chebyshev polynomials as smoothers. We precomputed the +required eigenvalue estimates with GMRES in order to build the bounding circle/ellipse given our asymmetric linear +systems. We found that smoothing with these polynomials was far less efficient/robust (and practically more difficult) +than using the GMRES polynomials; indeed using the GMRES polynomials directly is one of the key messages of +[60]. +10 + +4.1.2. Drop tolerance on Aff +If our original linear system Ax = b is not sparse, neglecting the fill-in with the fixed sparsity polynomials +in Section 4.1.1 may not be sufficient by itself to ensure a practical multigrid method. We can also introduce a +drop tolerance to Aff that is applied prior to constructing our polynomial approximations ˆA +−1 +ff . In general this is +not necessary in the streaming limit, but with scattering we find it helps keep the complexity low. This is similar +to the strong R threshold used in the hypre implementation of lAIR, which determines strong neighbours prior to +the construction of Z, and in Section 7 we denote it as such. We can either apply the dropping after the polynomial +coefficients are computed (in a similar fashion to how the fixed sparsity in Section 4.1.1 is applied), or before such that +we are forming a polynomial approximation to a sparsified Aff. With scattering, we did not find much of a difference +in convergence so we chose to apply the dropping before, as this helps reduce the cost of the matvec used to compute +the polynomial coefficients. +4.2. Approximations of A−1 +ff +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +(a) Operators formed from GMRES polynomials without fixed +sparsity. +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +(b) Operators formed from fixed sparsity GMRES polynomials +and relative drop tolerances applied to the resulting R and P. +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +(c) Operators formed from fixed sparsity GMRES polynomials, +relative drop tolerances applied to the resulting R and P and rela- +tive drop tolerances applied to the resulting Acoarse. +Figure 1: Eigenvalue distribution of A−1 +coarseS on the second level of a pure streaming problem, where S is the exact Schur complement formed +from the ideal restrictor and prolongator (i.e., the exact coarse-grid matrix) and Acoarse is the approximate coarse grid matrix formed from our +approximate ideal restrictor and prolongator. The colours correspond to different GMRES polynomial orders for approximating A−1 +ff , with m = 1, +m = 2, m = 3 and m = 4. +11 + +As mentioned in Section 4.1, the low-order polynomials we use and the fixed sparsity described in Section 4.1.1 +may impact the operators in our multigrid hierarchy. Given this we examine the impact of our approximations to +A−1 +ff and the resulting operators by considering the spectrum of A−1 +coarseS and Acoarse, where Acoarse is the coarse matrix +formed from our approximate operators and S is the exact coarse grid matrix. +As an example, we use AIRG on a pure streaming problem with an unstructured grid, with two levels of uniform +refinement in angle (giving 16 angles). Fig. 1 plots the the spectrum of A−1 +coarseS and ideally it should approach one. +Fig. 1a shows the result of constructing our coarse grid matrix with increasing orders of our GMRES polynomial, but +without fixing the sparsity of the matrix powers, so we are using q0(Aff), q1(Aff), q2(Aff) and q3(Aff) exactly. We can +see increasing the order increases the accuracy of our resulting coarse grid matrix, as would be expected, with the +eigenvalues converging to one. The radius of a circle that bounds the eigenvalues reduces from 0.1760 to 0.0072 with +m = 1 to m = 4, respectively. +Fig. 1b shows the result of introducing both the fixed sparsity matrix powers discussed in Section 4.1.1 and +introducing relative drop tolerances to the resulting R and P operators. For the restrictor we drop any entry in a row +that is less than 0.1 times the maximum absolute row entry, and for the prolongator we keep only the largest entry +in each row. We can see these approximations are detrimental to Acoarse compared with Fig. 1a, with the eigenvalues +further from one. The bounding circle at all orders has greater radius, with 0.2533 for m = 1 and 0.0913 for m = 4. +Finally, Fig. 1c shows the results from using the same fixed sparsity matrix powers and drop tolerances on R and +P while also introducing a relative drop tolerance on the resulting Acoarse, where we drop any entry in a row that is +less than 0.1 times the maximum absolute row entry. We again see this further degrades our approximate coarse grid +matrix, with the effect of increasing the order of our GMRES polynomials diminished; the bounding circle goes from +0.3539 to 0.3532 with m = 1 to m = 4, respectively. +It is clear that introducing additional sparsity into our GMRES polynomials and resulting operators degrades our +coarse matrix. We examine this further by plotting the smallest eigenvalues of Acoarse and S in Fig. 2. In the limit of +ideal operators we know the near-nullspace vectors are preserved, but we would like to verify that with an approximate +ideal restrictor and approximate ideal prolongator this is still the case. We can see that in Fig. 2a that the GMRES +polynomials with m = 4 and fixed sparsity does an excellent job capturing the smallest eigenvalues. Furthermore +introducing both fixed sparsity to the GMRES polynomial and drop tolerances on R and P results in reasonable +approximations. We can see in Fig. 2b that introducing the drop tolerances on the resulting Acoarse results in small +eigenvalues that do not match the exact coarse grid matrix. +These results indicate that our low-order GMRES polynomials with fixed sparsity and the introduction of drop +tolerances to our approximate ideal R and P results in an excellent approximation for the coarse grid streaming +operator. It is well known that introducing additional sparsity to multigrid operators can harm the resulting operators, +and it is clear from these results that introducing a drop tolerance to our coarse grid matrix has the biggest impact. In +a multilevel setting however, we find that doing this often result in the best performance, with the slight increase in +iterations balanced by the reduced complexity; care must be taken to not make the drop tolerance too high. +5. CF splitting +All multigrid/multilevel methods require the formation of a hierarchy of “grids”; LDU methods and reduction +multigrids like in this work require the selection of a subset of DOFs defined as “fine” and “coarse”. For asymmetric +linear systems, CF splitting algorithms often result in coarse grids with directionality (i.e., they result in a semi- +coarsening), typically through heuristic methods that identify strong connections in matrix entries (e.g, see [71]) with +algorithms like CLJP, PMIS, HMIS, etc [72]) or through compatible relaxation [73, 74, 75]. We would like the CF +splitting to produce a well-conditioned Aff on each level without giving a large grid or operator complexity across the +hierarchy. The effectiveness of some of the approximations used in the literature (described in Section 4) for A−1 +ff also +clearly depend on the sparsity of Aff produced by a CF splitting. +Previous works have used various CF splittings, including those that produce a maximally-independent set, giving +a diagonal Aff that is easily inverted [76]; or if a block-independent set is generated then Aff is block-diagonal and the +blocks can be inverted directly [77, 78]. With a more general CF splitting [52] used ILU factorisations to approximate +A−1 +ff . [79] produced CF splittings specifically for reduction multigrids and LDU methods that are targeted at producing +a diagonally dominant Aff. +12 + +0.03 +0.032 +0.034 +0.036 +0.038 +0.04 +0.042 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +#10 -3 +(a) The × are eigenvalues of Acoarse with fixed sparsity GMRES +polynomial, the ∗ are with fixed sparsity GMRES polynomial and +relative drop tolerances applied to the resulting R and P. +0.03 +0.032 +0.034 +0.036 +0.038 +0.04 +0.042 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +#10 -3 +(b) The . are eigenvalues of Acoarse with fixed sparsity GMRES +polynomial, relative drop tolerances applied to the resulting R and +P and relative drop tolerances applied to the resulting Acoarse. +Figure 2: Eigenvalue distribution for the smallest eigenvalues of Acoarse and S on the second level of a pure streaming problem, where S is the +exact Schur complement formed from the ideal restrictor and prolongator (i.e., the exact coarse-grid matrix) whose eigenvalues are denoted with +the black “o”. The red symbols are the eigenvalues of Acoarse, which is the approximate coarse grid matrix formed from our approximate ideal +restrictor and prolongator given a GMRES polynomial approximation of A−1 +ff with m = 4 and different sparsification applied to operators. +In this work we use traditional CF splitting algorithms (like in [28]) as we find they perform well enough and +parallel implementations are readily available. Section 7 presents the results from using the lAIR implementation in +hypre and we found that using the Falgout-CLJP algorithm in hypre resulted in good CF splittings. In order to make +fair comparisons, we show results from using AIRG with the same algorithm. +6. Work estimates +One of the key metrics we use to quantify the performance of the iterative methods tested is the number of Work +Units (WUs) required to solve our linear systems; this is a FLOP count scaled by the number of FLOPs required to +compute a matvec. We present several different WU calculations, each of which is scaled by a different matvec FLOP +count. This is in an attempt to show fair comparisons against other multigrid methods, along with source iteration. +To begin, we must first establish a FLOP count for all the different components of our iterative methods. We begin +with our definition of the Cycle Complexity (CC) of AIRG. The CC is the amount of work performed during a single +V-cycle, scaled by the number of nnzs in the top-grid matrix. Our calculation of the CC includes the work performed +during smoothing and grid-transfer operators; we use our definition of CC and WUs in all the results below. We define +the work required to compute a matvec with our matrices on each level as {.}l. For an assembled matrix we set this +as the nnzs. This assumes fused-multiply-add (FMA) instructions are available and hence the cost of multiplying by +Aff for example is nnzs rather than 2×nnzs (this cost scales out of the CC anyway). If we consider a general linear +system, Ax = b, the FLOP count for performing a single V-cycle with lmax levels of AIRG is given by +FLOPAIRG +V += { ˆA +−1}lmax + +l=lmax−1 +� +l=1 +vup{ ˆA +−1 +ff }l + vup{Aff}l + {Afc}l + {R}l + {P}l, +(31) +where vup = 2 is the number of up F-point smooths and we perform one application of a GMRES polynomial approx- +imation of ˆA +−1 as a coarse grid solve on l = lmax. +In Section 7, we also show the results from using lAIR in hypre with FCF-Jacobi smoothing; by default the CC +output by hypre doesn’t include all work associated with smoothing, residual calculation, etc, and hence we recompute +13 + +it. Due to the use of FCF-Jacobi, the result of Afcxc during the F smooths cannot be cached (similarly for the C-point +smooths). We therefore compute +FLOPhypre +V += { ˆA +−1}lmax + +l=lmax−1 +� +l=1 +vup(2nl +F + nl +C) + vup +� +2 +� +{Aff}l + {Afc}l� ++ {Acf}l + {Acc}l� ++ {R}l + {P}l, +(32) +where nl +F and nl +C are the number of F and C-points on level l, respectively and given we only use one FCF-Jacobi as +our smoother on each level vup = 1. The cycle complexity (for either hypre or AIRG) is then given by +CC = FLOPV +{A}1 . +(33) +Any matvec that involves scatter should be computed matrix-free and we denote that with an “mf” subscript. +Given our sub-grid scale discretisation, we need to account for the cost of computing the source on the rhs of (11), the +fine-scale solution Θ and the addition of the coarse and fine scale solutions to form ψ. These are given by +FLOPsource = {B}mf + { ˆD +−1}, +FLOPSGS = {C}mf + { ˆD +−1}, +FLOPψ = NDDOFs +(34) +The FLOP count of one iteration of our angular preconditioner (14) is given by +FLOPangle = 2 × NCDOFs + 4.5 × {Ddiff}. +(35) +We investigated using AIRG to invert the diffusion operator, but found it difficult to beat the default boomerAMG +implementation in hypre (i.e., not lAIR), which is unsurprising given hypre has been heavily optimised for such +elliptic operators. The factor of 4.5 comes from the cycle complexity of running boomerAMG on a heavily refined +spatial grid and as might be expected we see the cycle complexity plateaus to around this value (hence we have an +upper bound on work on less refined grids). +We must now quantify the cost of our matrix-free matvecs. As mentioned in Section 2.2 there are numerous ways +we could compute such a matvec, depending on how much memory we have available; Section 3 discussed that we +store the streaming/removal operator, MΩ to precondition with and hence we use that in our matvec. The additional +cost therefore comes with scattering (which we assume is isotropic in this work) and hence we have +{A − B ˆD +−1C}mf = {MΩ} + +� +i∈cg nodes +2 × δ(i) × NCDOFs(i)+ +� +i∈dg nodes +δ(i) × (2 × NDDOFs(i) + 2 × nnodes + 1) + +� +e∈eles +3 × δ(i) × { ˆD +−1 +e } +(36) +where nnodes is the number of spatial nodes on our DG elements (3 in 2D or 4 in 3D with tri/tets), { ˆD +−1 +e } is the number +of non-zeros in our stored block approximation on a given element (the factor of 3 comes from one application of ˆD +−1 +and one of BΩ and CΩ which have the same sparsity); this sparsity depends on the angles present on each DG node, +but is at a maximum when uniform angle is used and for each angle present on all nodes of an element we have a +nnodes × nnodes block. N*DOFS(i) is the number of DOFs on an individual (CG or DG) spatial node i and δ(i) is 1 or 0 +depending on the presence of scatter; for a CG node this is zero if every element connected to node i has a zero scatter +cross-section, and one otherwise, for a DG node this is zero if the element containing node i has a zero cross-section, +one otherwise. The calculation in (36) includes the work required to map to/from Legendre space on both our coarse +(CG) and fine (DG) spatial meshes in order to apply the scatter component in A, B and C. Similar expressions are +used for the individual {B}mf and {C}mf in (34). +We now have all the components required to calculate our WUs. We begin with a simple definition, where we use +AIRG as a preconditioner on the assembled matrix, A − B ˆD +−1C, that could include scatter and is hence non-scalable. +If nits is the number of outer GMRES iterations performed then the total FLOPs are +FLOPfull = nits +� +{A − B ˆD +−1C} + FLOPV +� ++ FLOPsource + FLOPSGS + FLOPψ, +(37) +14 + +and hence the WUs +WUsfull = +FLOPfull +{A − B ˆD +−1C} +. +(38) +If instead we use the additively preconditioned iterative method defined in Section 3 along with our matrix-free +matvec, our total FLOPs are +FLOPmf = nits +� +{A − B ˆD +−1C}mf + FLOPV + FLOPangle +� ++ FLOPsource + FLOPSGS + FLOPψ, +(39) +We then scale these FLOP counts in several different ways. Firstly there is the WUs required to compute a matrix-free +matvec of our sub-grid scale discretisation, namely +WUsmf = +FLOPmf +{A − B ˆD +−1C}mf +. +(40) +In order to make rough comparisons with a traditional DG FEM source iteration method, we can scale by the work +required to compute a matrix-free matvec with a DG FEM. In order to not unfairly disadvantage a DG discretisation, +we assume the DG streaming operator is stored in memory, and hence the FLOPs required to compute a single DG +matvec is +FLOPDG = 5 +3{ ˆD +−1} + +� +i∈dg nodes +δ(i) × (2 × NDDOFs(i) + nnodes). +(41) +The factor of 5/3 comes from the jump terms (in 2D) that are not included in the nnzs of our sparsified DG matrix +ˆD +−1. The work units scaled by this quantity are therefore +WUsDG = FLOPfull +FLOPDG +or +FLOPmf +FLOPDG +. +(42) +Again we note that with scattering we have {A − B ˆD +−1C}mf ≈ 1.8 × FLOPDG. +7. Results +Outlined below are several examples problems in both the streaming and scattering limits, designed to test the per- +formance of AIRG and our additively preconditioned iterative method. We solve our linear systems with GMRES(30) +to a relative tolerance of 1× 10-10, with an absolute tolerance of 1× 10-50 and use an initial guess of zero unless +otherwise stated. We should note that we use AIRG (and lAIR) on our matrices without relying on any (potential) +block structure, for example in the streaming limit with uniform angle we could use our multigrid on each of the angle +blocks separately. We also don’t scale our matrices; for example we could view a diagonal scaling as preconditioning +the outer GMRES iteration and/or the GMRES polynomials in our multigrid, but we did not find it necessary. +When using AIRG we perform zero down smooths and two up F-point smooths (our C-points remain unchanged). +On the bottom level of our multigrid we use one Richardson iteration to apply a GMRES polynomial approximation of +the coarse matrix. We use a 1-point prolongator which computes W and then keeps the biggest absolute entry per row. +Unless otherwise noted we use 3rd order (m = 4) GMRES polynomials with fixed sparsity as described in Section +4.1.1. We only use isotropic scatter in this work. For both AIRG and lAIR, we use the row-wise infinity norm to define +any drop tolerances. When using the lAIR implementation in hypre, we use zero down smooths and one iteration of +FCF-Jacobi for up smooths, while on the bottom level we use a direct-solve, as we found these options resulted in the +lowest cost/best scaling. We searched the parameter space to try and find the best values for drop tolerances, strength +of connections, etc when using lAIR and AIRG; these searches were not exhaustive however, and it is possible there +are more optimal values. +All timing results are taken from compiling our code, PETSc 3.15 and hypre with “-O3” optimisation flag. We +compare timing results between our PETSc implementation of AIRG and the hypre implementation of lAIR. As such +we try and limit the impact of different implementation details. Given this, when calculating the setup time we exclude +the CF splitting time, the time required to drop entries from matrices (as the PETSc interfaces require us to take copies +15 + +of matrices), and to extract the submatrices Aff, Afc, etc. These should all be relatively low cost parts of the setup, are +shared by both AIRG and lAIR and should scale with the nnzs. The setup time we compare is therefore that required to +form the restrictors, prolongators and coarse matrices. Also we should note that with AIRG our setup time is an upper +bound, as we have not built an optimised matmatmult for building our fixed sparsity GMRES polynomials (we know +the sparsity of the matrix powers is the same as the two input matrices). As such we compute a standard matmatmult +in PETSc and then drop entries (although we do provide a flop count for a matmatmult with fixed sparsity). When +timing lAIR, we use the PETSc interface to hypre and hence we run two solves (and set the initial condition to zero in +both). The hypre setup occurs on the 0th iteration of the first solve, so a second solve allows us to correctly measure +just the solve time. We only show solve times for problems with pure streaming, as our implementation of the P0 +matrix-free matvec with scattering is not well optimised. +All tabulations of memory used are scaled by the total NDOFs in ψ in (8), i.e., NDOFs=NCDOFs + NDDOFs. +Included in this figure is the memory required to store the GMRES space, both additive preconditioners (if required) +and hence the AIRG hierarchy, ˆA +−1 +ff , separate copies of Aff, Afc, Acf and Acc, the diffusion operator and temporary +storage. We do not report the memory use of hypre, although given the operator complexities it is similar to AIRG. +7.1. AIRG on the matrix A − B ˆD +−1C +In this section, we build the matrix A − B ˆD +−1C and use this matrix as a preconditioner, applied with 1 V-cycle of +either AIRG or lAIR. As such we do not use the iterative method described in Section 3, nor do we use the matrix-free +matvec described in Section 2.2. As mentioned using an iterative method that relies on the full operator is not practical +with scattering given the nonlinear increase in nnzs with angular refinement, but we wish to show our methods are +still convergent in the scattering (diffuse) limit. Instead, Section 7.2 shows the results from using the iterative method +in Section 3. +Our test problem is a 3× 3 box with a source of strength 1 and size 0.2 × 0.2 in the centre of the domain. We apply +vacuum conditions on the boundaries and discretise this problem with unstructured triangles and ensure that our grids +are not semi-structured (e.g., we don’t refine coarse grids by splitting elements). We use uniform level 1 refinement +in angle, with 1 angle per octant (similar to S2). +7.1.1. Pure streaming problem +For the pure streaming problem we set the total and scatter cross-sections to zero. To begin, we examine the +performance of AIRG and lAIR with spatial refinement and a fixed uniform level 1 angular discretisation (i.e, with +4 angles in 2D). Table 1 shows that using distance 1 lAIR in this problem, with Falgout-CLJP CF splitting results in +growth in both the iteration count and work. We could not find a combination of parameters that results in scalability +with lAIR; increasing the number of FCF smooths to 3 results in an iteration count with similar growth, namely 15, +14, 14, 17, 19 and 22, but with a cycle complexity at the finest spatial refinement of 11.2 and hence 271 WUs. Even +using distance 2 lAIR didn’t result in scalability, as shown in Table 2 where we can see a slightly decreased iteration +count, with higher cycle and operator complexities, resulting in a similar number of WUs. Using both distance 2 lAIR +and 3 FCF smooths still results in growth, with iteration counts of 15, 13, 14, 16, 17 and 19, with a cycle complexity +of 15.9 and hence 323 WUs at the finest spatial refinement. +CG nodes NDOFs +nits +CC Op Complx WUsfull WUsDG Memory +97 +2.4× 103 26 2.96 +1.5 +106 +26.3 +- +591 +1.6× 104 25 +3.5 +1.77 +116 +27.8 +- +2313 +6.3× 104 28 +3.8 +1.9 +138 +32.6 +- +9166 +2.5× 105 31 +4.1 +2.0 +159 +37.4 +- +35784 +9.9× 105 36 +4.2 +2.1 +189 +44.5 +- +150063 +4.2× 106 42 +4.3 +2.16 +225 +52.6 +- +Table 1: Results from using distance 1 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of +Falgout-CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025 and a strong R threshold of 0.25. +16 + +CG nodes NDOFs +nits CC Op Complx WUsfull WUsDG Memory +97 +2.4× 103 26 3.2 +1.58 +112 +27.9 +- +591 +1.6× 104 24 3.7 +1.85 +116 +28 +- +2313 +6.3× 104 27 4.1 +2.04 +141 +33.5 +- +9166 +2.5× 105 30 4.4 +2.16 +164 +38.6 +- +35784 +9.9× 105 34 4.6 +2.26 +192 +44.9 +- +150063 +4.2× 106 38 4.7 +2.32 +219 +51.1 +- +Table 2: Results from using distance 2 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of +Falgout-CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025 and a strong R threshold 0.25. +We also tried decreasing the strong R threshold to 1 × 10-7 in case some neighbours were being excluded, but this +resulted in very little change in the iteration count, while the number of nnzs in the restrictor (and hence the setup +time) grew considerably. Preliminary investigation suggests we would need to include neighbours at greater distance +than two (along with only F-point smoothing, rather than FCF), but this increases the setup cost considerably. We also +note that using nAIR (at several different orders) in this case results in a similar iteration count (even with diagonal +scaling of our operators); this is likely due to the lack of lower triangular structure in our discretisation. +CG nodes NDOFs +nits +CC Op. Complx WUsfull WUsDG Memory +97 +2.4× 103 11 5.58 +1.96 +83 +20.4 +11.9 +591 +1.6× 104 10 5.85 +2.48 +79 +18.9 +11.7 +2313 +6.3× 104 +8 +6.4 +2.88 +70 +16.6 +12.2 +9166 +2.5× 105 +8 +6.7 +3.16 +73 +17.1 +12.4 +35784 +9.9× 105 +9 +6.9 +3.36 +82 +19.3 +12.5 +150063 +4.2× 106 +9 +7.07 +3.48 +84 +19.6 +12.7 +Table 3: Results from using AIRG with m = 4 and without fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre +implementation of Falgout-CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025. +Tables 3 & 4 however shows that using AIRG with a third-order GMRES polynomial and Falgout-CLJP CF +splitting results in less work with smaller growth. Using GMRES polynomials without fixed sparsity requires 84 WUs +at the highest level of refinement, compared to 75 with fixed sparsity. The work in Table 3 has plateaued, however +the work with fixed sparsity is growing slightly with spatial refinement. Compared to lAIR, we see that AIRG with +sparsity control needs three times less work to solve our pure streaming problem. Rather than use our one-point +ideal prolongator, we also investigated using W with the same drop tolerances as applied to Z. With fixed-sparsity +this resulted in 11, 10, 9, 9, 9 and 10 iterations, with 67, 68, 66, 68, 70, and 78 WUs. The slight increase in cycle +complexity is compensated by using one fewer iterations at the finest level, but overall we see similar work. In an +attempt to ascertain the effect of using our one-point approximation to the ideal prolongator, we also constructed a +classical one-point prolongator as in [28, 26] where each F-point is injected from its strongest C-point neighbour. +With fixed-sparsity this results in the same number of iterations and WUs as in Table 4, which confirms that the +classical prolongator is not responsible for the poor performance of lAIR. The benefit of using a classical one-point +prolongator in this fashion is we require one fewer matmatmults on each level of our setup. For the rest of this paper +we use the one-point approximation to the ideal operator, but it is clear there is a range of practical operators (with +different sparsification strategies) we could use with AIRG. +Table 4 shows we can solve our streaming problem with the equivalent of approximately 18 DG matvecs. We +can also see that the plateau in the operator complexity results in almost constant memory use, at 10 copies of the +angular flux. We should also note that we can use AIRG as a solver, rather than as a preconditioner. We see the +same iteration count in pure streaming problems and the lack of the GMRES space means we only need memory +equivalent to approx. 5 copies of the angular flux; this is the amount of memory that would be required to store just +a DG streaming operator in 2D. This shows that our discretisation and iterative method are low-memory in streaming +17 + +problems. +CG nodes NDOFs +nits +CC Op. Complx WUsfull WUsDG Memory +97 +2.4× 103 12 +3.7 +1.97 +67 +16.5 +10 +591 +1.6× 104 10 +4.1 +2.5 +62 +14.7 +10 +2313 +6.3× 104 +9 +4.4 +2.87 +60 +14.1 +10.2 +9166 +2.5× 105 10 +4.6 +3.17 +67 +15.8 +10.3 +35784 +9.9× 105 10 4.74 +3.36 +69 +16 +10.4 +150063 +4.2× 106 11 4.84 +3.5 +75 +17.6 +10.4 +Table 4: Results from using AIRG with m = 4 and fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre implementation +of Falgout-CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025. +The cost of a solve must also be balanced by the cost of the setup of our multigrid. Our setup involves computing +our GMRES polynomial approximations, followed by standard AMG operations, namely computing matrix-matrix +products to form our restrictors/prolongators and coarse matrices on each level. Fig. 3 begins by showing the relative +amount of work required to compute Z for AIRG and distance 1 lAIR. For AIRG this is the sum of FLOPs required +to compute ˆA +−1 +ff and those to compute −Acf ˆA +−1 +ff . For lAIR, it is more difficult to calculate a FLOP count for the small +dense solves which locally enforce RA = 0, as it is dependent on the BLAS implementation; instead we take the +size of each of the n × n dense systems and compute the sum of n3. As such we do not try and compare an absolute +measure of the work required by both (we have timing results below). Fig. 3 instead shows that the growth in work for +both AIRG with fixed sparsity and lAIR begins to plateau with grid refinement; AIRG without fixed sparsity however +grows with refinement, showing the necessity of the sparsity control on the matrix powers. +Fig. 4a shows that the cost of computing our GMRES polynomial approximations to ˆA +−1 +ff with fixed sparsity is +constant, at around 8 FLOPs per DOF. The plateauing growth seen in Fig. 3 therefore comes from the matmatmult +required to compute −Acf ˆA +−1 +ff . The FLOP count with fixed sparsity relies on a custom matmatmult algorithm that +assumes the same sparsity in each component of the matmatmult, as in (29). Computing the matrix powers with a +standard matmatmult and then dropping any fill-in results in an almost constant number of FLOPs per DOF, at around +9.6, which is convenient as this makes it less necessary to build a custom matmatmult implementation. If we do not +fix the sparsity of our polynomial approximations we can see growth in the cost, due to the increasingly dense matrix +powers, as might be expected. +Fig. 4b shows that the cost of our setup with fixed sparsity plateaus, and the total cost of our setup plus solve grows, +due to the increase in work during the solve shown in Table 4, with 29% growth from lowest to highest refinement. +Without fixed sparsity we see less growth in Table 3 from the solve, but higher growth in the setup. This results in +similar growth in the total work, but the fixed sparsity case uses approximately 30% fewer FLOPs. +Fig. 5 shows results for the solve, Z, setup and total time taken to solve our system with AIRG and lAIR. We can +see in Fig. 5a that the results match those in Tables 1-5, with AIRG with Falgout-CLJP coarsening and fixed sparsity +giving the lowest solve times. We found a difference in the efficiency of our PETSc implementation vs the hypre +implementations however. If we modify the parameters used with AIRG and increase the work required to solve to +roughly match that distance 1 lAIR in this problem, we found that there was a factor of almost 2 difference in the +solve times. +Fig. 5b shows the total cost of computing the approximate ideal restrictor, Z. For AIRG we also show the time +for computing the GMRES polynomial approximation to A−1 +ff separately. We can see for AIRG with fixed sparsity +there is slight growth in the time to compute our polynomials, with much greater growth without fixed sparsity. Given +the FLOP count in Fig. 4a we would expect the time with fixed sparsity to be constant. The further growth we see +in the time to compute Z comes from the matmatmult to compute −AcfA−1 +ff , which Fig. 4b suggests should plateau. +We also see growth in the time taken to compute Z with lAIR, with higher growth for distance 2. Again Fig. 3 shows +the work estimate plateauing with spatial refinement. At the higher spatial refinements, the Z with AIRG, Falgout- +CLJP coarsening and fixed sparsity costs more to compute than distance 1 lAIR, but less than distance 2. An AIRG +implementation that takes advantage of the shared sparsity of the matrices when forming ˆA +−1 +ff could reduce this setup +further, with a reduction in the FLOPs required (by approx. 20% as shown in Fig. 4a), and also in the cost of any +18 + +102 +103 +104 +105 +1 +1.2 +1.4 +1.6 +1.8 +2 +CG Nodes +Relative work +(a) The × is for AIRG with fixed sparsity, × is for AIRG without +fixed sparsity and ⊗ is for lAIR. +Figure 3: Sum of FLOPs required across all levels to compute Z, scaled to the NDOFs, and then relative to the work required on the least refined +spatial grid, for AIRG with m = 4 (see Table 4) and distance 1 lAIR (see Table 1) with Falgout-CLJP in a 2D pure streaming problem. The relative +scaling is done separately for each line; they do not all cost the same at the coarsest resolution, we provide timings in Section 7.1.1. +symbolic computation. Given the substantially improved convergence shown in Table 4 when compared to distance 2 +lAIR in Table 2, this shows AIRG is an effective and relatively cheap way to compute approximate ideal operators in +convection problems. +Fig. 5c shows the setup times (which include the times to compute Z from Fig. 5b) and we can see that lAIR has +the cheapest overall setup, with fixed sparsity AIRG coming in the middle, and AIRG without fixed sparsity giving the +highest setup times. We expect AIRG without fixed sparsity to be increasing given the increased cost of computing +the GMRES polynomial with spatial refinement, as shown in Fig. 4a, but we see increased setup time with AIRG with +fixed sparsity and distance 1 lAIR, whereas the work estimates in Figures 3 and 4b again suggests the setup times +should plateau. We found that the time taken to compute the matmatmults used to build the transfer operators and +coarse matrices is increasing more than the FLOP count would imply. Of course a FLOP count is not necessarily +perfectly indicative of the time required in a matmatmult given the symbolic compute and memory accesses. It is +possible more efficient matmatmult implementations are available. +Fig. 5d shows the total time taken to setup and solve with our multigrid. Two of our AIRG results, namely +using GMRES polynomials without and without fixed sparsity manage to beat lAIR. This helps show the promise of +GMRES polynomials as part of an AIR-style multigrid; even without fixed sparsity they can be competitive. The fixed +sparsity AIRG however results in a considerably decrease in total time, taking approx. 3× less time than distance 1 +lAIR. Even if we try and remove the effect of implementation differences between lAIR and AIRG discussed above, +by equating the solve time of lAIR with an AIRG result that requires similar WUs, fixed sparsity AIRG still has a +total time of 0.65 × that of distance 1 lAIR. +We also examined using SAIs to build an approximation of A−1 +ff with the fixed sparsity of Aff, like in [46]. We +found approximations of A−1 +ff computed with SAI comparable to our fixed sparsity GMRES polynomials with m = 4, +with identical convergence behavior compared to the fixed sparsity AIRG shown in Table 4 (except with the first +spatial refinement). With spatial refinement using SAI had 11, 10, 9, 10, 10 and 11 iterations and hence the same +work units and solve times as AIRG given the matching fixed sparsity. However we found it was more expensive to +setup SAIs on each level when compared to our GMRES polynomials, by approximately 2×; we used the ParaSails +implementation in hypre for comparisons. In particular the fixed sparsity GMRES polynomials require fewer FLOPs +as the nnzs in Aff grow compared to fixed sparsity SAI. If nF is the number of F-points, we require m matvecs which +scale linearly with the nnzs and a single QR factorisation of size nF × (m + 1), which doesn’t depend on the nnzs. +If m > 2, we must also compute m − 2 fixed sparsity matrix powers at cost (m − 2)srscnF, where sr and sc are the +average number of nnzs per row and column of Aff, respectively. If we assume s = sr ≈ sc then the cost of computing +19 + +102 +103 +104 +105 +8 +10 +12 +14 +CG Nodes +FLOPs per DOF +(a) The × is the cost of computing ˆA−1 +ff for AIRG with fixed spar- +sity, the × is with fixed sparsity computed with a standard mat- +matmult followed by dropping entries and the × is without fixed +sparsity +102 +103 +104 +105 +50 +100 +150 +CG Nodes +FLOPs per DOF +(b) The dashed × is the cost of the setup for AIRG with fixed +sparsity, the × is without fixed sparsity, the × is the setup plus +solve with fixed sparsity, the × is without fixed sparsity. +Figure 4: Sum of FLOPs required across all levels during the setup and solve, scaled to the NDOFs, for AIRG with m = 4 and Falgout-CLJP in a +2D pure streaming problem (see Tables 4 and 3). +the matrix-powers can be written as s(m − 2) × nnzs(Aff). In comparison, the SAI algorithm with the fixed sparsity +of Aff requires solving nF (local) least-squares problems. If we just consider the required nF QR factorisations of +size s × s, this requires a FLOP count of s2 × nnzs(Aff) (we have dropped the constants). For many problems a low +polynomial order is acceptable and hence (m − 2) ≪ s; this is particularly true on lower multigrid levels where the +average row or column sparsity can grow. There may be a scale, however, at which the extra local cost of setting up +SAIs is balanced by the extra communication required by our fixed sparsity GMRES polynomials. With m = 4 we +require the communication associated with four matvecs, a single all-reduce and computing A2 +ff, whereas SAI only +requires that of A2 +ff. One of the benefits of using our GMRES polynomials however is that we can decrease the amount +of communication required by decreasing the polynomial order below 2. +Given this, we examine the role of changing the GMRES polynomial order with AIRG and fixed sparsity, from +zero to four (m = 1 to m = 5; the zeroth and first order polynomials implicitly have fixed sparsity) in Fig. 6. We +can see the zeroth order polynomial is very cheap to construct, but results in the highest total time. This is because +the iteration count is higher. The increasingly higher order polynomials take longer to setup, although the difference +between successive orders decreases, due to the shared sparsity. With increasing polynomial order, on the most refined +spatial grid we have 40, 16, 12, 11 and 11 iterations to solve, with cycle complexities of 3.25, 4.67, 4.81, 4.84 and +4.85, respectively. We can see that all the polynomials between first and third order result in similar total times; the +decreased cost of setup for the lower orders is balanced by the increase in iterations. We see however very similar +growth in iteration count with spatial refinement with the first through fourth order polynomials; for example with +first order GMRES polynomials (m = 2) in Fig. 6 we have 15, 14, 12, 14, 14 and 16 iterations. This gives a higher +overall amount of work than with m = 4 (up to approximately 100 WUs at the highest refinement), but given the +similar growth in iteration count and reduced communication in parallel during the setup, requiring only two matvecs +and a single all-reduce, this may be a good choice in parallel. +Given the results with spatial refinement above, we chose to examine the role of angular refinement with AIRG +with fixed sparsity and m = 4. Table 5 shows that with 3 levels of angular refinement on the third refined spatial +grid, the iteration count increases slightly from 9 to 11, but the cycle and operator complexity are almost identical and +hence we have fixed memory consumption. We do not show the timings as they scale as would be expected. Table 6 +shows that distance 1 lAIR requires a fixed amount of work with angular refinement, but this is roughly twice that of +AIRG. +20 + +CG nodes Angle lvl. NDOFs +nits CC Op. Complx WUsfull WUsDG Memory +2313 +1 +6.3× 104 +9 +4.4 +2.87 +60 +14.1 +10.2 +2313 +2 +2.5× 105 10 4.4 +2.88 +65 +15.4 +10.4 +2313 +3 +1× 106 +11 4.4 +2.87 +70 +16.7 +10.4 +Table 5: Results from using AIRG with m = 4 and fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre implementation +of Falgout-CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025 with different levels of angular refinement. +CG nodes Angle lvl. NDOFs +nits CC Op Complx WUsfull WUsDG Memory +2313 +1 +6.3× 104 28 3.8 +1.9 +138 +32.6 +- +2313 +2 +2.5× 105 27 3.7 +2.36 +133 +31.7 +- +2313 +3 +1× 106 +28 3.8 +2.35 +133 +31.7 +- +Table 6: Results from using distance 1 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout- +CLJP with a strong threshold of 0.2, drop tolerance on A of 0.0075 and R of 0.025 and a strong R threshold of 0.25 with different levels of angular +refinement. +7.1.2. Scattering problem +To test the performance of AIRG with diffusion, we set the total and scattering cross-section to 10.0. Tables 7 and +8 show that both distance 1 and distance 2 lAIR, respectively, perform similarly, with approximately 3× growth in +WUs from the least to most refined spatial grid. Both the cycle and operator complexities have plateaued though. +CG nodes NDOFs +nits CC Op Complx WUsfull WUsDG Memory +97 +2.4× 103 23 2.2 +1.3 +77 +49 +- +591 +1.6× 104 27 3.0 +1.9 +112 +69 +- +2313 +6.3× 104 30 3.3 +2.2 +134 +82 +- +9166 +2.5× 105 34 3.4 +2.2 +153 +93 +- +35784 +9.9× 105 41 3.4 +2.2 +183 +110 +- +150063 +4.2× 106 54 3.3 +2.2 +238 +144 +- +Table 7: Results from using distance 1 lAIR in hypre on a pure scattering problem in 2D with CF splitting by the hypre implementation of +Falgout-CLJP with a strong threshold of 0.9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.4. +AIRG performs simliarly to lAIR in this problem, as shown in Tables 9 without fixed sparsity and 10 with fixed +sparsity, with around 3× growth and similar number of WUs. AIRG with fixed sparsity results in slightly lower +operator complexities, but both methods result in memory use of around 20 copies of the angular flux; this is higher +than that in the streaming limit as the full matrix with scattering and a uniform angular discretisation at one level of +refinement has 4× the nnzs as that in the streaming limit. There is not a great deal of difference in the cycle complexity +between AIRG with and without fixed sparsity; this is because the strong R threshold of 0.4 results in a very sparse +version of Aff being used to construct the GMRES polynomials (and hence very little fill-in relative to the top grid +matrix). We can decrease the strong R tolerance to decrease the iteration count (in both AIRG and lAIR), but the +nnzs in (or equivalently the number of neighbours used to construct) Z grows considerably, as might be expected with +scattering. +Fig. 7 shows the timing results from AIRG and lAIR in this problem. Again we see a curious implementation +difference in solve times in Fig. 7a, as both lAIR and AIRG require simliar number of WUs, but the hypre imple- +mentation of lAIR requires roughly twice the time to solve. Fig. 7b shows that the time to compute the GMRES +polynomial for AIRG is largely constant, with the time to compute Z again between distance 1 and distance 2 lAIR; +this is also true for the total setup time in Fig. 7c. Fig. 7d shows that our AIRG implementation is the cheapest method +overall, taking around 0.4× the amount of time as lAIR to solve this problem. If we again equate the solve time be- +tween lAIR and AIRG given the similar amount of work required, lAIR and AIRG perform similarly, each requiring +21 + +CG nodes NDOFs +nits CC Op Complx WUsfull WUsDG Memory +97 +2.4× 103 22 2.5 +1.5 +80 +51 +- +591 +1.6× 104 25 3.5 +2.2 +117 +72 +- +2313 +6.3× 104 27 3.8 +2.4 +133 +81 +- +9166 +2.5× 105 31 3.7 +2.4 +150 +91 +- +35784 +9.9× 105 37 3.7 +2.5 +178 +108 +- +150063 +4.2× 106 47 3.7 +2.5 +225 +136 +- +Table 8: Results from using distance 2 lAIR in hypre on a pure scattering problem in 2D with CF splitting by the hypre implementation of +Falgout-CLJP with a strong threshold of 0.9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.4. +CG nodes NDOFs +nits CC Op. Complx WUsfull WUsDG Memory +97 +2.4× 103 22 1.9 +1.3 +68 +43 +17.3 +591 +1.6× 104 26 2.2 +2.1 +88 +54 +18.0 +2313 +6.3× 104 34 2.5 +2.4 +122 +74 +18.7 +9166 +2.5× 105 41 2.7 +2.7 +155 +94 +19.5 +35784 +9.9× 105 45 2.8 +2.8 +175 +106 +19.9 +150063 +4.2× 106 61 2.7 +2.9 +232 +140 +19.5 +Table 9: Results from using AIRG with m = 4 and without fixed sparsity on a pure scattering problem in 2D with CF splitting by the hypre +implementation of Falgout-CLJP with a strong threshold of 0.9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.4. +approximately 2.9 µs total time per DOF. +Fig. 8 shows the results from changing the GMRES polynomial order and similar trends to that in the streaming +limit can be seen, namely the 0th order polynomial is very cheap to setup, but results in the highest total time. Indeed +the 0th order polynomial did not converge at the two highest spatial refinements. The first through fourth order +polynomials all result in similar total times, with 62, 61, 60 and 64 iterations, respectively. This result indicates that +the higher polynomial order does not necessarily help decrease the iteration count with scattering. This is because the +strong R threshold is so high; decreasing this makes the effect of the polynomial order (and the fixed sparsity) much +more pronounced, but we found the lowest overall total times by allowing heavy dropping. Given using the full matrix +is not scalable with angular refinement, we don’t show convergence results in that case; instead the next section uses +the iterative method defined in Section 3 on this scattering problem. +7.2. Additively preconditioned iterative method +In this section we show the performance of the iterative method from Section 3 in the scattering limit. The results +in this section are not designed to show the performance of a standard DSA method with different scattering ratios, +optical cell lengths, etc; we appeal to the wealth of literature on the topic. Instead we wish to show that the additive +combination of our preconditioners is effective and that multigrid methods can be used to invert these operators +scalably. As discussed, this means forming the streaming/removal operator MΩ, a CG diffusion operator Ddiff and the +streaming/removal components BΩ and/or CΩ, all of which can be done scalably. We use 1 V-cycle of AIRG with the +same drop/strong tolerances as in Section 7.1.1 to apply M−1 +Ω and 1 V-cycle of boomerAMG (with default options) to +apply D−1 +diff per outer GMRES iteration. +For our iterative method to be effective, 1 V-cycle of both methods must reduce the error by a fixed amount with +space/angle refinement (which is equivalent to a solve with a fixed tolerance taking a fixed amount of work). We +can assume that multigrid methods such as boomerAMG can invert the diffusion operator with fixed work (we also +scale the diffusion operator by its inverse diagonal prior to use), but in Section 7.1.1 we only showed that AIRG can +invert the streaming operator with fixed work in the solve, rather than the streaming/removal operator. Thankfully the +removal term results in a better conditioned matrix given extra term on the (block) diagonals; the streaming limit is +the most difficult to solve. Fig. 9 shows (part of) the spectrum of the streaming operator vs the streaming/removal +22 + +CG nodes NDOFs +nits CC Op. Complx WUsfull WUsDG Memory +97 +2.4× 103 22 1.9 +1.3 +67 +43 +17.3 +591 +1.6× 104 26 2.2 +2.0 +88 +54 +18.0 +2313 +6.3× 104 34 2.5 +2.4 +122 +74 +18.7 +9166 +2.5× 105 38 2.7 +2.7 +144 +87 +19.4 +35784 +9.9× 105 51 2.8 +2.8 +196 +118 +19.7 +150063 +4.2× 106 60 2.7 +2.8 +224 +135 +19.3 +Table 10: Results from using AIRG with m = 4 and fixed sparsity on a pure scattering problem in 2D with CF splitting by the hypre implementation +of Falgout-CLJP with a strong threshold of 0.9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.4. +operator for the 2D source problem with the third refined grid, level one angular refinement and total cross-section of +10.0. We can see that the smallest eigenvalues of the streaming/removal operator are (slightly) further from the origin. +The convergence of AIRG relies on the convergence of our GMRES polynomial approximations to A−1 +ff . We can +also see in Fig. 9 that the eigenvalues of Aff are more compact than that of the full operators, confirming that the CF +splitting is helping produce a better conditioned Aff in both cases. We know that our operators are non-normal, so the +spectrum does not completely determine the convergence of our GMRES polynomials [63]. Given this, Fig. 9 also +plots the field of values (a.k.a., the numerical range) of our operators, given by +F (A) = {x∗Ax | xx∗ = 1, x ∈ Cn}, +(43) +which is a convex set that contains the eigenvalues. To give some insight into the convergence of the GMRES +polynomials, we define µ to be the distance from the origin, or +µ = min +z∈F (A) |z|. +(44) +[80] show that (see also [63]) if the field of values doesn’t contain the origin, β ∈ (0, π/2) such that cos(β) = µ/||A|| +and the Hermitian part of A, namely (A + A∗)/2 is positive definite then the residual at step m is bounded by +||rm|| ≤ ||r0|| +� +2 + 2√ +3 +� +(2 + γβ)γm +β , +(45) +where +γβ = 2 sin +� +β +4 − 2β/π +� +. +(46) +We confirmed numerically that none of our operators or their fine-fine sub-matrices touch the origin (and hence the +field of values are all in the right-half of the complex plane) and that their Hermitian parts are positive definite. +For the Aff component of the streaming operator on the top grid, pictured in Fig. 9 we found that with m = 4, (45) +gives ||rm|| ≤ 9.41||r0||, while for the Aff component of the streaming/removal operator we have ||rm|| ≤ 9.40||r0|| (the +disk bound in [81] gives a similar conclusion). These bounds are not particularly tight, but they do indicate that a 3rd +order GMRES polynomial should result in a smaller residual for the streaming/removal operator and hence we would +expect AIRG to perform better. +We should note that as µ → 0, β gets closer to π/2 and the asymptotic convergence factor, γm +β → 1. In general this +means the further the minimum field of values is from the origin, the better the convergence; this is also demonstrated +by considering the disk bound in [81], given by |δ/c| = (1−cos(β))/(1+cos(β)) < γβ, where δ and c are the radius and +centre of a disk, respectively, that covers F (A). This helps explain why using GMRES polynomials to approximate +Aff can be effective even when GMRES polynomial preconditioning of the full operators may not be; Fig. 9a shows +that the field of values for the streaming operator almost touches the origin, with µ ≈ 3.2 × 10-5 and hence single- +level GMRES polynomial preconditioning would likely not be effective in this problem (this is backed by numerical +experiments; we find considerable growth in the iteration count with refinement). This hints at the importance of +combining GMRES polynomials with a reduction multigrid. +23 + +CG nodes NDOFs +nits CC Op. Complx WUsmf WUsDG Memory +97 +2.4× 103 23 3.0 +1.6 +33 +61 +16.5 +591 +1.6× 104 24 4.0 +1.0 +36 +67 +17.2 +2313 +6.3× 104 25 4.1 +1.7 +38 +69 +17.2 +9166 +2.5× 105 26 4.2 +2.4 +39 +72 +17.2 +35784 +9.9× 105 26 4.4 +2.8 +39 +73 +17.3 +150063 +4.2× 106 26 4.6 +3.2 +39 +73 +17.5 +Table 11: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 10.0 in 2D. The cycle +and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP. +CG nodes Angle lvl. NDOFs +nits CC Op. Complx WUsmf WUsDG Memory +2313 +1 +6.3× 104 25 4.1 +1.7 +38 +69 +17.2 +2313 +2 +2.5× 105 27 4.0 +1.4 +40 +71 +16.5 +2313 +3 +1× 106 +28 4.1 +1.4 +41 +73 +16.2 +Table 12: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 10.0 in 2D with angle +refinement. The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP. +We see in Table 11 our additively preconditioned iterative method is effective with a total and scatter cross-section +of 10, with the iteration count growing from 23 to a plateau of 26 with spatial refinement. The work is very close +to constant, with fixed iteration count and slight growth in the cycle complexity, even though we used AIRG with +fixed sparsity. As such we didn’t investigate using AIRG without fixed sparsity, as the fixed sparsity was sufficient +to give plateauing work. This helps confirm the observations above, namely that AIRG is more effective on the +streaming/removal operator. +Comparing to the results in Section 7.1.2, it uses roughly 73 DG WUs, compared to around 135 when using either +AIRG or lAIR as a preconditioner on the full matrix. It also uses less memory at approximately 18 copies of the +angular flux, even though we have to store the diffusion operator and several extra temporary vectors. Compared to +the pure streaming problem, this method requires approximately 4.1× more work; we can see in Table 11 that this +work largely comes from computing the matrix-free matvec with scattering, with 26 iterations at the highest spatial +refinement requiring 39 WUsmf. The split of work is 26 WUsmf in the matvec required by the outer GMRES, 11 +WUsmf to apply the additive preconditioners and around 2 WUsmf to compute the source and Θ. Similarly, Table 13 +shows a (lower) constant iteration count with a lower total and scatter cross-section of 1.0. +Given the streaming/removal operator is easier to solve, lAIR performs better when used additively to invert MΩ, +compared with just the streaming operator. With a total and scatter cross-section of 10.0, we find both distance 1 and +2 lAIR give 30, 33, 31, 34, 36 and 39 iterations with spatial refinement, with similar grid and operator complexities to +AIRG. Increasing the number of FCF smooths from 1 to 3 results in an iteration count with less growth, namely 30, +25, 25, 26, 26 and 27 iterations, but the cycle complexity at the finest level of refinement is large at 10.7, compared +with AIRG at 4.6. We also know from Section 7.1.1 that the iteration count of lAIR grows in the streaming limit, so +we do not test lAIR any further as part of our additive method. +Tables 12 and 14 show that the additive method with AIRG and angular refinement perform well, as the iteration +count and the work required is very close to constant. Importantly we can also see the memory use is fixed. These +results show that as might be expected, using an (inconsistent) CG DSA can form an effective preconditioner in +scattering problems when used with an outer GMRES iteration. Importantly the combination of a single V-cycle of +AIRG used to apply the streaming/removal operator and a single V-cycle of a traditional multigrid on the diffusion +operator can be used additively and results in almost constant work with spatial and angular refinement on unstructured +grids. +24 + +CG nodes NDOFs +nits CC Op. Complx WUsmf WUsDG Memory +97 +2.4× 103 18 3.6 +1.9 +27 +50 +17.1 +591 +1.6× 104 18 4.0 +2.4 +27 +50 +17.2 +2313 +6.3× 104 18 4.3 +2.8 +28 +51 +17.4 +9166 +2.5× 105 19 4.5 +3.1 +29 +54 +17.5 +35784 +9.9× 105 19 4.7 +3.3 +30 +55 +17.6 +150063 +4.2× 106 19 4.8 +3.5 +30 +55 +17.7 +Table 13: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 1.0 in 2D. The cycle +and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP. +CG nodes Angle lvl. NDOFs +nits CC Op. Complx WUsmf WUsDG Memory +2313 +1 +6.3× 104 18 4.3 +2.8 +28 +51 +17.4 +2313 +2 +2.5× 105 18 4.3 +2.8 +27 +49 +16.7 +2313 +3 +1× 106 +19 4.3 +2.8 +29 +51 +16.5 +Table 14: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 1.0 in 2D with angle +refinement. The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP. +8. Conclusions +This paper presented a new reduction multigrid based on approximate ideal restrictors (AIR) combined with +GMRES polynomials (AIRG) with excellent performance in advection-type problems. Matrix polynomial methods +have been used for many years in multilevel methods but we believe we are the first to use GMRES polynomials in +this fashion. Reduction multigrids and LDU methods in particular benefit from using GMRES polynomials, as the +improved conditioning of Aff, when compared to A, can allow the formation of good approximate inverses with low +polynomial orders. This allowed us to easily build both approximate ideal restrictors, approximate ideal prolongators +(without the need to compute near-nullspace vectors) and perform F-point smoothing (without the need to compute +additional dampening parameters). +GMRES polynomials share many advantages with other polynomial methods; in particular their coefficients can +be computed very simply; low-order polynomials don’t require additional work to ensure stability (like in [61, 62]); +explicitly forming approximate matrix inverses is simple and only involves matrix-matrix products or if desired; the +polynomials can be applied matrix-free; their application is highly parallel with their setup able to use communication- +avoiding techniques; and they also work well across a range of symmetric and asymmetric problems. +When applied to the time independent Boltzmann Transport Equation (BTE) we could solve pure streaming prob- +lems (i.e., in the pure advection limit) on unstructured spatial grids with space/angle refinement with fixed memory +use. The time-independent streaming limit is the most challenging to solve and we found we could either get fixed +work in the solve and growth in the setup, or by introducing fixed sparsity into the matrix-powers of our GMRES +polynomials, we found fixed work in the setup with growth in the solve. We found good performance from using +between first to fourth order GMRES polynomials on each level of our multigrid. Fixing the sparsity of our third- +order (m = 4) GMRES polynomials resulted in a fixed FLOP count in the setup, and building an implementation of a +matmatmult A = BC where the three matrices share the same sparsity would reduce the implementation costs of our +setup for second order polynomials and higher. With fixed sparsity we found at most 20% growth in the work to solve +with either 6 levels of spatial refinement or three levels of uniform angular refinement. +A balance must be struck between the scalability of the solve vs expense of the setup, but we believe this is the +first method to show scalable solves with a stable spatial discretisation that doesn’t feature lower-triangular structure +in the streaming limit of the BTE. We did not spend much effort tweaking parameters and we have found that that we +can get better performance in these problems with standard AMG tweaks such as level specific drop tolerances. +We also compared AIRG to two different reduction multigrids and found performance advantages; one where +sparse approximation inverses (SAIs) are used to approximate A−1 +ff , and the lAIR implementation in hypre. We used +ParaSails in hypre to form SAIs with the fixed sparsity of Aff and found almost identical convergence behavior +25 + +to fixed sparsity AIRG (and hence the same solve time) in the streaming limit. We found however that the setup +of the SAIs took twice as long as our GMRES polynomials with m = 4. For lAIR, we could not find a set of +parameters that resulted in fixed work in the solve. Our further investigations suggest the combination of distance 3 +or 4 lAIR plus (only) F-point smooths are required to get scalable results with lAIR, but this is not practical given the +setup/communication costs. +In comparison to distance 1 or 2 lAIR, AIRG took roughly two to three times less work to solve. Timing the setup +showed that computing Z with our third-order GMRES polynomial approximations cost between that of computing +Z with distance 1 and 2 lAIR. The total time of our setup at the highest level of spatial refinement matched that of +distance 2 lAIR. The total time (setup plus solve) for AIRG was roughly 3× less than lAIR, though implementation +differences make this comparison difficult. We then investigated using AIRG and lAIR on the full matrix formed +with scattering. Forming this matrix cannot be done scalably with angular refinement, but we showed that AIRG is +applicable in the diffuse limit, performing about as well as lAIR. +We then built an iterative method that used the additive combination of two preconditioners applied to the angular +flux; 1 V-cycle of AIRG was used to invert the streaming/removal operator and 1 V-cycle of boomerAMG was used +to invert a CG diffusion operator. The streaming/removal operator is easier to solve than the streaming operator +and hence we found the work in the solve plateaued with fixed sparsity AIRG. Using distance 1 or 2 lAIR to invert +the streaming/removal operator resulted in cycle complexities over twice that of fixed sparsity AIRG. Given the +performance shown here it would be worth investigating the use of AIRG as part of a standard DG FEM source +iteration; preliminary work reveals AIRG performs similarly when used with DG streaming or streaming/removal +operators. +The only remaining consideration is how we can apply this method with our previously developed angular adap- +tivity and the parallel performance, which we will investigate in future work. AIRG should be performant in parallel, +as the entire multigrid hierarchy can be applied with only matrix-vector products (i.e., no reductions). The CF split- +ting algorithm used has a parallel implementation available in hypre. The GMRES polynomial coefficients on each +level must be computed once during the setup and can be trivially stored for multiple solves. Furthermore given our +use of low-order GMRES polynomials in AIRG, we found a single step method based on a QR factorisation of the +Krylov basis could be used to generate these coefficients stably. In parallel we could therefore use a tall-skinny QR +and generate the coefficients of a polynomial of order m − 1 with m matvecs and a single all-reduce on each level. +For zero and first order polynomials, there is no other communication required. For second order and higher, the +remainder of the GMRES polynomial setup uses m − 2 matmatmults and matmatadds to compute matrix-powers. If +we impose the aforementioned fixed sparsity we only need to communicate the required off-processor rows of Aff in +the matmatmults once in order to compute those matrix powers, regardless of the order of the polynomial. +Given the results in this paper, we believe the combination of our low-memory sub-grid scale discretisation, +AIRG with low-order GMRES polynomials, and an iterative method that additively preconditions with the stream- +ing/removal operator and an inconsistent CG DSA forms an excellent method for solving transport problems on +unstructured grids in both the streaming and scattering limit. +Acknowledgments +The authors would like to acknowledge the support of the EPSRC through the funding of the EPSRC grants +EP/R029423/1 and EP/T000414/1. +References +References +[1] J. S. Warsa, T. A. Wareing, J. E. 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ArXiv:1211.5969 [cs, math]. +29 + +102 +103 +104 +105 +0.5 +1 +1.5 +2 +2.5 +CG Nodes +Time (µs) per DOF +(a) Solve time +102 +103 +104 +105 +0.2 +0.4 +0.6 +0.8 +CG Nodes +Time (µs) per DOF +(b) Dashed are time to compute ˆA−1 +ff for AIRG, solid are time to +compute Z +102 +103 +104 +105 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +CG Nodes +Time (µs) per DOF +(c) Setup time +102 +103 +104 +105 +0.5 +1 +1.5 +2 +2.5 +CG Nodes +Time (µs) per DOF +(d) Total time +Figure 5: Timings per DOF for AIRG with m = 4 and lAIR in a 2D pure streaming problem. The × is AIRG with fixed sparsity with Falgout-CLJP, +the × is AIRG without fixed sparsity and Falgout-CLJP, the ⊗ is distance 1 lAIR with Falgout-CLJP, the ⊗ is distance 2 lAIR with Falgout-CLJP. +30 + +102 +103 +104 +105 +0.2 +0.4 +0.6 +CG Nodes +Time (µs) per DOF +(a) Setup time +102 +103 +104 +105 +0.4 +0.6 +0.8 +1 +1.2 +CG Nodes +Time (µs) per DOF +(b) Total time +Figure 6: Timings per DOF for AIRG with fixed sparsity, Falgout-CLJP and with varying GMRES polynomial order in a 2D pure streaming +problem. The × is AIRG with m = 1, o is m = 2, ⊗ is m = 3, □ is m = 4, ⋄ is m = 5. +31 + +102 +103 +104 +105 +1 +2 +3 +4 +5 +CG Nodes +Time (µs) per DOF +(a) Solve time +102 +103 +104 +105 +0.1 +0.2 +0.3 +0.4 +0.5 +CG Nodes +Time (µs) per DOF +(b) Dashed are time to compute ˆA−1 +ff for AIRG, solid are time to +compute Z +102 +103 +104 +105 +0.2 +0.4 +0.6 +0.8 +CG Nodes +Time (µs) per DOF +(c) Setup time +102 +103 +104 +105 +1 +2 +3 +4 +5 +6 +CG Nodes +Time (µs) per DOF +(d) Total time +Figure 7: Timings per DOF for AIRG with m = 4 and lAIR in a 2D pure scattering problem. The × is AIRG with fixed sparsity with Falgout-CLJP, +the × is AIRG without fixed sparsity and Falgout-CLJP, the ⊗ is distance 1 lAIR with Falgout-CLJP, the ⊗ is distance 2 lAIR with Falgout-CLJP. +32 + +102 +103 +104 +105 +0.3 +0.4 +0.5 +0.6 +CG Nodes +Time (µs) per DOF +(a) Setup time +102 +103 +104 +105 +1 +1.5 +2 +2.5 +3 +CG Nodes +Time (µs) per DOF +(b) Total time +Figure 8: Timings per DOF for AIRG with fixed sparsity, Falgout-CLJP and with varying GMRES polynomial order in a 2D pure scattering +problem. The × is AIRG with m = 1, o is m = 2, ⊗ is m = 3, □ is m = 4, ⋄ is m = 5. +(a) Streaming operator +(b) Streaming/removal operator with a total cross-section of 10.0. +Figure 9: The 10 biggest and smallest eigenvalues (by real, imaginary parts and magnitude) (dots) and field of values (solid lines) of different +operators, with the red the equivalent for Aff with CF splitting by Falgout-CLJP. 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+Robust statistical properties of T1 transitions in +confluent cell tissues +Harish P Jain1,*, Axel Voigt2,3,4, and Luiza Angheluta1 +1Njord Centre, Department of Physics, University of Oslo, Oslo, 0371, Norway +2Institute of Scientific Computing, Technische Universit¨at Dresden, Dresden, 01062, Germany +3Center of Systems Biology Dresden, Pfotenhauerstr. 108, 01307 Dresden, Germany +4Cluster of Excellence - Physics of Life, TU Dresden, 01062 Dresden, Germany +*harishpj@fys.uio.no +ABSTRACT +Large-scale tissue deformation which is fundamental to tissue development hinges on local cellular rearrangements, such as +T1 transitions. In the realm of the multi-phase field model, we analyse the statistical and dynamical properties of T1 transitions +in a confluent tissue. We identify an energy profile that is robust to changes in several model parameters. It is characterized by +an asymmetric profile with a fast increase in energy before the T1 transition and a sudden drop after the T1 transition, followed +by a slow relaxation. The latter being a signature of the fluidity of the cell tissue. We show that T1 transitions are sources of +localised large deformation of the cells undergoing the neighbour exchange and induce other T1 transitions in the nearby cells +through a chaining of events that propagate local cell deformation to large scale tissue flows. +1 Introduction +Collective motion of cells is essential to several processes including development of an embryo, tissue morphogenesis, wound +healing, homeostasis and cancer metastasis1–3. These biological processes are highly complex and orchestrate mechanical, +chemical and biochemical interactions across multiple scales4–7. Through the interplay between directed motion, neighbour +alignment and mechanical interactions, cell tissues exhibit emergent structures and dynamics that are crucial for their biological +function. A fundamental underlying process for emergent large-scale behavior is the topological rearrangement of neighbouring +cells, also known as T1 transition. It is a local, dissipative event that leads to remodelling of the tissue architecture and +influences the large-scale flow properties of cell tissues that affects tissue homeostasis and epithelial morphogenesis8–10. +In confluent tissues, the tissue architecture can change in several ways. To isolate the tissue dynamics driven by spontaneous +T1 transitions, we consider an idealised situation where apoptosis and cell division are neglected, cells have a constant volume, +identical mechanical properties, and their total number is fixed. During a T1 transition, typically two neighbouring cells move +apart, while two of their neighbours come towards each other and make contact as illustrated in Figure 1. The average number +of neighbours before and after the T1 transition is invariant. Through T1 transitions, cells undergo large deformations and +shape changes, and encounter an energy barrier that they have to overcome through their activity11,12. Albeit, there are several +competing scenarios of the mechanical-chemical-biological feedback involved in a T1 transition, our understanding of these +coupled processes is still elusive13. +T1 transitions are common features also in granular matter and foams under external forcing14–16. The energy relaxation +after a T1 transition has been studied in foams by measuring the length of T1 junctions17. This concept was adapted for active +tissues 18, where the length of the T1 junction before and after a T1 transition has been measured. During a T1 transition in +dry foams15, the cells form a rosette where either four or more edges meet. It has been shown that a junction is energetically +stable for three edges incident at 120 degrees. So, while undergoing a T1 transition, the cells pass from one metastable state +to another via an unstable state comprising of a rosette. In confluent tissue extracellular spaces (gaps) change this process19. +Rosettes and tri-junctions can no longer be defined by the number of edges meeting but are placed where gaps are formed. +Also, various mathematical models have been used to study different facets of T1 transitions in foams and tissues11,19–23. +They are mainly based on vertex models, which approximate cells by polygonal shapes. However, cell shape plays a crucial role +in T1 transitions and the ability to accurately describe complex cell shapes, beyond polygons, might be advantageous19,24,25. +We consider a multi-phase field model that allows for spontaneous T1 transitions while capturing the cell shape at a high +resolution and allowing for large shape deformations. Multi-phase field models have been used to probe several questions +pertaining collective motion of cells26–34. These models consider cells as active incompressible droplets and unlike vertex +models, T1 transitions emerge spontaneously as a result of shape deformations. In vertex models, also extracellular spaces +(gaps) need explicit modelling with ad-hoc assumptions19, whereas in multi-phase field models they are emergent. +arXiv:2301.11758v1 [physics.bio-ph] 27 Jan 2023 + +In this paper, we focus on characterising the energy profile preceding and succeeding T1 transitions. We show that this +energy profile is statistically robust to changes in several model parameters. It is characterized by an asymmetric profile with +a fast increase in energy before the T1 transition, a sudden drop after the T1 transition, followed by a slow relaxation. The +relaxation profile provides insights into the flow properties of the tissue. Previously, the relaxation has been indirectly studied +in tissues by examining the relaxation of an ellipsoid droplet immersed in a tissue17,19,35. The relaxation profile was attributed +to yield stress due to limitations in the measurement timescales35, and also associated with the fluidization of the tissue19,36. +We further consider the duration of T1 transitions and find that the average duration scales inversely to the maximum average +energy attained during the T1 transition. Also, we show that T1 transitions may trigger the creation of other T1 transitions +nearby and the chaining of T1 transitions leads to large-scale deformation and fluid like behaviour. +We introduce the multi-phase field model in Section 2 and discuss results on local statistical properties of T1 transitions in +Section 3. We further analyse the dependency of these statistical properties on various model parameters. The effect of cell +deformability and activity is considered in detail. We study the impact of chaining of T1 transitions on flow at larger scales. In +Section 4 we relate these finding to mechanical and rheological properties of the tissue and postulate that they can be used to +characterize fluidization. Details on the numerical methods, the initialization and characterization of T1 transitions are provided +in Section 5. +2 Multi-phase field model +We represent a two-dimensional confluent cell tissue within a multi-phase field model following formulations28,32–34. We +consider a system of N cells of equal area occupying a square domain of size [0,L]×[0,L] and use periodic boundary conditions. +Each cell is represented by a scalar phase field φi(x,t) as an indicator function of the domain occupied by each cell labeled by +i = 1,2,··· ,N. Namely, the bulk phase values φi ≈ 1 and φi ≈ −1 indicate the interior and exterior of the cell, respectively. +The cell boundary is defined by the localised transition region between the two bulk values. The time evolution of the i-th phase +field follows a conservative dynamics which preserves the cell areas and is given by +∂tφi +vi ·∇φi = ∆δF +δφi +, +(1) +where ∆ is the two-dimensional Laplacian applied to the variational derivative of a free energy functional F with respect to the +phase field φi. The free energy F = FCH +FINT contains the Cahn-Hilliard energy +FCH = 1 +Ca +N +∑ +i=1 +� +Ω +�ε +2||∇φi||2 + 1 +4ε (φ 2 +i −1)2 +� +dx, +(2) +and the interaction energy28,34 +FINT = 1 +In +N +∑ +i=1 +� +Ω B(φi)∑ +j̸=i +w(φj)dx. +(3) +The capillary number Ca and interaction number In are tuning parameters for the cell deformability and the strength of mutual +repulsion/attraction interactions, respectively. In equation 2, the Cahn-Hilliard energy has a local free energy density given by +the double well potential with the minima corresponding to the two bulk values and a gradient energy. The parameter ε controls +the width of the diffuse interface. The Cahn-Hilliard energy ensures phase separation into two bulk regions which are separated +by a thin, diffusive interface. This energy alone is minimised by cells with circular shapes. In equation 3, each cell’s interior +and interface (B(φi) = (φi +1)/2) is coupled with every other cell through a local interaction potential, +w(φj) = 1−(a+1) +�φ j −1 +2 +�2 ++a +�φj −1 +2 +�4 +, +where the parameter a = 1 models repulsion, while a > 1 models attraction and repulsion (see34 for a detailed analysis of role +of a). +Cell activity is introduced through the advection velocity vi(x,t) in equation 1 and is given by +vi(x,t) = v0B(φi)ei(t), +(4) +where v0 is a constant parameter that controls the magnitude of the activity, ei = [cosθi(t),sinθi(t)] where θi is the orientation +of the self-propulsion which evolves as +dθi = +� +2DrdWi(t)+α(βi(t)−θi(t))dt. +(5) +2/14 + +Figure 1. Successive time snapshots of tissue section undergoing a T1 transition, a finite-time neighbour exchange process +between cells A, B, C and D. The transition starts when cells B and D lose contact and is completed when cells A and C make +contact. During the T1 transition an extracellular space (gap) is formed between cells A, B, C and D. Also see Supplementary +Movie 1 +The first term on the right side of equation 5 is a rotational diffusion term with a Wiener process Wi. The second term is a +relaxation to the orientation of the cell’s shape elongation. The cell elongation is identified by the principal eigenvector of the +shape deformation tensor29,32 +Si = +� +Si,0 +Si,1 +Si,1 +−Si,0 +� +(6) +which is symmetric and traceless and has the two components +Si,0 = 1 +8 +� +Ω +��∂φi +∂y +�2 +− +�∂φi +∂x +�2� +dx +and +Si,1 = −1 +4 +� +Ω +�∂φi +∂x +∂φi +∂y +� +dx. +Its corresponding eigenvalues are λ ± +i = ± +� +S2 +i,0 +S2 +i,1 and eigenvectors are η± +i = ( Si,0+λ ± +i +Si,1 +,1). The vector η+ +i is parallel to the +elongation axis of the cell and determines the preferred self-propulsion direction as +βi(t) = +� +arg(η+ +i (t)) +: ei(t)·η+ +i (t) > 0 +−arg(η+ +i (t)) +: ei(t)·η+ +i (t) < 0 +(7) +Therefore, the second term on the right hand side of equation (5) aligns θi(t) with βi(t). The parameter α controls the time +scale of this alignment of the self-propulsion direction with the elongation axis of the cell. There are different possibilities +to define the advection velocity vi(x,t) (see Ref.32 for an overview and comparison). The current form includes approaches +of Ref.29 and, as the elongation is a result of the interaction with neighbouring cells, it accounts for contact inhibition of +locomotion37,38. The model leads to properties appropriate to describe, e.g., Madin-Darby canine kidney (MDCK) cells32,39. +(a) +(b) +(c) +Figure 2. (a). Free energy density, in a region surrounding a T1 transition. (b) and (c) Coarse-grained energy density in a +linear and log-scale, respectively. The white dot represents the epicenter of the T1 transition while the green dotted circle +represents the coarse graining radius ravg, the estimated core of the T1 transition. +3/14 + +A +A +A +A +B +D +D +D +B +B +D +B +c +c +c +c36 +32 +28 +24 +20 +16 +12 +8 +4 +05.4 +4.8 +4.2 +e grained) +3.6 +3.0 +(coarse +2.4 +energy +1.8 +1.2 +0.6 +0.05.4 +4.8 +4.2 +3.6 +3.0 +2.4 +energy ( +1.8 +1.2 +0.6 +0.03 Results +3.1 Energy profile of T1 transitions +Within our multi-phase field approach, T1 transitions are neighbour exchange processes with a finite duration. A prototypical +time sequence of a T1 transition is illustrated in Figure 1. Four cells A, B, C and D are involved. Before the T1 transition, the +cell junction shared by cells B and D shrink. The T1 transition starts when the cells B and D break contact and move apart. +This results in the formation of an extracellular space which we call ’gap’. Cells A and C move towards each other, close the +gap, and form a new contact concluding the T1 transition. After the T1 transition, this new junction between cells A and D +expands. The junctions that shrink and expand are called T1 junctions. We refer to Section 5 for the procedure to detect T1 +transitions and their durations. A T1 transition not only leads to topological rearrangements of the four neighbouring cells, it +also involves deformation of the cells. While details, such as the specific shape of the cells and their deformation, the duration +of the T1 transition and the relaxation process differ between T1 transitions, we will demonstrate that robust statistical features +of T1 transitions exist. +Figure 3. (a) Evolution of energy (averaged for 158 T1 transitions) at the epicenter of the T1 transitions. Negative time +corresponds to time before a T1 transition and positive time corresponds to time after a T1 transition. The shaded region +denotes a width of 1 standard deviation. The gray dashed line is the average energy across the whole domain. (b) Average +energy profile during a T1 transition as function of percentage of T1 duration. The standard deviation is also indicated. (c), (d) +and (e) Montages of deformed cells involved in a T1 transition. Each montage is made up of 5 images, that capture the cells at +equidistant times, stacked over each other. The darkest colored overlay represents the latest time. (c) Cell shapes before the +start of the T1 transition, (d) during the T1 transition, and (e) after the end of the T1 transition. Also see Supplementary Movie +2 for corresponding simulation +We define the epicenter of a T1 transition as the point with the minimum total distance from the centers of the involved cells +in the neighbour exchange process midway through the T1 transition. We define the immediate region around the epicenter as +the core of a T1 transition, which is of essence because it is the region where T1 junctions shrink and expand, and the gap +appears and disappears. +Figure 2a shows the total free energy density midway through a T1 transition. The epicentre is shown by the white dot and +4/14 + +4.85 +energy before T1 +(a) +energy during T1 +0.36 +5 +(b) +energy after T1 +maximum + standard deviation of energy +mean global energy +4.80 +4 +T +~75% of maximum ! +~75% of maximum +4.75 +buunp +0.32 +rgy standar +2 +0.30 +ener +4.65 +1 +0.28 +4.60 +0 +-30 +-20 +-10 +0(T1) +10 +20 +30 +0%(start) +20% +40% +60% +80% +100%(end) +time +percent of T1 duration +(d) +e +tT1 +5 to 0 +c +0 to 100% +tt1=0the estimated core is highlighted by the green circle. It has a radius ravg = 0.02L, where L is the side length of the computational +domain. We compute a coarse-grained energy whose value at any point in the domain is the average of the energy density in a +circular region centered at that point with radius ravg. Figure 2b shows this coarse grained energy field fravg, which we will call +’energy’ field in the following. The signature of triple-junctions and T1 transitions already becomes appealing due to their +higher energy. The difference between both is enhanced by using a log scale, see Figure 2c. Considering this energy field in the +epicenter over time provides a spatial-temporal description of T1 transitions. For discussions on the sensitivity of this procedure +on ravg we refer to Section 5. +Figure 3a shows the time evolution of this energy averaged over 158 T1 transitions. The time is negative before a T1 +transition and is positive after a T1 transition, and is denoted by tT1. The energy during the T1 transitions is excluded, which +leads to a discontinuity at tT1 = 0. The two values at tT1 = 0 correspond to the averaged energies at the start and the end of +the T1 transitions. As the duration of T1 transitions differs, an averaged energy as a function of time during the T1 transition +does not provide any meaningful information. Details on the energy during the T1 transition are shown in Figure 3b using a +normalized time. The energy profile in Figure 3a, 3b has a peak at the T1 transition. The profile is asymmetric with a strong +increase in energy before the T1 transition and a sudden decrease after the T1 transition followed by a slow relaxation. The +asymmetry can be quantified by considering the 75% of the maximum value, which is marked in Figure 3a. Figures 3c-3e +illustrate the evolution for one T1 transition, the one depicted in Figure 1. These figures contain overlays of several snapshots +as per the time marked in the figures. The darkest of these snapshots pertains to the latest time. The yellow region marks the +estimated core of the T1 transition. The asymmetry before and after the T1 transition, Figure 3c, 3e, respectively, is clearly +visible. The T1 junctions are longer at tT1 = 5 compared with tT1 = −5. During the T1 transition, Figure 3d, the asymmetry is +less pronounced. Most of the deformations are concentrated in the core. These deformations arise as a result of the formation of +the gap, and subsequently its disappearance. The shrinking and formation of T1 junctions and the deformations within the core +are a signature of the T1 transition. However, they also influence the deformation of the four cells outside of the core, and their +neighbours, which can be perceived by the overlayed cell shapes. Interestingly in the depicted T1 transition, the deformations +of each of the four cells seems to be persistent before, during and after the T1 transition (see the arrows indicating the direction +of deformations). We will elaborate on this and other coarse grained effects in Section 3.4. The energy profile indicates an +accumulation of energy to reach the energy barrier at the T1 transition. This is due to probing several possibilities in local +movement and cell shape deformation, which are coupled by the definition of activity, taking into account cell elongation and +contact inhibition of locomotion. After the energy barrier has been overcome the fast relaxation of the energy can be associated +with a steep gradient in the energy landscape in one direction. +The asymmetric shape of the energy profile is robust to changes in most model parameters, as demonstrated in Figure 4 +where α, Ca, a, D and v0 are varied and the energy profile associated with passive sheared foams is included for comparison. +Figure 4b shows the energy rescaled by the maximum energy as changes in Ca directly affect the free energy, see equation +(2). Within the range of parameters explored, the changes in the values of alignment parameter α, interaction coefficient a, +and diffusivity D have minimal effects on the energy profile. We see that the profile is robust even in absence of noise (D = 0) +(Figure 4d). On the other hand, the profile deviates from Figure 3a for low values of v0 and Ca. Figure 4e shows that the cell +activity v0 affects the rate at which the cells approach a T1 transition which is indicated by the slower accumulation of energy +for low v0. However, change in v0 has a minor effect on the energy relaxation immediately after a T1 transition. The slow +relaxation afterwards is largest for large values of v0. This can be associated with the definition of activity, which is related to +cell elongation and at least on average cells elongate in the direction of movement after the T1 transition. The characteristic +profile of the accumulation of energy before the T1 transition and the fast relaxation of energy after the T1 transition is also +present for low values of Ca, see Figure 4b. However, as Figure 4b considers a rescaled energy the actual rates depend on Ca. +The slow relaxation after the sudden decrease only slightly depends on Ca. We would like to point out that the results for low +values of v0 and Ca should be considered with care, as the number of T1 transitions considered in these cases is much lower. +While the system is still in the fluid phase, the extreme values for v0 = 0.1 and Ca = 0.05 already approach the transition to the +solid phase. +In passive foams T1 transitions can be induced by applying shear. This is considered by an advection velocity field +vi(x,t) = 0.5|x1 −L/2| and the resulting energy profile is compared with the profile from Figure 3a, see Figure 4f. The profiles +differ before the T1 transition and within the slow relaxation, but are similar in the sudden drop of energy right after the T1 +transition. The latter reiterates that the energy relaxation right after a T1 transition is independent on activity. The differences +in the accumulation of the energy can be associated with the persistent orientation of advection velocity due to shear, which +results in collective deformation and a more deterministic approach of the T1 transition. Also the termination of the decay in +the passive case results from the restricted possibilities of relaxation due to the applied shear. +5/14 + +Figure 4. Evolution of energy for different parameter values. The pink and cyan shaded region are used to denote time before +and after the T1 transitions, respectively. The number of T1 transition used to obtain these results in indicated. (a) The aligning +parameter α is varied. (b) The parameter to control cell deformability, Ca is varied. As Ca is a parameter that influences the +overall total energy, for better comparison the energy is rescaled by division with the maximum energy. (c) Adhesion and +repulsion corresponds to a = 1.5 and repulsion corresponds to a = 1. (d) The diffusivity D is varied. (e) The magnitude of the +activity v0 is varied. (f) The passive shear corresponds to advection field vi(x,t) = 0.5|x1 − L +2| while the active case +corresponds to parameters in Table 1. +3.2 Duration and other properties of T1 transitions +As mentioned earlier, the duration of T1 transitions strongly depends on the specific cell arrangements. We now discuss the +statistical properties of the duration. Figure 5a shows the probability distributions of the duration of T1 transitions. The +distributions peak at smaller values and have a long tail for larger values. The profiles corresponds to repulsive and adhesive +(a > 1), and only repulsive interactions (a = 1), and are fitted by Gamma distributions. The average duration of T1 transitions +for repulsive interactions (3.418 measured for 539 T1 transitions across 3 simulations) is smaller compared to that for repulsive +and adhesive interactions (3.826 for 631 T1 transitions across 4 simulations). Keeping other parameters fixed, the average +number of T1 transitions in the repulsive and adhesive case was 157.75 while for the repulsive case was 179.66, respectively. +Therefore, in the repulsive case, cells undergo neighbour exchanges faster and more often. Figure 5b shows the duration of T1 +6/14 + +α: 0.0, Total Tl: 173 +(b) +1.0 +5.0 +(a) +α: 0.001, Total T1: 194 +α: 0.01, Total T1: 159 +0.9 +4.5 +α: 0.1, Total T1: 158 +:: +: +α: 1.0, Total T1: 140 +0 +:: +4.0 +:: +8 +: +8: +: +: +: +. +::: +ner +led +1: +. + 3.0- +0.6 +: +:: +Ca: 0.05, Total T1: 25 +2.5 +:::: +Ca: 0.1, Total T1: 96 +: +. +Ca: 0.15, Total T1: 160 +0.4 +2.0 +: +Ca: 0.2, Total T1: 151 +Ca: 0.25, Total T1: 193 +0 +0.3 +1.5 +Ca: 0.3, Total T1: 182 +-30 +-20 +20 +-10 +0(T1) +10 +30 +-30 +-10 +0(T1) +-20 +10 +20 +30 +time +time +(c) +D: 0.0 , Total T1: 182 +5.0 +(d) +5.0- +D: 0.01 , Total T1: 158 +.. +D: 0.02 , Total T1: 177 +4.5 +4.5 +D: 0.03, Total T1: 145 +8 +: +: +4.0. +: +0 +4.0 +8 +:: +: +: +ner + 3.0 +8 + 3.0 +:. +2.5 +2.5. +1...: +2.0 +2.0 +8888888 +adhesion & repulsion, Total T1: 158 +repulsion, Total T1: 188 +1.5 +1.5 +-30 +-20 +-10 +0(T1) +-30 +-20 +10 +20 +30 +-10 +0(T1) +10 +20 +30 +time +time +Vo: 0.1 , Total T1: 8 +active, Total T1: 158 +(4) +5.0 +5.0 +(e) +Vo: 0.2 , Total T1: 27 +passive shear, Total T1: 64 +Vo: 0.3 , Total T1: 72 +. +4.5 +Vo: 0.4 , Total T1: 98 +4.5 +. +Vo: 0.5 , Total T1: 158 +0 +. +o: +Vo: 0.6 , Total Tl: 268 +4.0 +4.0 +. +Vo: 0.7 , Total T1: 266 +: +8 +: +: +. +: +.. + 3.0 +.. +....... + 3.0 +: +2.5- +2.5 +2.0 +8.8 +2.0 +1.5 +1.5 +-30 +-20 +-10 +0(T1) +10 +20 +30 +-30 +-20 +-10 +0(T1) +10 +20 +30 +time +time(a) +(b) +(c) +(d) +Figure 5. (a) Probability distributions of the duration of T1 transitions for only repulsion interactions (magenta dots) and for +both repulsion and adhesion (cyan dots). Both data sets are fitted by Gamma distributions highlighting the exponential tails. (b) +Scatter plot of duration of T1 transition as function of the maximum energy reached during a T1 transition. (c) Evolution of +average shape index and (d) Evolution of the average velocity of center of mass of the cells involved in the T1 transitions as +function of time relative to a T1 transition. The shaded regions mark the standard deviations of both quantities. +transitions as a function of the maximum energy reached during a T1 transition. While the data is scattered, it qualitatively +shows that high energy T1 transitions are faster. This qualitative result holds for both cases and can be explained by a larger +accumulation of energy in the core, which increases the spatial energy gradients and in turn speeds up the relaxation of the +energy which leads to the shorter duration. +Figure 5c shows the averaged shape index (perimeter/√area) of the four cells involved in a T1 transition as function of +time relative to a T1 transition. The asymmetry found for the energy profile and the discontinuity at tT1 = 0 is also present +for this quantity. The cells deform and elongate as they approach a T1 transition and relax afterwards. This increases and +decreases their shape index, respectively. The faster relaxation leads to the asymmetry in the evolution of the shape index. +The asymmetry around a T1 transition is also seen in the average velocity of the center of mass of the cells involved in a T1 +transition as shown in Figure 5d. While the velocity is almost constant before the T1 transition, the velocity peaks at the +T1 transition and slows down afterwards until it reaches the average value before the T1 transition. The peak in the average +velocity of the center of mass is due to the large deformations of the portions of cells within the core and their fast relaxation +after the T1 transition. Both quantities, the shape index and the cell velocity of the four cells involved in a T1 transition are also +experimentally accessible. These quantities can be related to the energy considered above. +3.3 Effect of cell deformability, activity and gaps on T1 transitions +The asymmetric energy profile in Figure 3a is robust to tuning of most of the model parameters. Significant variations only +occur for low values of Ca and v0, see Figure 4b, 4e. We now analyse the effect of cell deformability and activity on T1 +transitions in more detail. This requires a detailed analysis of the influence of gaps. The gap fraction is related to the confluency +as confluency = 100(1−gap fraction). It essentially is a fixed quantity set by the initial data. We fix all parameters as per table +1 and compare two different initial cell sizes, denoted by ’low gap’ with gap fraction 0.00048 and ’high gap’ with gap fraction +0.00212. Both can be considered as confluent. The number of T1 transitions within the considered time frame is not influenced +by this variation. The total numbers of T1 transitions are 162 and 158 for low and high gap cases, respectively. However, the +7/14 + +0.22 +before Tl +after T1 +0.20 +S +cell +0.18 +0.16 +f +velocity +0.14 +0.12 +0.10 +0.08 +-30 +-20 +-10 +0(T1) +10 +20 +30 +timerepulsion gamma fit +adhesion & repulsion gamma fit +repulsion +0.3 +adhesion & repulsion +probability +0.2 +0.1 +0.0 +0 +2 +4 +6 +8 +10 +Tl duration10 +adhesion & repulsion +repulsion +8 + duration +6 +4 +2 +4.0 +4.5 +5.0 +5.5 +max energy4.15 +before Tl +after T1 +shape index of Tl cells +4.10 +4.05 +O +4.00 +000: +0000 +3.95 +3.90 +-30 +-20 +-10 +0(T1) +10 +20 +30 +timeFigure 6. Dependency of various properties on deformability Ca ((a) - (f)) and activity v0 ((g) - (l)). Total T1 considers the +total number of T1 transitions within the considered time frame, T1 duration is the averaged time from start to end of all T1 +transitions, Gap fraction is the extracellular space, considered as ∑i B(φi) below a fixed threshold, again averaged over time, +Shape index considers the averaged shape index of the four cells involved in the T1 transitions. Time between T1 is the average +time a cell spends between successive T1 transitions, Max energy is the maximum energy reached at a T1 transition and vavg is +the average velocity of center of mass of all cells. +average duration of T1 transitions is reduced by reducing the gap fraction. The values are 2.559 and 3.794 for low and high gap +cases, respectively. We measure the gap fraction as the fraction of domain where ∑i B(φi) is less than a fixed threshold which is +set to 0.2. This essentially excludes possible partial overlap of the diffuse interface region of cells and only accounts for gaps at +tri-junctions and rosettes. This makes the measured gap fraction to depend on deformability and activity. For the considered +cases low Ca leads to rounder cells with stronger overlap of the diffuse interfaces of the cells, which are in contact. This leads +8/14 + +300 +8 +0.012 +(a) +(b) +(c) +250 +duration +6 +0.009 +4 +2 +% 0.003 +50 +0 +0 +0.0 +0.05 0.10 0.15 0.20 0.25 0.30 +0.05 0.10 0.15 0.20 0.25 0.30 +0.05 0.10 0.15 0.20 0.25 0.30 +Ca +Ca +Ca +4.2 +125 +(d) +(f) +(e) +Regular pentagon +15 +4.1 +Regular hexagon +energy +index +100 +between +4.0 +75 +10 +1/Ca fit +pe +3.9 + xeu +50 +hal +time +5 +S +3.8 +25 +3.7 +0.05 0.10 0.15 0.20 0.25 0.30 +0.05 0.10 0.15 0.20 0.25 0.30 +0.05 0.10 0.15 0.20 0.25 0.30 +Ca +Ca +Ca +300 +8 +0.012 +.(g). +(h) +(i) +250 +6 +150 +4 +b +2 +50 +0 +0.0 +0 +0.2 0.3 0.4 0.5 0.6 0.7 +0.2 +¥0.7 +0.4 0.5 0.6 0.7 +0.1 +0.1 ( +0.3 0.4 0.5 0.6 +0.1 +0.2 +0.3 +Vo +Vo +Vo +4.2 +125 +(k) +·(I) +Regular pentagon +0.20 +4.1 +index +Regular hexagon +100 +between +0.15 +4.0 +75 +Vavg +0.10 +50 +leus +time +S +3.8 +25 +0.05 +3.7 +0 +0.00 +0.2 0.3 0.4 0.5 +0.60.7 +0.1 0.2 0.3 +0.40.50.60.7 +0.1 +0.2 +0.1 +0.3 0.4 0.50.60.7 +Vo +Vo +Voto an increase in the measured gap fraction, see Figure 6c. A similar dependency, but smaller in magnitude, is found for activity. +Larger v0 lead to stronger interactions between cells and thus more overlap of the diffuse interface region of cells in contact +which again leads to an increase in measured gap fraction, see 6i. The gap fraction in both figures is the average quantity over +the considered time frame. Both results and the dependencies discussed below are considered for the ’high gap’ setting. +As shown in Figure 6a, the number of T1 transitions increases with increasing cell deformability parameter Ca. Cells that +are more deformable can more easily acquire the shape deformations associated with T1 transitions. When Ca is low, these +deformations are energetically more expensive resulting in fewer T1 transitions. Also the duration of T1 transitions depends on +Ca, as shown in Figure 6b. T1 transitions are shorter when cells are more deformable. We suspect that this might be due to the +presence of smaller gaps at T1 transitions, as this requires less shape deformation. Figure 6d shows the average cell shape index +of the four involved cells in a T1 transition as function of cell deformability Ca. The shape index increases as deformability +increases. The shape index of Ca = 0.05 is less than that of a regular pentagon. The shape index of regular pentagon (3.813) +was attributed as the critical shape index for jamming transition in classical vertex models40 without gaps. It has been argued +that gaps influence the mechanical properties and solid-liquid transition17, which might explain this discrepancy, as our system +is still within the fluid phase. Further details, which are related to the previous dependencies are shown in Figures 6e and 6f. +Figure 6e shows the average time a cell spends between successive T1 transitions as function of Ca. This quantity is large for +low Ca but decreases and plateaus to low values upon increasing Ca. Figure 6f shows the maximum energy reached during a T1 +transition against Ca. We see from the dotted curve that the maximum energy is proportional to 1/Ca. Recall that 1/Ca scales +the Cahn-Hilliard energy as per equation (2). This means that Fravg is primarily affected by the Cahn-Hilliard energy, which +explains the correspondence of our results with the length of T1 junctions discussed earlier and considered in17,18. +The dependency on v0 shows qualitatively similar behaviour for the number of T1 transitions, the duration of T1 transitions, +the shape index of the cells involved in T1 transitions and the time a cell spends between successive T1 transitions, see Figures +6g, 6h, 6j, 6k, respectively. The increase in T1 transitions and decrease in the time between T1 transitions with activity is a +property of active systems, which are driven out of equilibrium. T1 transitions are topological defects and thus an indication of +out of equilibrium. The decrease in duration with increasing activity can again be associated with the decrease in measured +gap fraction, see 6i, and also the increasing shape index with activity is a direct consequence of the form of active forcing +considered. Figure 6l shows the average velocity of center of mass of all cells as a function of v0. As expected, activity is +primarily converted into motion with an almost linear dependency. +3.4 Chaining of T1 transitions +So far, we have analysed robust statistical properties of T1 transitions within their cores. However, we have also seen that these +local features influence the position and shape of the four cells involved in a T1 transition, and their neighbours. This can +induce new T1 transitions and lead to the formation of chains of T1 transitions as illustrated in Figure 7. Each of these images +consists of 10 tissue states captured at equally-spaced time instants and overlaid on top of each other. The cell shapes outlined +in the darkest colors correspond to the latest time. The yellow circles mark the cores of the T1 transitions at those time instants. +The chaining of T1 transitions is a result of the assumptions on constant cell area and a confluent tissue. Any cell deformation +associated with a T1 transition induces deformation of the neighbouring cells and thereby increases the possibility of new +T1 transitions. This is further enhanced by activity and the considered propulsion mechanism which favours the direction of +elongation. +This chaining of T1 transitions is also observed experimentally in sheared foams41 and in our simulations of passive foams +which are sheared with a constant shear velocity profile. For v0 = 0, typically one or two T1 transitions occur due to the initial +non-equilibrium configuration of the tissue. As cells relax toward an equilibrium state, their motility is reduced which prevents +any further T1 transitions. The situation for small v0 is similar. The tissue becomes jammed by cells being caged amongst +their neighbours and no T1 transitions occur32. Furthermore, when cell deformability (Ca) is low, the energetic cost for cell +deformations that are necessary to undergo T1 transitions is high, which prevents or at least reduces T1 transitions and the +tissue also becomes jammed32. This corresponds to the low number of T1 transitions in Figure 4b, 4e for low Ca and low v0, +respectively. +However, in the considered case in Figure 7 we are far away from jamming and the chaining of T1 transitions leads to +cell deformation propagating to larger scales. This is highlighted in Figure 8a, which shows the evolution of the cell tissue in +the whole time window considered in Figure 7 together with the trajectory of the center of mass of the colored cells, which +highlights the movement on larger spatial scales. The chaining of T1 transitions is also a source of large-scale flows as evidenced +in Figure 8b. We consider the velocities of the centers of mass of all cells, average this quantity with the neighboring cells and +construct a continuous velocity field by interpolating in space. The velocity field is shown together with the cell boundaries at +t = 52. The mean direction corresponds with the direction of the black path shown in Figure 8a. However, as the variations in +magnitude and direction of the flow field in Figure 8b indicate, T1 transitions can also induce fluctuations and could play an +important role in sustaining chaotic flows (active turbulence) in cell tissues42–44. +9/14 + +(a) +(b) +(c) +(d) +(e) +(f) +Figure 7. Chaining of T1 transitions. Each panel is a montage of 10 snapshots of tissue configurations taken successively at +constant times intervals. Latest time is represented by the cell shapes marked in the darkest color shades. The cores of the T1 +transitions are highlighted in yellow. Also see Supplementary Movie 3. +Figure 8. (a) Montage of tissue snapshots from time t = 25 to t = 79 (see figure 7). The black path is the trajectory of the +center of mass of the 11 coloured cells. (b) LIC visualization of streamlines, magnitude (color) and direction (black arrows) of +the flow velocity. The velocity and the cell boundaries correspond to time t = 52. +10/14 + +time: 25 to 34time: 34 to 43time: 43 to 52time: 52 to 61time: 61 to 70time: 70 to 794 Discussion +Large-scale tissue deformation requires cellular rearrangements. The simplest rearrangement in confluent cell tissue is a T1 +transition. We have analysed these neighbour exchanges among cells in detail using a multi-phase field model and identified +a characteristic asymmetric energy profile, see Figure 3. The energy profile has a peak at the T1 transition. The profile is +asymmetric with a strong increase in energy before the T1 transition and a sudden decrease after the T1 transition which is +followed by a slow relaxation. Detailed studies on the dependency of this profile on model parameters show robustness to +variations in most parameters. They also allowed to associate the strong energy increase before the T1 transition with the +strength in activity. This region is characterized by an accumulation of energy to reach the energy barrier at the T1 transition. +This is achieved by probing several possibilities of direction of movement and shape deformation. This process is enhanced +by activity, which is quantified by Figure 4e. In contrast to this the sudden relaxation after the T1 transition can clearly be +associated with energy relaxation. It is almost independent of activity, see Figure 4e, and cell deformability, see Figure 4b, and +also present in sheared foams, see Figure 4f. We would like to remark that the behaviour is independent but the actual slope and +duration of this regime depends on deformability, as the energy is scaled in Figure 4b. The sudden decrease is associated with a +steep gradient in the energy landscape in one direction set by the deformation of the cells in the core of the T1 transition. The +third characteristic region, the slow relaxation, depends on activity and cell deformability. This relaxation profile provides +insight in the mechanical properties of the tissue. Similar energy profiles have been obtained by actuation and relaxation of +magnetic microdroplets which are injected into the tissue17,19,35. In these experiments a slow relaxation is associated with the +fluidization of the tissue19,35, while stagnation of the relaxation indicates more solid-like behaviour17 and is associated with +irreversible (plastic) tissue rearrangements. We postulate that these mechanical characterizations can also be obtained from the +energy decay of the T1 transitions. +In the considered confluent tissue the type of interaction between the cells, if repulsive or repulsive and attractive, seems to +play a minor role on the characteristic energy profile of a T1 transition, see Figure 4c. However, the degree of confluency is +known to influence the solid-fluid phase transition35. Increasing the extracellular space enhances fluidization. While we only +consider the fluid phase, we observe an increased duration of T1 transitions for larger extracellular space. A finite duration +of T1 transitions in cell tissues has been associated with molecular processes and is considered in an adhoc manner in vertex +models22. Within the multi-phase field model a finite duration is a result of the mechanical properties of the cells and the their +interactions. An increased duration of T1 transitions is observed for low deformability and low activity, see Figures 6b and 6h, +respectively. Both indicating more solid-like behaviour, which is consistent with22, where increased duration of T1 transitions +leads to decreasing the overall number of T1 transitions and a possible stiffening of the global tissue mechanics. However, +these results don’t take extracellular space into account. +Even if characterized locally, due to the confluent cell tissue, large enough deformations induced by T1 transitions lead to +permanent cell deformations in the neighbourhood, which can trigger other T1 transitions, leading to a chaining effect. This +behaviour is associated with the foam-like architecture and consistent with previously reported nonlinear tissue mechanics35. It +is this chaining of T1 transitions which allows for large-scale tissue deformations and flow patterns which can be associated +with sustaining chaotic flows, see Figure 8b. +We believe these results also to hold in more general situations, e.g. for varying cell sizes and varying mechanical cell +properties. +5 Numerical Methods +Model Parameters +Unless otherwise specified, we use the model parameters as per Table 1 +τ +τsave +T +L +ε +v0 +a +Ca +In +Dr +α +0.005 +0.5 +150 +100 +0.15 +0.5 +1.5 +0.2 +0.1 +0.1 +0.1 +Table 1. Default values of the model parameters. +Finite element simulations +The simulations are run for time interval [0,T] discretised into Nt units with a uniform timestep size τ, i.e. T = Ntτ. We employ +a semi-implicit discretization in time. Discretization in space follows the finite element method. We adaptively refine the diffuse +interface and employ a parallelization approach which scales with the number of cells. For details we refer to28,32–34,45,46. The +algorithm is implemented in the open-source library AMDiS47,48. +11/14 + +Detecting T1 transitions +The T1 transitions are detected by tracking the neighbour relations of all cells. If two cells A and B are in contact, their neighbour +relation is denoted by (A,B) or (B,A), both of which are equivalent. Suppose, there are four cells as in the Figure 1. The set of +neighbour relations between these four cells before, during and after a T1 transition are {(A,B),(B,C),(C,D),(D,A),(B,D)}, +{(A,B),(B,C),(C,D),(D,A)} and {(A,B),(B,C),(C,D),(D,A),(A,C)} respectively. Before and after a T1 transition, there +are 5 distinct neighbour relations between the four cells. The sets of relations before and after a T1 transition have four elements +in common. These common elements make up the set of relations during a T1 transition. The duration of a T1 transition is time +difference when the number of neighbour relations between the four cells change from 5 to 4 and back to 5. +Sensitivity of fravg on ravg +The coarse graining region of a point p is the region with all points x such that |p−x| < ravg. As the free energy is high at the +cell edges, the points which include the edges within its coarse graining region around it would have high fravg. Moreover, +points with triple junctions (where 3 edges meet) within its coarse graining region would have a higher fravg due to the presence +of longer total length of cell edges. Usually at a given time, fravg has peaks near the T1 epicenter. This is because, the region +around it would have either two triple junctions along with a gap as seen in the snapshots of Figure 1. Also, it is clear that points +that do not have any cell edges within its coarse graining region, would have zero fravg. We have found that increasing the ravg +loses information about the T1 transition in the value of fravg at the epicenter. A larger coarse graining region would entail a +larger contribution from the bulk of the interior of the cell and would reduce fravg at the epicenters such that fravg at epicenters +would not be uniquely discerned as a signature of a T1 transition. On the other hand, reducing ravg would mean that we might +not encompass the information of the two triple junctions and the gap formed during the T1 transition. It also increases the +deviations in the statistics that we describe. Moreover, if the energy along the length of the edge is uniform then the energy +field fravg at a point gives an approximate measure of length of edges within the coarse graining region around that point. +Data availability +All data are available from the corresponding author upon reasonable request. 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Sci. 10, 57–67, DOI: 10.1007/ +s00791-006-0048-3 (2007). +48. Witkowski, T., Ling, S., Praetorius, S. & Voigt, A. Software concepts and numerical algorithms for a scalable adaptive +parallel finite element method. Adv. Comput. Math. 41, 1145–1177, DOI: 10.1007/s10444-015-9405-4 (2015). +Acknowledgements +This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the +Marie Skłodowska-Curie grant agreement No 945371. We acknowledge computing resources provided within project WIR at +ZIH at TU Dresden. +Author contributions statements +H.P.J. implemented the codes, performed all simulations, analysed data and contributed to conceptual development and +manuscript writing. A.V. and L.A. contributed to supervision, conceptual development, data analysis and manuscript writing. +Additional information +Competing interests The authors declare no competing interests. +14/14 + diff --git a/09FKT4oBgHgl3EQfOS13/content/tmp_files/load_file.txt b/09FKT4oBgHgl3EQfOS13/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8c2952f369d10e9ca190ed523eeca124baaff30 --- /dev/null +++ b/09FKT4oBgHgl3EQfOS13/content/tmp_files/load_file.txt @@ -0,0 +1,1148 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf,len=1147 +page_content='Robust statistical properties of T1 transitions in confluent cell tissues Harish P Jain1,*, Axel Voigt2,3,4, and Luiza Angheluta1 1Njord Centre, Department of Physics, University of Oslo, Oslo, 0371, Norway 2Institute of Scientific Computing, Technische Universit¨at Dresden, Dresden, 01062, Germany 3Center of Systems Biology Dresden, Pfotenhauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 108, 01307 Dresden, Germany 4Cluster of Excellence - Physics of Life, TU Dresden, 01062 Dresden, Germany harishpj@fys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='no ABSTRACT Large-scale tissue deformation which is fundamental to tissue development hinges on local cellular rearrangements, such as T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In the realm of the multi-phase field model, we analyse the statistical and dynamical properties of T1 transitions in a confluent tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We identify an energy profile that is robust to changes in several model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It is characterized by an asymmetric profile with a fast increase in energy before the T1 transition and a sudden drop after the T1 transition, followed by a slow relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The latter being a signature of the fluidity of the cell tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We show that T1 transitions are sources of localised large deformation of the cells undergoing the neighbour exchange and induce other T1 transitions in the nearby cells through a chaining of events that propagate local cell deformation to large scale tissue flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 1 Introduction Collective motion of cells is essential to several processes including development of an embryo, tissue morphogenesis, wound healing, homeostasis and cancer metastasis1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These biological processes are highly complex and orchestrate mechanical, chemical and biochemical interactions across multiple scales4–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Through the interplay between directed motion, neighbour alignment and mechanical interactions, cell tissues exhibit emergent structures and dynamics that are crucial for their biological function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A fundamental underlying process for emergent large-scale behavior is the topological rearrangement of neighbouring cells, also known as T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It is a local, dissipative event that leads to remodelling of the tissue architecture and influences the large-scale flow properties of cell tissues that affects tissue homeostasis and epithelial morphogenesis8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In confluent tissues, the tissue architecture can change in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' To isolate the tissue dynamics driven by spontaneous T1 transitions, we consider an idealised situation where apoptosis and cell division are neglected, cells have a constant volume, identical mechanical properties, and their total number is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' During a T1 transition, typically two neighbouring cells move apart, while two of their neighbours come towards each other and make contact as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The average number of neighbours before and after the T1 transition is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Through T1 transitions, cells undergo large deformations and shape changes, and encounter an energy barrier that they have to overcome through their activity11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Albeit, there are several competing scenarios of the mechanical-chemical-biological feedback involved in a T1 transition, our understanding of these coupled processes is still elusive13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' T1 transitions are common features also in granular matter and foams under external forcing14–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The energy relaxation after a T1 transition has been studied in foams by measuring the length of T1 junctions17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This concept was adapted for active tissues 18, where the length of the T1 junction before and after a T1 transition has been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' During a T1 transition in dry foams15, the cells form a rosette where either four or more edges meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It has been shown that a junction is energetically stable for three edges incident at 120 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' So, while undergoing a T1 transition, the cells pass from one metastable state to another via an unstable state comprising of a rosette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In confluent tissue extracellular spaces (gaps) change this process19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Rosettes and tri-junctions can no longer be defined by the number of edges meeting but are placed where gaps are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also, various mathematical models have been used to study different facets of T1 transitions in foams and tissues11,19–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' They are mainly based on vertex models, which approximate cells by polygonal shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, cell shape plays a crucial role in T1 transitions and the ability to accurately describe complex cell shapes, beyond polygons, might be advantageous19,24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We consider a multi-phase field model that allows for spontaneous T1 transitions while capturing the cell shape at a high resolution and allowing for large shape deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Multi-phase field models have been used to probe several questions pertaining collective motion of cells26–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These models consider cells as active incompressible droplets and unlike vertex models, T1 transitions emerge spontaneously as a result of shape deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In vertex models, also extracellular spaces (gaps) need explicit modelling with ad-hoc assumptions19, whereas in multi-phase field models they are emergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='11758v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='bio-ph] 27 Jan 2023 In this paper, we focus on characterising the energy profile preceding and succeeding T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We show that this energy profile is statistically robust to changes in several model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It is characterized by an asymmetric profile with a fast increase in energy before the T1 transition, a sudden drop after the T1 transition, followed by a slow relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The relaxation profile provides insights into the flow properties of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Previously, the relaxation has been indirectly studied in tissues by examining the relaxation of an ellipsoid droplet immersed in a tissue17,19,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The relaxation profile was attributed to yield stress due to limitations in the measurement timescales35, and also associated with the fluidization of the tissue19,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We further consider the duration of T1 transitions and find that the average duration scales inversely to the maximum average energy attained during the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also, we show that T1 transitions may trigger the creation of other T1 transitions nearby and the chaining of T1 transitions leads to large-scale deformation and fluid like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We introduce the multi-phase field model in Section 2 and discuss results on local statistical properties of T1 transitions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We further analyse the dependency of these statistical properties on various model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The effect of cell deformability and activity is considered in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We study the impact of chaining of T1 transitions on flow at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In Section 4 we relate these finding to mechanical and rheological properties of the tissue and postulate that they can be used to characterize fluidization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Details on the numerical methods, the initialization and characterization of T1 transitions are provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 2 Multi-phase field model We represent a two-dimensional confluent cell tissue within a multi-phase field model following formulations28,32–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We consider a system of N cells of equal area occupying a square domain of size [0,L]×[0,L] and use periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Each cell is represented by a scalar phase field φi(x,t) as an indicator function of the domain occupied by each cell labeled by i = 1,2,··· ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Namely, the bulk phase values φi ≈ 1 and φi ≈ −1 indicate the interior and exterior of the cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The cell boundary is defined by the localised transition region between the two bulk values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The time evolution of the i-th phase field follows a conservative dynamics which preserves the cell areas and is given by ∂tφi +vi ·∇φi = ∆δF δφi , (1) where ∆ is the two-dimensional Laplacian applied to the variational derivative of a free energy functional F with respect to the phase field φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The free energy F = FCH +FINT contains the Cahn-Hilliard energy FCH = 1 Ca N ∑ i=1 � Ω �ε 2||∇φi||2 + 1 4ε (φ 2 i −1)2 � dx, (2) and the interaction energy28,34 FINT = 1 In N ∑ i=1 � Ω B(φi)∑ j̸=i w(φj)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (3) The capillary number Ca and interaction number In are tuning parameters for the cell deformability and the strength of mutual repulsion/attraction interactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In equation 2, the Cahn-Hilliard energy has a local free energy density given by the double well potential with the minima corresponding to the two bulk values and a gradient energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The parameter ε controls the width of the diffuse interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The Cahn-Hilliard energy ensures phase separation into two bulk regions which are separated by a thin, diffusive interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This energy alone is minimised by cells with circular shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In equation 3, each cell’s interior and interface (B(φi) = (φi +1)/2) is coupled with every other cell through a local interaction potential, w(φj) = 1−(a+1) �φ j −1 2 �2 +a �φj −1 2 �4 , where the parameter a = 1 models repulsion, while a > 1 models attraction and repulsion (see34 for a detailed analysis of role of a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Cell activity is introduced through the advection velocity vi(x,t) in equation 1 and is given by vi(x,t) = v0B(φi)ei(t), (4) where v0 is a constant parameter that controls the magnitude of the activity, ei = [cosθi(t),sinθi(t)] where θi is the orientation of the self-propulsion which evolves as dθi = � 2DrdWi(t)+α(βi(t)−θi(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (5) 2/14 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Successive time snapshots of tissue section undergoing a T1 transition, a finite-time neighbour exchange process between cells A, B, C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The transition starts when cells B and D lose contact and is completed when cells A and C make contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' During the T1 transition an extracellular space (gap) is formed between cells A, B, C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also see Supplementary Movie 1 The first term on the right side of equation 5 is a rotational diffusion term with a Wiener process Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The second term is a relaxation to the orientation of the cell’s shape elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The cell elongation is identified by the principal eigenvector of the shape deformation tensor29,32 Si = � Si,0 Si,1 Si,1 −Si,0 � (6) which is symmetric and traceless and has the two components Si,0 = 1 8 � Ω ��∂φi ∂y �2 − �∂φi ∂x �2� dx and Si,1 = −1 4 � Ω �∂φi ∂x ∂φi ∂y � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Its corresponding eigenvalues are λ ± i = ± � S2 i,0 +S2 i,1 and eigenvectors are η± i = ( Si,0+λ ± i Si,1 ,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The vector η+ i is parallel to the elongation axis of the cell and determines the preferred self-propulsion direction as βi(t) = � arg(η+ i (t)) : ei(t)·η+ i (t) > 0 −arg(η+ i (t)) : ei(t)·η+ i (t) < 0 (7) Therefore, the second term on the right hand side of equation (5) aligns θi(t) with βi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The parameter α controls the time scale of this alignment of the self-propulsion direction with the elongation axis of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' There are different possibilities to define the advection velocity vi(x,t) (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='32 for an overview and comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The current form includes approaches of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='29 and, as the elongation is a result of the interaction with neighbouring cells, it accounts for contact inhibition of locomotion37,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The model leads to properties appropriate to describe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=', Madin-Darby canine kidney (MDCK) cells32,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a) (b) (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Free energy density, in a region surrounding a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (b) and (c) Coarse-grained energy density in a linear and log-scale, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The white dot represents the epicenter of the T1 transition while the green dotted circle represents the coarse graining radius ravg, the estimated core of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3/14 A A A A B D D D B B D B c c c c36 32 28 24 20 16 12 8 4 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 e grained) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 (coarse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 energy ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='03 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 Energy profile of T1 transitions Within our multi-phase field approach, T1 transitions are neighbour exchange processes with a finite duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A prototypical time sequence of a T1 transition is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Four cells A, B, C and D are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Before the T1 transition, the cell junction shared by cells B and D shrink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The T1 transition starts when the cells B and D break contact and move apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This results in the formation of an extracellular space which we call ’gap’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Cells A and C move towards each other, close the gap, and form a new contact concluding the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' After the T1 transition, this new junction between cells A and D expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The junctions that shrink and expand are called T1 junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We refer to Section 5 for the procedure to detect T1 transitions and their durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A T1 transition not only leads to topological rearrangements of the four neighbouring cells, it also involves deformation of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' While details, such as the specific shape of the cells and their deformation, the duration of the T1 transition and the relaxation process differ between T1 transitions, we will demonstrate that robust statistical features of T1 transitions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a) Evolution of energy (averaged for 158 T1 transitions) at the epicenter of the T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Negative time corresponds to time before a T1 transition and positive time corresponds to time after a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shaded region denotes a width of 1 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The gray dashed line is the average energy across the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (b) Average energy profile during a T1 transition as function of percentage of T1 duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The standard deviation is also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (c), (d) and (e) Montages of deformed cells involved in a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Each montage is made up of 5 images, that capture the cells at equidistant times, stacked over each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The darkest colored overlay represents the latest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (c) Cell shapes before the start of the T1 transition, (d) during the T1 transition, and (e) after the end of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also see Supplementary Movie 2 for corresponding simulation We define the epicenter of a T1 transition as the point with the minimum total distance from the centers of the involved cells in the neighbour exchange process midway through the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We define the immediate region around the epicenter as the core of a T1 transition, which is of essence because it is the region where T1 junctions shrink and expand, and the gap appears and disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 2a shows the total free energy density midway through a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The epicentre is shown by the white dot and 4/14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='85 energy before T1 (a) energy during T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='36 5 (b) energy after T1 maximum standard deviation of energy mean global energy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='80 4 T ~75% of maximum !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' ~75% of maximum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='75 buunp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='32 rgy standar 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 ener 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='65 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='60 0 30 20 10 0(T1) 10 20 30 0%(start) 20% 40% 60% 80% 100%(end) time percent of T1 duration (d) e tT1 5 to 0 c 0 to 100% tt1=0the estimated core is highlighted by the green circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It has a radius ravg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='02L, where L is the side length of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We compute a coarse-grained energy whose value at any point in the domain is the average of the energy density in a circular region centered at that point with radius ravg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 2b shows this coarse grained energy field fravg, which we will call ’energy’ field in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The signature of triple-junctions and T1 transitions already becomes appealing due to their higher energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The difference between both is enhanced by using a log scale, see Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Considering this energy field in the epicenter over time provides a spatial-temporal description of T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' For discussions on the sensitivity of this procedure on ravg we refer to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 3a shows the time evolution of this energy averaged over 158 T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The time is negative before a T1 transition and is positive after a T1 transition, and is denoted by tT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The energy during the T1 transitions is excluded, which leads to a discontinuity at tT1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The two values at tT1 = 0 correspond to the averaged energies at the start and the end of the T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As the duration of T1 transitions differs, an averaged energy as a function of time during the T1 transition does not provide any meaningful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Details on the energy during the T1 transition are shown in Figure 3b using a normalized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The energy profile in Figure 3a, 3b has a peak at the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The profile is asymmetric with a strong increase in energy before the T1 transition and a sudden decrease after the T1 transition followed by a slow relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The asymmetry can be quantified by considering the 75% of the maximum value, which is marked in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figures 3c-3e illustrate the evolution for one T1 transition, the one depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These figures contain overlays of several snapshots as per the time marked in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The darkest of these snapshots pertains to the latest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The yellow region marks the estimated core of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The asymmetry before and after the T1 transition, Figure 3c, 3e, respectively, is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The T1 junctions are longer at tT1 = 5 compared with tT1 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' During the T1 transition, Figure 3d, the asymmetry is less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Most of the deformations are concentrated in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These deformations arise as a result of the formation of the gap, and subsequently its disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shrinking and formation of T1 junctions and the deformations within the core are a signature of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, they also influence the deformation of the four cells outside of the core, and their neighbours, which can be perceived by the overlayed cell shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Interestingly in the depicted T1 transition, the deformations of each of the four cells seems to be persistent before, during and after the T1 transition (see the arrows indicating the direction of deformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We will elaborate on this and other coarse grained effects in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The energy profile indicates an accumulation of energy to reach the energy barrier at the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is due to probing several possibilities in local movement and cell shape deformation, which are coupled by the definition of activity, taking into account cell elongation and contact inhibition of locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' After the energy barrier has been overcome the fast relaxation of the energy can be associated with a steep gradient in the energy landscape in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The asymmetric shape of the energy profile is robust to changes in most model parameters, as demonstrated in Figure 4 where α, Ca, a, D and v0 are varied and the energy profile associated with passive sheared foams is included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 4b shows the energy rescaled by the maximum energy as changes in Ca directly affect the free energy, see equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Within the range of parameters explored, the changes in the values of alignment parameter α, interaction coefficient a, and diffusivity D have minimal effects on the energy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We see that the profile is robust even in absence of noise (D = 0) (Figure 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' On the other hand, the profile deviates from Figure 3a for low values of v0 and Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 4e shows that the cell activity v0 affects the rate at which the cells approach a T1 transition which is indicated by the slower accumulation of energy for low v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, change in v0 has a minor effect on the energy relaxation immediately after a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The slow relaxation afterwards is largest for large values of v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This can be associated with the definition of activity, which is related to cell elongation and at least on average cells elongate in the direction of movement after the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The characteristic profile of the accumulation of energy before the T1 transition and the fast relaxation of energy after the T1 transition is also present for low values of Ca, see Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, as Figure 4b considers a rescaled energy the actual rates depend on Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The slow relaxation after the sudden decrease only slightly depends on Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We would like to point out that the results for low values of v0 and Ca should be considered with care, as the number of T1 transitions considered in these cases is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' While the system is still in the fluid phase, the extreme values for v0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 and Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 already approach the transition to the solid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In passive foams T1 transitions can be induced by applying shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is considered by an advection velocity field vi(x,t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5|x1 −L/2| and the resulting energy profile is compared with the profile from Figure 3a, see Figure 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The profiles differ before the T1 transition and within the slow relaxation, but are similar in the sudden drop of energy right after the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The latter reiterates that the energy relaxation right after a T1 transition is independent on activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The differences in the accumulation of the energy can be associated with the persistent orientation of advection velocity due to shear, which results in collective deformation and a more deterministic approach of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also the termination of the decay in the passive case results from the restricted possibilities of relaxation due to the applied shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 5/14 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Evolution of energy for different parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The pink and cyan shaded region are used to denote time before and after the T1 transitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The number of T1 transition used to obtain these results in indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a) The aligning parameter α is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (b) The parameter to control cell deformability, Ca is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As Ca is a parameter that influences the overall total energy, for better comparison the energy is rescaled by division with the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (c) Adhesion and repulsion corresponds to a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 and repulsion corresponds to a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (d) The diffusivity D is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (e) The magnitude of the activity v0 is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (f) The passive shear corresponds to advection field vi(x,t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5|x1 − L 2| while the active case corresponds to parameters in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 Duration and other properties of T1 transitions As mentioned earlier, the duration of T1 transitions strongly depends on the specific cell arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We now discuss the statistical properties of the duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 5a shows the probability distributions of the duration of T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The distributions peak at smaller values and have a long tail for larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The profiles corresponds to repulsive and adhesive (a > 1), and only repulsive interactions (a = 1), and are fitted by Gamma distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The average duration of T1 transitions for repulsive interactions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='418 measured for 539 T1 transitions across 3 simulations) is smaller compared to that for repulsive and adhesive interactions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='826 for 631 T1 transitions across 4 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Keeping other parameters fixed, the average number of T1 transitions in the repulsive and adhesive case was 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='75 while for the repulsive case was 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='66, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Therefore, in the repulsive case, cells undergo neighbour exchanges faster and more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 5b shows the duration of T1 6/14 α: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0, Total Tl: 173 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 (a) α: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='001, Total T1: 194 α: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='01, Total T1: 159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 α: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1, Total T1: 158 :: : α: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0, Total T1: 140 0 :: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 :: 8 : 8: : : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' ::: ner led 1: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 : :: Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05, Total T1: 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 :::: Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1, Total T1: 96 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15, Total T1: 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 : Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2, Total T1: 151 Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='25, Total T1: 193 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 Ca: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3, Total T1: 182 30 20 20 10 0(T1) 10 30 30 10 0(T1) 20 10 20 30 time time (c) D: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 , Total T1: 182 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 (d) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0- D: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='01 , Total T1: 158 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='. D: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='02 , Total T1: 177 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 D: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='03, Total T1: 145 8 : : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' : 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 8 :: : : ner 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=': 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 8888888 adhesion & repulsion, Total T1: 158 repulsion, Total T1: 188 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 30 20 10 0(T1) 30 20 10 20 30 10 0(T1) 10 20 30 time time Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 , Total T1: 8 active, Total T1: 158 (4) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 (e) Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 , Total T1: 27 passive shear, Total T1: 64 Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3 , Total T1: 72 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 , Total T1: 98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 , Total T1: 158 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' o: Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='6 , Total Tl: 268 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Vo: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='7 , Total T1: 266 : 8 : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 30 20 10 0(T1) 10 20 30 30 20 10 0(T1) 10 20 30 time time(a) (b) (c) (d) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a) Probability distributions of the duration of T1 transitions for only repulsion interactions (magenta dots) and for both repulsion and adhesion (cyan dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Both data sets are fitted by Gamma distributions highlighting the exponential tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (b) Scatter plot of duration of T1 transition as function of the maximum energy reached during a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (c) Evolution of average shape index and (d) Evolution of the average velocity of center of mass of the cells involved in the T1 transitions as function of time relative to a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shaded regions mark the standard deviations of both quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' transitions as a function of the maximum energy reached during a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' While the data is scattered, it qualitatively shows that high energy T1 transitions are faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This qualitative result holds for both cases and can be explained by a larger accumulation of energy in the core, which increases the spatial energy gradients and in turn speeds up the relaxation of the energy which leads to the shorter duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 5c shows the averaged shape index (perimeter/√area) of the four cells involved in a T1 transition as function of time relative to a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The asymmetry found for the energy profile and the discontinuity at tT1 = 0 is also present for this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The cells deform and elongate as they approach a T1 transition and relax afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This increases and decreases their shape index, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The faster relaxation leads to the asymmetry in the evolution of the shape index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The asymmetry around a T1 transition is also seen in the average velocity of the center of mass of the cells involved in a T1 transition as shown in Figure 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' While the velocity is almost constant before the T1 transition, the velocity peaks at the T1 transition and slows down afterwards until it reaches the average value before the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The peak in the average velocity of the center of mass is due to the large deformations of the portions of cells within the core and their fast relaxation after the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Both quantities, the shape index and the cell velocity of the four cells involved in a T1 transition are also experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These quantities can be related to the energy considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3 Effect of cell deformability, activity and gaps on T1 transitions The asymmetric energy profile in Figure 3a is robust to tuning of most of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Significant variations only occur for low values of Ca and v0, see Figure 4b, 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We now analyse the effect of cell deformability and activity on T1 transitions in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This requires a detailed analysis of the influence of gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The gap fraction is related to the confluency as confluency = 100(1−gap fraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It essentially is a fixed quantity set by the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We fix all parameters as per table 1 and compare two different initial cell sizes, denoted by ’low gap’ with gap fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='00048 and ’high gap’ with gap fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='00212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Both can be considered as confluent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The number of T1 transitions within the considered time frame is not influenced by this variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The total numbers of T1 transitions are 162 and 158 for low and high gap cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, the 7/14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='22 before Tl after T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 S cell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='16 f velocity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='08 30 20 10 0(T1) 10 20 30 timerepulsion gamma fit adhesion & repulsion gamma fit repulsion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3 adhesion & repulsion probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 0 2 4 6 8 10 Tl duration10 adhesion & repulsion repulsion 8 duration 6 4 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 max energy4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 before Tl after T1 shape index of Tl cells 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='00 000: 0000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='90 30 20 10 0(T1) 10 20 30 timeFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Dependency of various properties on deformability Ca ((a) - (f)) and activity v0 ((g) - (l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Total T1 considers the total number of T1 transitions within the considered time frame, T1 duration is the averaged time from start to end of all T1 transitions, Gap fraction is the extracellular space, considered as ∑i B(φi) below a fixed threshold, again averaged over time, Shape index considers the averaged shape index of the four cells involved in the T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Time between T1 is the average time a cell spends between successive T1 transitions, Max energy is the maximum energy reached at a T1 transition and vavg is the average velocity of center of mass of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' average duration of T1 transitions is reduced by reducing the gap fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The values are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='559 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='794 for low and high gap cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We measure the gap fraction as the fraction of domain where ∑i B(φi) is less than a fixed threshold which is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This essentially excludes possible partial overlap of the diffuse interface region of cells and only accounts for gaps at tri-junctions and rosettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This makes the measured gap fraction to depend on deformability and activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' For the considered cases low Ca leads to rounder cells with stronger overlap of the diffuse interfaces of the cells, which are in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This leads 8/14 300 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='012 (a) (b) (c) 250 duration 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='009 4 2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='003 50 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='30 Ca Ca Ca 300 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='012 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='7 Vo Vo Voto an increase in the measured gap fraction, see Figure 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A similar dependency, but smaller in magnitude, is found for activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Larger v0 lead to stronger interactions between cells and thus more overlap of the diffuse interface region of cells in contact which again leads to an increase in measured gap fraction, see 6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The gap fraction in both figures is the average quantity over the considered time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Both results and the dependencies discussed below are considered for the ’high gap’ setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As shown in Figure 6a, the number of T1 transitions increases with increasing cell deformability parameter Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Cells that are more deformable can more easily acquire the shape deformations associated with T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' When Ca is low, these deformations are energetically more expensive resulting in fewer T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also the duration of T1 transitions depends on Ca, as shown in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' T1 transitions are shorter when cells are more deformable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We suspect that this might be due to the presence of smaller gaps at T1 transitions, as this requires less shape deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 6d shows the average cell shape index of the four involved cells in a T1 transition as function of cell deformability Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shape index increases as deformability increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shape index of Ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='05 is less than that of a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The shape index of regular pentagon (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='813) was attributed as the critical shape index for jamming transition in classical vertex models40 without gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It has been argued that gaps influence the mechanical properties and solid-liquid transition17, which might explain this discrepancy, as our system is still within the fluid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Further details, which are related to the previous dependencies are shown in Figures 6e and 6f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 6e shows the average time a cell spends between successive T1 transitions as function of Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This quantity is large for low Ca but decreases and plateaus to low values upon increasing Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 6f shows the maximum energy reached during a T1 transition against Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We see from the dotted curve that the maximum energy is proportional to 1/Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Recall that 1/Ca scales the Cahn-Hilliard energy as per equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This means that Fravg is primarily affected by the Cahn-Hilliard energy, which explains the correspondence of our results with the length of T1 junctions discussed earlier and considered in17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The dependency on v0 shows qualitatively similar behaviour for the number of T1 transitions, the duration of T1 transitions, the shape index of the cells involved in T1 transitions and the time a cell spends between successive T1 transitions, see Figures 6g, 6h, 6j, 6k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The increase in T1 transitions and decrease in the time between T1 transitions with activity is a property of active systems, which are driven out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' T1 transitions are topological defects and thus an indication of out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The decrease in duration with increasing activity can again be associated with the decrease in measured gap fraction, see 6i, and also the increasing shape index with activity is a direct consequence of the form of active forcing considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 6l shows the average velocity of center of mass of all cells as a function of v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As expected, activity is primarily converted into motion with an almost linear dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='4 Chaining of T1 transitions So far, we have analysed robust statistical properties of T1 transitions within their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, we have also seen that these local features influence the position and shape of the four cells involved in a T1 transition, and their neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This can induce new T1 transitions and lead to the formation of chains of T1 transitions as illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Each of these images consists of 10 tissue states captured at equally-spaced time instants and overlaid on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The cell shapes outlined in the darkest colors correspond to the latest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The yellow circles mark the cores of the T1 transitions at those time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The chaining of T1 transitions is a result of the assumptions on constant cell area and a confluent tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Any cell deformation associated with a T1 transition induces deformation of the neighbouring cells and thereby increases the possibility of new T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is further enhanced by activity and the considered propulsion mechanism which favours the direction of elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This chaining of T1 transitions is also observed experimentally in sheared foams41 and in our simulations of passive foams which are sheared with a constant shear velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' For v0 = 0, typically one or two T1 transitions occur due to the initial non-equilibrium configuration of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As cells relax toward an equilibrium state, their motility is reduced which prevents any further T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The situation for small v0 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The tissue becomes jammed by cells being caged amongst their neighbours and no T1 transitions occur32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Furthermore, when cell deformability (Ca) is low, the energetic cost for cell deformations that are necessary to undergo T1 transitions is high, which prevents or at least reduces T1 transitions and the tissue also becomes jammed32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This corresponds to the low number of T1 transitions in Figure 4b, 4e for low Ca and low v0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, in the considered case in Figure 7 we are far away from jamming and the chaining of T1 transitions leads to cell deformation propagating to larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is highlighted in Figure 8a, which shows the evolution of the cell tissue in the whole time window considered in Figure 7 together with the trajectory of the center of mass of the colored cells, which highlights the movement on larger spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The chaining of T1 transitions is also a source of large-scale flows as evidenced in Figure 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We consider the velocities of the centers of mass of all cells, average this quantity with the neighboring cells and construct a continuous velocity field by interpolating in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The velocity field is shown together with the cell boundaries at t = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The mean direction corresponds with the direction of the black path shown in Figure 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, as the variations in magnitude and direction of the flow field in Figure 8b indicate, T1 transitions can also induce fluctuations and could play an important role in sustaining chaotic flows (active turbulence) in cell tissues42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 9/14 (a) (b) (c) (d) (e) (f) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Chaining of T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Each panel is a montage of 10 snapshots of tissue configurations taken successively at constant times intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Latest time is represented by the cell shapes marked in the darkest color shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The cores of the T1 transitions are highlighted in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also see Supplementary Movie 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (a) Montage of tissue snapshots from time t = 25 to t = 79 (see figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The black path is the trajectory of the center of mass of the 11 coloured cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' (b) LIC visualization of streamlines, magnitude (color) and direction (black arrows) of the flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The velocity and the cell boundaries correspond to time t = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 10/14 time: 25 to 34time: 34 to 43time: 43 to 52time: 52 to 61time: 61 to 70time: 70 to 794 Discussion Large-scale tissue deformation requires cellular rearrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The simplest rearrangement in confluent cell tissue is a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We have analysed these neighbour exchanges among cells in detail using a multi-phase field model and identified a characteristic asymmetric energy profile, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The energy profile has a peak at the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The profile is asymmetric with a strong increase in energy before the T1 transition and a sudden decrease after the T1 transition which is followed by a slow relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Detailed studies on the dependency of this profile on model parameters show robustness to variations in most parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' They also allowed to associate the strong energy increase before the T1 transition with the strength in activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This region is characterized by an accumulation of energy to reach the energy barrier at the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is achieved by probing several possibilities of direction of movement and shape deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This process is enhanced by activity, which is quantified by Figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In contrast to this the sudden relaxation after the T1 transition can clearly be associated with energy relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It is almost independent of activity, see Figure 4e, and cell deformability, see Figure 4b, and also present in sheared foams, see Figure 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We would like to remark that the behaviour is independent but the actual slope and duration of this regime depends on deformability, as the energy is scaled in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The sudden decrease is associated with a steep gradient in the energy landscape in one direction set by the deformation of the cells in the core of the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The third characteristic region, the slow relaxation, depends on activity and cell deformability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This relaxation profile provides insight in the mechanical properties of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Similar energy profiles have been obtained by actuation and relaxation of magnetic microdroplets which are injected into the tissue17,19,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In these experiments a slow relaxation is associated with the fluidization of the tissue19,35, while stagnation of the relaxation indicates more solid-like behaviour17 and is associated with irreversible (plastic) tissue rearrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We postulate that these mechanical characterizations can also be obtained from the energy decay of the T1 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' In the considered confluent tissue the type of interaction between the cells, if repulsive or repulsive and attractive, seems to play a minor role on the characteristic energy profile of a T1 transition, see Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, the degree of confluency is known to influence the solid-fluid phase transition35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Increasing the extracellular space enhances fluidization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' While we only consider the fluid phase, we observe an increased duration of T1 transitions for larger extracellular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A finite duration of T1 transitions in cell tissues has been associated with molecular processes and is considered in an adhoc manner in vertex models22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Within the multi-phase field model a finite duration is a result of the mechanical properties of the cells and the their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' An increased duration of T1 transitions is observed for low deformability and low activity, see Figures 6b and 6h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Both indicating more solid-like behaviour, which is consistent with22, where increased duration of T1 transitions leads to decreasing the overall number of T1 transitions and a possible stiffening of the global tissue mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' However, these results don’t take extracellular space into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Even if characterized locally, due to the confluent cell tissue, large enough deformations induced by T1 transitions lead to permanent cell deformations in the neighbourhood, which can trigger other T1 transitions, leading to a chaining effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This behaviour is associated with the foam-like architecture and consistent with previously reported nonlinear tissue mechanics35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It is this chaining of T1 transitions which allows for large-scale tissue deformations and flow patterns which can be associated with sustaining chaotic flows, see Figure 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We believe these results also to hold in more general situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' for varying cell sizes and varying mechanical cell properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 5 Numerical Methods Model Parameters Unless otherwise specified, we use the model parameters as per Table 1 τ τsave T L ε v0 a Ca In Dr α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 150 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Default values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Finite element simulations The simulations are run for time interval [0,T] discretised into Nt units with a uniform timestep size τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' T = Ntτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We employ a semi-implicit discretization in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Discretization in space follows the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We adaptively refine the diffuse interface and employ a parallelization approach which scales with the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' For details we refer to28,32–34,45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The algorithm is implemented in the open-source library AMDiS47,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 11/14 Detecting T1 transitions The T1 transitions are detected by tracking the neighbour relations of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' If two cells A and B are in contact, their neighbour relation is denoted by (A,B) or (B,A), both of which are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Suppose, there are four cells as in the Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The set of neighbour relations between these four cells before, during and after a T1 transition are {(A,B),(B,C),(C,D),(D,A),(B,D)}, {(A,B),(B,C),(C,D),(D,A)} and {(A,B),(B,C),(C,D),(D,A),(A,C)} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Before and after a T1 transition, there are 5 distinct neighbour relations between the four cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The sets of relations before and after a T1 transition have four elements in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' These common elements make up the set of relations during a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The duration of a T1 transition is time difference when the number of neighbour relations between the four cells change from 5 to 4 and back to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Sensitivity of fravg on ravg The coarse graining region of a point p is the region with all points x such that |p−x| < ravg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' As the free energy is high at the cell edges, the points which include the edges within its coarse graining region around it would have high fravg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Moreover, points with triple junctions (where 3 edges meet) within its coarse graining region would have a higher fravg due to the presence of longer total length of cell edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Usually at a given time, fravg has peaks near the T1 epicenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' This is because, the region around it would have either two triple junctions along with a gap as seen in the snapshots of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Also, it is clear that points that do not have any cell edges within its coarse graining region, would have zero fravg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We have found that increasing the ravg loses information about the T1 transition in the value of fravg at the epicenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A larger coarse graining region would entail a larger contribution from the bulk of the interior of the cell and would reduce fravg at the epicenters such that fravg at epicenters would not be uniquely discerned as a signature of a T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' On the other hand, reducing ravg would mean that we might not encompass the information of the two triple junctions and the gap formed during the T1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' It also increases the deviations in the statistics that we describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Moreover, if the energy along the length of the edge is uniform then the energy field fravg at a point gives an approximate measure of length of edges within the coarse graining region around that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Data availability All data are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' The AMDiS implementation and additional codes for pre- and postprocessing are available from the corresponding author upon reasonable request Supplementary Information Supplementary Movie 1 Supplementary Movie 2 Supplementary Movie 3 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Ladoux, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' & Mège, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Mechanobiology of collective cell behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} 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+page_content=', Praetorius, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' & Voigt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Software concepts and numerical algorithms for a scalable adaptive parallel finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 41, 1145–1177, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='1007/s10444-015-9405-4 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Acknowledgements This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' We acknowledge computing resources provided within project WIR at ZIH at TU Dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Author contributions statements H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' implemented the codes, performed all simulations, analysed data and contributed to conceptual development and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' contributed to supervision, conceptual development, data analysis and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' Additional information Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} +page_content=' 14/14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FKT4oBgHgl3EQfOS13/content/2301.11758v1.pdf'} diff --git a/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/2301.02599v1.pdf.txt b/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/2301.02599v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a84a778d1137155afb04cb9e4a87a719c2fb1fa2 --- /dev/null +++ b/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/2301.02599v1.pdf.txt @@ -0,0 +1,828 @@ +arXiv:2301.02599v1 [math.GM] 2 Jan 2023 +Wigner–Yanase–Dyson function and logarithmic mean +Shigeru Furuichi1∗ +1Department of Information Science, +College of Humanities and Sciences, Nihon University, +3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan +Abstract. +The ordering between Wigner–Yanase–Dyson function and logarithmic mean +is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse +inequalities for Wigner–Yanase–Dyson function and logarithmic mean. We also compare the +obtained results with the known bounds of the logarithmic mean. +Finally we give operator +inequalities based on the obtained results. +Keywords : +Wigner–Yanase–Dyson function, logarithmic mean, Kantorovich constant, +Specht ratio and reverse inequalities +2020 Mathematics Subject Classification : +Primary 26E60, Secondary 26D07. +1 +Introduction +In this paper, we study the ordering of the symmetric homogeneous means N(x, y) for x, y > 0. +The mean N(x, y) is called the symmetric homogeneous mean if the following conditions are +satisfied ([8]): +(i) N(x, y) = N(y, x). +(ii) N(kx, ky) = kN(x, y) for k > 0. +(iii) min{x, y} ≤ N(x, y) ≤ max{x, y}. +(iv) N(x, y) is non–decreasing in x and y. +Since we do not treat the weighted means, a symmetric homogeneous mean is often called a +mean simply in this paper. In order to determine the ordering of two means such as N1(x, y) ≤ +N2(x, y) for x, y > 0, it is sufficient to show the ordering N1(x, 1) ≤ N2(x, 1) for x > 0 by +homogeneity such that yN (x/y, 1) = N(x, y) for a symmetric homogeneous mean N(·, ·) and +x, y > 0. Throughout this paper, we use the standard symbol A(x, y) := x + y +2 +, L(x, y) := +x − y +log x − log y, (x ̸= y > 0) with L(x, x) := x, G(x, y) := √xy and H(x, y) := +2xy +x + y as the +arithmetic mean, logarithmic mean, geometric mean and harmonic mean, respectively. +The +Wigner–Yanase–Dyson function is given by +Wp (x, y) := +p (1 − p) (x − y)2 +(xp − yp) (x1−p − y1−p), (x ̸= y > 0, p ∈ R) , with Wp(x, x) = x +∗E-mail:furuichi.shigeru@nihon-u.ac.jp +1 + +which was firstly appeared in [13]. Since Wp(x, 1) is matrix monotone function on x ∈ (0, ∞) +when −1 ≤ p ≤ 2 [16], the parameter p is often considered to be −1 ≤ p ≤ 2. We mainly +consider the case of 0 ≤ p ≤ 1 in this paper, as it was done so in [4, 5, 6] to study the +Wigner–Yanase–Dyson metric with Morozova–Chentsov function or the Wigner–Yanase–Dyson +skew information. It is easily seen that W1−p(x, y) = Wp(x, y) and W1/2(x, y) = +�√x + √y +2 +�2 +which is called the Wigner–Yanase function or the binomial mean Bp(x, y) := +�xp + yp +2 +�1/p +with p = 1/2. It is also known that +H(x, y) ≤ G(x, y) ≤ L(x, y) ≤ Wp(x, y) ≤ W1/2(x, y) ≤ A(x, y), (x, y > 0, 0 ≤ p ≤ 1). +The set M(n, C) represents all n×n matrices on complex field. The set M+(n, C) represents +all positive semi–definite matrices in M(n, C). The stronger ordering N1(x, y) ⪯ N2(x, y) for +means N1 and N2 have been studied in [3, 7, 8, 11, 14] for the study of the unitarily invariant +norm inequalities and recent advances on the related topics. +It is known [8, 11] that the ordering N1(x, y) ⪯ N2(x, y) is equivalent to the unitarily +invariant norm inequality |||N1(S, T)X||| ≤ |||N1(S, T)X||| for S, T ∈ M+(n, C) and arbitrary +X ∈ M(n, C), implies the usual ordering N1(x, y) ≤ N2(x, y) which is equivalent to the Hilbert– +Schmidt (Frobenius) norm inequality ∥N1(S, T)X∥2 ≤ ∥N2(S, T)X∥2. See [8, 11] the precise +definition and equivalent conditions on the stronger ordering N1(x, y) ⪯ N2(x, y). We study the +usual ordering for some means in this paper. The following propositions are known. +Proposition 1.1. ([9]) For S, T ∈ M+(n, C) and any X ∈ M(n, C), if 1/2 ≤ p ≤ 1 ≤ q ≤ 2 or +−1 ≤ q ≤ 0 ≤ p ≤ 1/2, then we have +|||H(S, T)X||| ≤ |||Wq(S, T)X||| ≤ |||L(S, T)X||| ≤ |||Wp(S, T)X||| ≤ +������B1/2(S, T)X +������. +In particular, p ∈ [0, 1] =⇒ |||L(S, T)X||| ≤ |||Wp(S, T)X|||. +Proposition 1.2. ([1]) For S, T ∈ M+(n, C) and any X ∈ M(n, C),if |p| ≤ 1, then +��� +��� +��� ˆGp(S, T)X +��� +��� +��� ≤ |||L(S, T)X||| ≤ +��� +��� +��� ˆAp(S, T)X +��� +��� +���, +where ˆGp(x, y) := p(xy)p/2(x − y) +xp − yp +and ˆAp(x, y) := p(xp + yp)(x − y) +2(xp − xp) +for |p| ≤ 1 and x ̸= y. +See [1] for the details on ˆGp(x, y) and ˆAp(x, y). From Proposition 1.1, we see L(x, y) ≤ +Wp(x, y) for 0 ≤ p ≤ 1. In Section 2, we study the reverse inequalities of L(x, y) ≤ Wp(x, y). In +addition, we compare the obtained results in Section 2 with the bounds in Proposition 1.2, in +Section 3. +2 +Reverse inequalities +For x > 0, t > 0,we have ln−t x ≤ log x ≤ lnt x, where lnt x := xt − 1 +t +, (x > 0, t ̸= 0). Thus we +have the simple bounds of Wp, (0 ≤ p ≤ 1) as +Wp(x, 1) ≤ L(x, 1)2, (x ≥ 1), +Wp(x, 1) ≥ L(x, 1)2, (0 < x ≤ 1). +2 + +Since fp(t) := xpt log x is convex in t when x ≥ 1, 0 ≤ p ≤ 1, taking an account for +� 1 +0 fp(t)dt = lnp x, we have xp/2 log x ≤ lnp x ≤ +�xp + 1 +2 +� +log x from Hermite–Hadamard in- +equality. Thus the slightly improved upper bound was obtained under the condition x ≥ 1: +4 +(xp + 1)(x1−p + 1)L(x, 1)2 ≤ Wp(x, 1) ≤ +1 +√xL(x, 1)2, (x ≥ 1). +Also,we have the reverse inequality of the above for 0 < x ≤ 1 since fp(t) concave in t when +0 < x ≤ 1.In this section, we study the reverse inequalities of L(x, y) ≤ Wp(x, y) for all x > 0 +not restricted as x ≥ 1 or 0 < x ≤ 1. +We firstly consider the difference type reverse inequality of L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ +p ≤ 1). From the simple calculations, we have +Wp(x, 1) ≤ +�√x + 1 +2 +�2 +≤ r +�√x − 1 +�2+√x ≤ r +�√x − 1 +�2+L(x, 1), (x > 0, 0 ≤ p ≤ 1, r ≥ 1/4) +(1) +Considering the parameter p, we can obtain the first inequality in the following as a general +result. +Theorem 2.1. Let x > 0. For 0 ≤ p ≤ 1, we have +Wp(x, 1) ≤ p(1 − p) +�√x − 1 +�2 + L(x, 1) ≤ A(x, 1). +(2) +Proof. If the inequality (2) holds for x ≥ 1, then the inequality (2) holds for 0 < x ≤ 1. It is +easily seen by putting x := 1/y ≥ 1 in the proven inequality (2) for x ≥ 1. Thus it is sufficient +to prove inequality (2) for x ≥ 1 to show the inequality (2) for x > 0. +In (2), put x instead of √x. Then the denominator is +2p(1 − p)(x − 1)(x2p − 1)(x2(1−p) − 1) log x ≥ 0 +for x ≥ 1 when we reduce the difference right hand side minus the left hand side to a common +denominator. Also we set the numerator as f(x, p), namely +f(x, p) := (x + 1)(x2p − 1)(x2(1−p) − 1) + 2p(1 − p) +� +(x2p − 1)(x2(1−p) − 1) − (x + 1)2� +log x. +Since f(x, 1 − p) = f(x, p),we have only to prove f(x, p) ≥ 0 for x ≥ 1 and 0 ≤ p ≤ 1/2. We +calculate +df(x, p) +dp += 4(x1−2p + 1)(log x)g(x, p), +g(x, p) := h(x, p) + p(1 − p)(x − 1)(x − x2p) log x +h(x, p) := px2 − (1 − p)x2p+1 − px + x − px2p, +dh(x, p) +dx += 2px − (1 − p)(1 + 2p)x2p − 2p2x2p−1 + 1 − p, +d2h(x, p) +dx2 += 2p +� +−(1 − p)(1 + 2p)x2p−1 + p(1 − 2p)x2p−2 + 1 +� +, +d3h(x, p) +dx3 += 2p(1 − p)(1 − 2p)x2p−3 {2p(x − 1) + x} ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2), +so that we have +d2h(x, p) +dx2 +≥ d2h(1, p) +dx2 += 0 =⇒ dh(x, p) +dx +≥ dh(1, p) +dx += 0 =⇒ h(x, p) ≥ h(1, p) = 0. +3 + +From p(1 − p)(x − 1)(x − x2p) log x ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2) with the above results, we have +df(x, p) +dp +≥ 0 which implies f(x, p) ≥ f(x, 0) = 0. +To prove the second inequality, we set +k(x, p) := x + 1 +2 +− p(1 − p) +�√x − 1 +�2 − x − 1 +log x , (x > 1). +Then we have +k(x, p) ≥ k(x, 1/2) = 4 − 4x + (√x + 1)2 log x +4 log x +≥ 0. +Indeed, we have x − 1 +log x ≤ +�√x + 1 +2 +�2 +which implies 4−4x+(√x +1)2 log x ≥ 0 for x > 1. This +completes the proof with k(1, p) = 0. +For the special case p = 1/2 in Theorem 2.1, the inequalities in (2) are reduced to G(x, 1) ≤ +L(x, 1) ≤ B1/2(x, 1). Note that the right hand side of the second inequality in (2) can not be +replaced by W1/2(x, 1) which is less than or equal to A(x, 1). +Secondly we consider the ratio type reverse inequality of L(x, y) ≤ Wp(x, y). From the known +wesults, we have +Wp(x, 1) ≤ +�√x + 1 +2 +�2 +≤ A(x, 1) ≤ S(x)G(x, 1) ≤ S(x)L(x, 1), (x > 0, 0 ≤ p ≤ 1). +(3) +Where S(x) := +x +1 +x−1 +e log x +1 +x−1 +is Specht ratio [15]. From the relation K(x) := (x + 1)2 +4x +≥ S(x), +Specht ratio in (3) can be replaced by Kantorovich constant K(x) [10]. See [2, Chapter 2] and +references therein for the recent results on the inequalities with Specht ratio and Kantorovich +constant. Moreover we have the following inequality if we use Kantorovich constant K(x). +Wp(x, 1) ≤ +�√x + 1 +2 +�2 += K(√x)√x ≤ K(√x)L(x, 1), (x > 0, 0 ≤ p ≤ 1) +(4) +From (3) and (4), it may be expected that +�√x + 1 +2 +�2 +≤ S(√x)√x. However, this fails. +Indeed we have the following proposition. In this point, we see that the ordering K(x) ≥ S(x) +is effective. +Proposition 2.2. For x > 0, +�√x + 1 +2 +�2 +≥ S(√x)√x. +(5) +Proof. When x = 1, we have equality of (5) since S(1) = 1. The inequality (5) is equivalent to +the following inequality: +(x − 1)x +x +x−1 +e log x +≤ +�x + 1 +2 +�2 +. +(6) +By the similar reason as we stated in the beginning of the proof in Theorem 2.1, it is sufficient +to prove (6) for x > 1. Taking a logarithm of both sides in (6) and considering its difference: +f(x) := 2 log +�x + 1 +2 +� +− log(x − 1) − +x +x − 1 log x + 1 + log (log x) . +4 + +Since L(x, 1) ≥ H(x, 1) and L(x, 1)−1 ≥ A(x, 1)−1 for x > 0, we have +f ′(x) = +1 +x(x − 1) +�x − 1 +log x + x log x +x − 1 − +4x +x + 1 +� +≥ 0, (x > 1). +Thus we have f(x) ≥ f(1) = 0. +It is notable that the inequality (5) can be also obtaind by putting v = 1/2 in [2, Theorem +2.10.1], taking a square the both sides and then replacing x by √x. +The following result is the ratio type reverse inequality of L(x, y) ≤ Wp(x, y) for 0 ≤ p ≤ 1. +Theorem 2.3. For x > 0, 0 ≤ p ≤ 1, we have +Wp(x, 1) ≤ K(x)p(1−p)L(x, 1). +(7) +Proof. For x = 1, we have equality in (7). So it is sufficient to prove (7) for x > 1. Take a +logarithm of the both sides in (7) and put the function f(x, p) as its difference, namely +f(x, p) +:= +− log(x − 1) − log (log x) + 2p(1 − p) log(x + 1) − p(1 − p) log 4x +− log p − log(1 − p) + log(xp − 1) + log(x1−p − 1). +We calculate +df(x, p) +dx += +− +1 +x − 1 − p(1 − p) +x ++ 2p(1 − p) +x + 1 ++ pxp−1 +xp − 1 + p − 1 +xp − x + +1 +x log x += +− +1 +x − 1 + p(1 − p) (x − 1) +x(x + 1) + +1 +x1−p lnp x + +1 +xp ln1−p x + +1 +x log x +≥ +− +1 +x − 1 + +1 +x1−p lnp x + +1 +xp ln1−p x =: g(x, p) +and +g(x, p) = +h(x, p) +(x − 1)(xp − 1)(x − xp) ≥ 0, (x > 1, 0 ≤ p ≤ 1). +Indeed, +h(x, p) : += +−(xp − 1)(x − xp) + pxp−1(x − 1)(x − xp) + (1 − p)(x − 1)(x − xp) += +(1 − p) + px − 2xp + (1 − p)x2p + px2p−1 += +(1 − p)(1 + x2p) + p(x + x2p−1) − 2xp +≥ +(1 − p) × 2xp + p × 2xp − 2xp = 0. +Therefore we have f(x, p) ≥ f(1, p) = 0. +Remark 2.4. It is natural to consider the replacement K(x) by S(x) in (7). However we have +not prove +Wp(x, 1) ≤ S(x)p(1−p)L(x, 1), +(x > 0, 0 ≤ p ≤ 1). +We also have not found any counter-example of the above inequality. +It is known that S(x) ≤ K(x), S(xr) ≤ S(x)r and K(xr) ≤ K(x)r for x > 0 and 0 ≤ r ≤ 1 +[2, Section 2.10]. Also it is known that both K(x) and S(x) are decreasing for 0 < x < 1 and +increasing x > 1 with S(1) = K(1) = 1. +We are interested to find the smaller constant depending p ∈ [0, 1] than K(x)p(1−p). By the +numerical computations we found the counter-example for +Wp(x, 1) ≤ K(xp)L(x, 1), +(x > 0, 0 ≤ p ≤ 1). +Thus the inequalities Wp(x, 1) ≤ S(xp)L(x, 1) and Wp(x, 1) ≤ K(xp(1−p))L(x, 1) do not hold in +general. +5 + +3 +Comparisons of the bounds for logarithmic mean +From Proposition 1.1 and Theorem 2.3, we have +K(x)−p(1−p)Wp(x, 1) ≤ L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ p ≤ 1). +(8) +On the other hand, from Proposition 1.2, we have +ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1), (x > 0, 0 ≤ p ≤ 1). +(9) +In this section, we compare the bounds of L(x, 1) in (8) and (9). The first result is the +comparison on the upper bounds of L(x, 1). +Theorem 3.1. Let x > 0 and p ∈ R. +(i) If p ∈ [0, 1/2], then Wp(x, 1) ≥ ˆAp(x, 1). +(ii) If p /∈ (0, 1/2), then Wp(x, 1) ≤ ˆAp(x, 1). +Proof. It is sufficient to prove for x ≥ 1. Taking account of x1−p − 1 +1 − p +≥ 0 for x ≥ 1, we set +fp(x) := 2(x − 1) − (xp + 1)(x1−p − 1) +(1 − p) += +� +2 − +1 +1 − p +� +(x − 1) + xp − x1−p +1 − p +. +Since we have +f ′ +p(x) = +� +2 − +1 +1 − p +� ++ pxp−1 − (1 − p)x−p +1 − p +, f ′′ +p (x) = px−p−1(1 − x2p−1), +we have f ′′ +p (x) ≥ 0 for p ∈ [0, 1/2].Thus we have f ′ +p(x) ≥ f ′ +p(1) = 0 which implies fp(x) ≥ +fp(1) = 0. Therefore we obtain (i). Similarly we have f ′′ +p (x) ≤ 0 for p /∈ (0, 1/2). Thus we have +f ′ +p(x) ≤ f ′ +p(1) = 0 which implies fp(x) ≤ fp(1) = 0. Therefore we obtain (ii). +The second result is the comparison on the lower bounds of L(x, 1). To prove it, we prepare +the following lemma which is interesting itself. +Lemma 3.2. For x > 0, we have +L(x, 1)2 − G(x, 1)2 +2L(x, 1)2 +≤ A(x, 1) − L(x, 1) +L(x, 1) +≤ log K(x). +(10) +Proof. It is sufficient to prove (10) for x ≥ 1. We firstly prove the second inequality. To this +end, we set +u(x) := 2(x − 1) − (x + 1) log x + 4(x − 1) log(x + 1) − 2(x − 1) log(4x), (x ≥ 1). +We calculate +u′(x) = +4x +x + 1 − +4 +x + 1 + 1 +x − 1 − 3 log x + 4 log(x + 1) − 4 log 2 +u′′(x) = (x − 1)(x2 + 6x + 1) +x2(x + 1)2 +≥ 0. +Thus we have +u′(x) ≥ u′(1) = 0 =⇒ u(x) ≥ u(1) = 0. +6 + +We secondly prove the first inequality. To this end, we set +v(x) := (x2 − 1) log x + x(log x)2 − 3(x − 1)2, +(x ≥ 1). +We calculate +v′(x) = 2(x + 1) log x + (log x)2 − 5x + 6 − 1 +x, +v′′(x) = 1 +x2 +� +2x(x + 1) log x − 3x2 + 2x + 1 +� +v(3)(x) = 2 +x3 w(x), +w(x) := x2 − 1 − x log x, +w′(x) = 2x − 1 − log x ≥ 0. +Thus we have +w(x) ≥ w(1) = 0 =⇒ v(3)(x) ≥ 0 =⇒ v′′(x) ≥ v′′(1) = 0 =⇒ v′(x) ≥ v′(1) = 0 =⇒ v(x) ≥ v(1) = 0. +Therefore we obtain 3L(x, 1) ≤ 2A(x, 1) + G(x, 1)2/L(x, 1), (x > 0) which is equivalent to the +first inequality of (10). +Here, the second inequality of (10) is equivalent to the inequality: +1 − (x + 1) log x +2(x − 1) ++ log +�(x + 1)2 +4x +� +≥ 0 +(11) +Also the inequality L(x, 1)2 − G(x, 1)2 +2L(x, 1)2 +≤ log K(x) is equivalent to the inequality: +1 − x(log x)2 +(x − 1)2 + 2 log +� +4x +(x + 1)2 +� +≤ 0. +(12) +The inequalities (11) and (12) will be used in the proof of Theorem 3.3 below. +Theorem 3.3. Let x > 0 and p ∈ R. +(i) If 0 ≤ p ≤ 1 +2, then ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1). +(ii) If p ≤ 0 or p ≥ 1, then ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1). +Proof. It is sufficient to prove for x ≥ 1. Since K(x) ≥ 1, in order to prove (i), +ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1) ⇐⇒ ˆGp(x, 1)K(x)p(1−p) ≥ Wp(x, 1) +⇐⇒ x +p +2 +�(x + 1)2 +4x +�p(1−p) +≥ (1 − p)(x − 1) +x1−p − 1 +we set +fp(x) := p +2 log x + p(1 − p) {2 log(x + 1) − log(4x)} − log(1 − p) − log(x − 1) + log(x1−p − 1). +Then we have +f ′ +p(x) += +− +1 +x − 1 + p +2x + p(1 − p)(x − 1) +x(x + 1) ++ 1 − p +x − xp += +gp(x) +2(x − 1)(x + 1)(x − xp) +7 + +where +gp(x) := 2(x + 1)(xp − 1 − p(x − 1)) + p(x − 1)(1 − xp−1) ((3 − 2p)x + (2p − 1)) . +When x ≥ 1 and 0 ≤ p ≤ 1/2, we have 1 ≥ xp−1, −xp−3 ≥ −xp−2 and (1−p)(1−2p)(p−2) ≤ 0. +Also when x ≥ 1 and 0 ≤ p ≤ 1/2, we have xp−1 ≥ xp−2. Thus we calculate +g′ +p(x) = (p + 1)(2p2 − 3p + 2)xp − 2p(2p2 − 2p − 1)xp−1 + p(1 − p)(1 − 2p)xp−2 ++2p(1 − 2p)x + 2(2p2 − 2p − 1) +g′′ +p(x) = p +� +(p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2 ++(1 − p)(1 − 2p)(p − 2)xp−3 + 2(1 − 2p) +� +≥ p +� +(p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − 2p)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2 ++(1 − p)(1 − 2p)(p − 2)xp−2� += (2p3 − p2 − 5p + 4)(xp−1 − xp−2) = 2(1 − p) {(1 − p)(1 + p) + 1 − p/2} (xp−1 − xp−2) ≥ 0. +Thus we have g′ +p(x) ≥ g′ +p(1) = 0 so that gp(x) ≥ gp(1) = 0. Therefore we have f ′ +p(x) ≥ 0 for +x ≥ 1 and taking an accout for lim +x→1 +(1 − p)(x − 1) +x1−p − 1 += 1, we havefp(x) ≥ fp(1) = 0 which proves +(i). +It is also sufficent to prove (ii) for x ≥ 1. For the special cases p = 0 or p = 1 we have +equality. Since +ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1) ⇐⇒ x +p +2 +�(x + 1)2 +4x +�p(1−p) +≤ (1 − p)(x − 1) +x1−p − 1 +, +we have only to prove fp(x) ≤ 0 for x ≥ 1. +(a) We consider the case p > 1. We set g′′ +p(x) = p · hp(x), namely +hp(x) := (p+1)(2p2−3p+2)xp−1+2(1−p)(2p2−2p−1)xp−2+(1−p)(1−2p)(p−2)xp−3+2(1−2p). +Then +h′ +p(x) = (p − 1)xp−4kp(x), +where +kp(x) := 2p3(x − 1)2 − p2(x − 1)(x − 11) − p(x2 + 6x − 17) + 2(x − 3)(x + 1) +and we have +k′ +p(x) = 4p3(x−1)−2p2(x−6)−2p(x+3)+4(x−1), k′′ +p(x) = 2(p+1)(2p2−3p+2) > 0, (p > 1). +Thus we have k′ +p(x) ≥ k′ +p(1) = 10p2 − 8p > 0, (p > 1) so that kp(x) ≥ kp(1) = 10p − 8 > +0, (p > 1). Therefore we have h′ +p(x) ≥ 0, (p > 1) which implies hp(x) ≥ hp(1) = 0. Thus +we have g′′ +p(x) ≥ 0 so that we have g′ +p(x) ≥ g′ +p(1) = 0 which implies gp(x) ≥ gp(1) = 0. +Taking account of x − xp ≤ 0 when x ≥ 1, p > 1, we have f ′ +p(x) ≤ 0 Therefore we have +fp(x) ≤ fp(1) = 0 which proves (ii) for the case p > 1. +(b) We consider the case p < 0. We calculate +dfp(x) +dp += +1 +1 − p + +�1 +2 + +x +xp − x +� +log x + (2p − 1) log(4x) + (2 − 4p) log(x + 1), +d2fp(x) +dp2 += −xp+1(log x)2 +(x − xp)2 ++ +1 +(p − 1)2 + 2 log(4x) − 4 log(x + 1), +d3fp(x) +dp3 += xp+1 (xp + x) (log x)3 +(xp − x)3 +− +2 +(p − 1)3 . +8 + +We further calculate +d +dx +�d3fp(x) +dp3 +� += −xp(log x)2 +(x − xp)4 s(x, p), +s(x, p) := 3(x2 − x2p) + (p − 1) +� +x2 + x2p + 4x1+p� +log x, +ds(x, p) +dp += +� +x2 − 5x2p + 4xp+1 + 2(p − 1) +� +x2p + 2xp+1� +log(x) +� +log x, +d2s(x, p) +dp2 += 4xp(log x)2 {2(x − xp) + (p − 1)(x + xp) log x} ≤ 0 (p ≤ 1, x ≥ 1). +Indeed, putting a := x, b := xp in the inequality +a − b +log a − log b ≤ a + b +2 +, (a, b > 0), we have +2(x−xp)+(p−1)(x+xp) log x ≤ 0 for p < 1 and x ≥ 1. (The equality holds when p = 1.) +From d2s(x, p) +dp2 +≤ 0, we have ds(x, p) +dp +≥ ds(x, p) +dp +���� +p=1 += 0 which implies s(x, p) ≤ s(x, 1) = +0 so that we have d +dx +�d3fp(x) +dp3 +� +≥ 0. From this, we have +d3fp(x) +dp3 +≥ d3fp(1) +dp3 += − +2 +(p − 1)3 > 0, (p < 1). +By the inequality (12), we have +p ≤ 0 =⇒ d2fp(x) +dp2 +≤ d2fp(x) +dp2 +���� +p=0 += 1 − x(log x)2 +(x − 1)2 + 2 log +� +4x +(x + 1)2 +� +≤ 0. +(13) +Thus we have by the inequality (11), +p ≤ 0 =⇒ dfp(x) +dp +≥ dfp(x) +dp +���� +p=0 += 1 − (x + 1) log x +2(x − 1) ++ log +�(x + 1)2 +4x +� +≥ 0. +Therefore p ≤ 0 =⇒ fp(x) ≤ f0(x) = 0 which proves (ii) for the case p < 0. +Remark 3.4. +(i) Since fp(x) = log +��(x + 1)2 +4x +�p(1−p) +× xp/2(x1−p − 1) +(1 − p)(x − 1) +� +appeared in the +proof of Theorem 3.3 and comparing the maximum degree of the numerator and the +denominator insides of the logarithmic function, we found that if p < 0 or p > 1/2, then +we have +lim +x→∞ +��(x + 1)2 +4x +�p(1−p) +× xp/2(x1−p − 1) +(1 − p)(x − 1) +� += 0. +Thus we have lim +x→∞ fp(x) = −∞ if p < 0 or p > 1/2. On the other hand, by the muner- +ical computations, we have f3/4(e) ≃ 0.0063209. Therefore there is no ordering between +K(x)−p(1−p)Wp(x, 1) and ˆGp(x, 1) for x > 0 and 1/2 < p < 1. +(ii) From Theorem 3.1 (i) and Theorem 3.3 (i), we have for x > 0 and 0 ≤ p ≤ 1/2, +K(x)−p(1−p)Wp(x, 1) ≤ ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1) ≤ Wp(x, 1). +(iii) From the proof (b) in Theorem 3.3, we obtained d3fp(x) +dp3 +≥ 0, (p < 1). From this with +simple calculations, we have +H(x, xp)3 ≤ xp+1A(x, xp) ≤ L(x, xp)3, (p ≤ 1, x > 0). +9 + +4 +Conclusion +As we have seen, we studied the inequalities on the relations between the Wigner–Yanase– +Dyson function Wp(·, ·) and the logarithmic mean L(·, ·). As one of main results, we obtained +two kinds of the reverse inequalities for L(x, y) ≤ Wp(x, y) for x, y > 0 and 0 ≤ p ≤ 1. That +is, the inequalities (2) and (7) shown in Theorem 2.1 and 2.3 are respectively equivalent to the +following inequalities for x, y > 0 and 0 ≤ p ≤ 1: +Wp(x, y) ≤ p(1 − p) +�√x − √y +�2 + L(x, y), +(14) +Wp(x, y) ≤ K (x/y)p(1−p) L(x, y). +(15) +The inequality (14) and (15) are the difference type reverse inequality and the ratio type reverse +inequality for L(x, y) ≤ Wp(x, y), (0 ≤ p ≤ 1), respectively. +In addition, we compared the obtained inequality (15) with the known result in Section 3. +It is summerized in the following. The inequalities given in Remark 3.4 (ii) are equivalent to +the following inequalities for x, y > 0 and 0 ≤ p ≤ 1/2: +K (x/y)−p(1−p) Wp(x, y) ≤ ˆGp(x, y) ≤ L(x, y) ≤ ˆAp(x, y) ≤ Wp(x, y). +(16) +We conclude this paper by giving operator inequalities baesd on Theorem 2.1 and 2.3. For +positive operators S, T and 0 ≤ p ≤ 1, we define the operator version of the logarithmic mean +and the Wigner–Yanase–Dyson function as +L(S, T) := +� 1 +0 +S♯tTdt, +Wp(S, T) := p(1 − p) +2 +(S − T) (S∇T − Hzp(S, T))−1 (S − T), (S ̸= T), +Wp(S, S) := S. +where +S♯pT := S1/2 � +S−1/2TS−1/2�p +S1/2, S∇T := S + T +2 +, Hzp(S, T) := 1 +2 (S♯pT + S♯1−pT) . +We often use the symbol S♯T := S♯1/2T for short. Hzp(S, T) is often called the Heinz mean. It +is notable that we have the relation for the validity in the case p := 1/2, +�S − T +2 +� +(S∇T + S♯T)−1 +�S − T +2 +� += S∇T − S♯T, +which can be confirmed by multiplying S−1/2 to both sides. +From Theorem 2.1, we have the following corollary. +Corollary 4.1. Let S and T be positive operators and let 0 ≤ p ≤ 1. Then we have +Wp(S, T) ≤ 2p(1 − p) (S∇T − S♯T) + L(S, T). +From Theorem 2.3, we also have the following corollary. +Corollary 4.2. Let S and T be positive operators with αS ≤ T ≤ βS for 0 < α ≤ β and let +0 ≤ p ≤ 1. Then we have +Wp(S, T) ≤ kp · L(S, T), +kp := max +α≤x≤β K(x)p(1−p). +10 + +Acknowledgement +The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 21K03341. +References +[1] S.Furuichi, +Unitarily +invariant +norm +inequalities +for +some +means, +J.Inequal.Appl.,2014(2014), Art.158. +[2] S.Furuichi and H.R.Moradi, Advances in mathematical inequalities, De Gruyter, 2020. +[3] S.Furuichi and M.E.Amlashi, On bounds of logarithmic mean and mean inequality chain, +arXiv:2203.01134. +[4] S.Furuichi and K.Yanagi, Schr¨odinger uncertainty relation, Wigner–Yanase–Dyson skew +information and metric adjusted correlation measure, J. Math. Anal. Appl.,388(2)(2012), +1147–1156. +[5] P. Gibilisco, F.Hansen and T. Isola, On a correspondence between regular and non-regular +operator monotone functions, Linear Alg. Appl., 430 (2009) 2225–2232. +[6] F.Hansen, Metric adjusted skew information, Proc. Nat.Acad. Sci.,105(2008), 9909–9916. +[7] F. Hiai and H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ. +Math. J. 48 (1999) 899–936. +[8] F.Hiai and H.Kosaki, Means of Hilbert space operators, Springer–Verlag, 2003. +[9] F.Hiai, H.Kosaki,D.Petz and B.Ruskai Families of completely positive maps associated with +monotone metrics, Linear Alg. Appl., 48(439)(2013), 1749–1791. +[10] L. V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat. Nauk, +3:6(28) (1948), 89–185. http://mi.mathnet.ru/eng/umn/v3/i6/p89. +[11] H. Kosaki, Positive definiteness of functions with applications to operator norm inequalities, +Mem. Amer. Math. Soc., 212(997), 2011. +[12] H.Kosaki, Strong monotonicity for various means, J.Func.Anal.,267(2014),1917–1958. +[13] D.Petz and H.Hasegawa, On the Riemannian metric of α–entropies of density matrices, +Lett.Math.Phys.,38(1996), 221-225. +[14] H.Kosaki, Positive definiteness and infinite divisibility of certain functions of hyperbolic +cosine function, Int. J. Math.,33(7) (2022), 2250050. +[15] W.Specht, +Zur +Theorie +der +elementaren +Mittel, +Math.Z, +74 +(1960), +91–98. +10.1007/BF01180475. +[16] V.E.S.Szab´o, A class of matrix monotone functions, Linear Alg. Appl.,420(2007), 79–85. +11 + diff --git a/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/load_file.txt b/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..475861eb2e086cdf254d1c6d632499601c9e681d --- /dev/null +++ b/0dE0T4oBgHgl3EQfuAFm/content/tmp_files/load_file.txt @@ -0,0 +1,354 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf,len=353 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='02599v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='GM] 2 Jan 2023 Wigner–Yanase–Dyson function and logarithmic mean Shigeru Furuichi1∗ 1Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The ordering between Wigner–Yanase–Dyson function and logarithmic mean is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Also bounds for logarithmic mean are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In this paper, we give two reverse inequalities for Wigner–Yanase–Dyson function and logarithmic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We also compare the obtained results with the known bounds of the logarithmic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Finally we give operator inequalities based on the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Keywords : Wigner–Yanase–Dyson function, logarithmic mean, Kantorovich constant, Specht ratio and reverse inequalities 2020 Mathematics Subject Classification : Primary 26E60, Secondary 26D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 1 Introduction In this paper, we study the ordering of the symmetric homogeneous means N(x, y) for x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The mean N(x, y) is called the symmetric homogeneous mean if the following conditions are satisfied ([8]): (i) N(x, y) = N(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (ii) N(kx, ky) = kN(x, y) for k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (iii) min{x, y} ≤ N(x, y) ≤ max{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (iv) N(x, y) is non–decreasing in x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since we do not treat the weighted means, a symmetric homogeneous mean is often called a mean simply in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In order to determine the ordering of two means such as N1(x, y) ≤ N2(x, y) for x, y > 0, it is sufficient to show the ordering N1(x, 1) ≤ N2(x, 1) for x > 0 by homogeneity such that yN (x/y, 1) = N(x, y) for a symmetric homogeneous mean N(·, ·) and x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Throughout this paper, we use the standard symbol A(x, y) := x + y 2 , L(x, y) := x − y log x − log y, (x ̸= y > 0) with L(x, x) := x, G(x, y) := √xy and H(x, y) := 2xy x + y as the arithmetic mean, logarithmic mean, geometric mean and harmonic mean, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The Wigner–Yanase–Dyson function is given by Wp (x, y) := p (1 − p) (x − y)2 (xp − yp) (x1−p − y1−p), (x ̸= y > 0, p ∈ R) , with Wp(x, x) = x ∗E-mail:furuichi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='shigeru@nihon-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='jp 1 which was firstly appeared in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since Wp(x, 1) is matrix monotone function on x ∈ (0, ∞) when −1 ≤ p ≤ 2 [16], the parameter p is often considered to be −1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We mainly consider the case of 0 ≤ p ≤ 1 in this paper, as it was done so in [4, 5, 6] to study the Wigner–Yanase–Dyson metric with Morozova–Chentsov function or the Wigner–Yanase–Dyson skew information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is easily seen that W1−p(x, y) = Wp(x, y) and W1/2(x, y) = �√x + √y 2 �2 which is called the Wigner–Yanase function or the binomial mean Bp(x, y) := �xp + yp 2 �1/p with p = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is also known that H(x, y) ≤ G(x, y) ≤ L(x, y) ≤ Wp(x, y) ≤ W1/2(x, y) ≤ A(x, y), (x, y > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The set M(n, C) represents all n×n matrices on complex field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The set M+(n, C) represents all positive semi–definite matrices in M(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The stronger ordering N1(x, y) ⪯ N2(x, y) for means N1 and N2 have been studied in [3, 7, 8, 11, 14] for the study of the unitarily invariant norm inequalities and recent advances on the related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is known [8, 11] that the ordering N1(x, y) ⪯ N2(x, y) is equivalent to the unitarily invariant norm inequality |||N1(S, T)X||| ≤ |||N1(S, T)X||| for S, T ∈ M+(n, C) and arbitrary X ∈ M(n, C), implies the usual ordering N1(x, y) ≤ N2(x, y) which is equivalent to the Hilbert– Schmidt (Frobenius) norm inequality ∥N1(S, T)X∥2 ≤ ∥N2(S, T)X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' See [8, 11] the precise definition and equivalent conditions on the stronger ordering N1(x, y) ⪯ N2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We study the usual ordering for some means in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The following propositions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' ([9]) For S, T ∈ M+(n, C) and any X ∈ M(n, C), if 1/2 ≤ p ≤ 1 ≤ q ≤ 2 or −1 ≤ q ≤ 0 ≤ p ≤ 1/2, then we have |||H(S, T)X||| ≤ |||Wq(S, T)X||| ≤ |||L(S, T)X||| ≤ |||Wp(S, T)X||| ≤ ������B1/2(S, T)X ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In particular, p ∈ [0, 1] =⇒ |||L(S, T)X||| ≤ |||Wp(S, T)X|||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' ([1]) For S, T ∈ M+(n, C) and any X ∈ M(n, C),if |p| ≤ 1, then ��� ��� ��� ˆGp(S, T)X ��� ��� ��� ≤ |||L(S, T)X||| ≤ ��� ��� ��� ˆAp(S, T)X ��� ��� ���, where ˆGp(x, y) := p(xy)p/2(x − y) xp − yp and ˆAp(x, y) := p(xp + yp)(x − y) 2(xp − xp) for |p| ≤ 1 and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' See [1] for the details on ˆGp(x, y) and ˆAp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1, we see L(x, y) ≤ Wp(x, y) for 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In Section 2, we study the reverse inequalities of L(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In addition, we compare the obtained results in Section 2 with the bounds in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 2 Reverse inequalities For x > 0, t > 0,we have ln−t x ≤ log x ≤ lnt x, where lnt x := xt − 1 t , (x > 0, t ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have the simple bounds of Wp, (0 ≤ p ≤ 1) as Wp(x, 1) ≤ L(x, 1)2, (x ≥ 1), Wp(x, 1) ≥ L(x, 1)2, (0 < x ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 2 Since fp(t) := xpt log x is convex in t when x ≥ 1, 0 ≤ p ≤ 1, taking an account for � 1 0 fp(t)dt = lnp x, we have xp/2 log x ≤ lnp x ≤ �xp + 1 2 � log x from Hermite–Hadamard in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus the slightly improved upper bound was obtained under the condition x ≥ 1: 4 (xp + 1)(x1−p + 1)L(x, 1)2 ≤ Wp(x, 1) ≤ 1 √xL(x, 1)2, (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Also,we have the reverse inequality of the above for 0 < x ≤ 1 since fp(t) concave in t when 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='In this section, we study the reverse inequalities of L(x, y) ≤ Wp(x, y) for all x > 0 not restricted as x ≥ 1 or 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We firstly consider the difference type reverse inequality of L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From the simple calculations, we have Wp(x, 1) ≤ �√x + 1 2 �2 ≤ r �√x − 1 �2+√x ≤ r �√x − 1 �2+L(x, 1), (x > 0, 0 ≤ p ≤ 1, r ≥ 1/4) (1) Considering the parameter p, we can obtain the first inequality in the following as a general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Let x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For 0 ≤ p ≤ 1, we have Wp(x, 1) ≤ p(1 − p) �√x − 1 �2 + L(x, 1) ≤ A(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' If the inequality (2) holds for x ≥ 1, then the inequality (2) holds for 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is easily seen by putting x := 1/y ≥ 1 in the proven inequality (2) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus it is sufficient to prove inequality (2) for x ≥ 1 to show the inequality (2) for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In (2), put x instead of √x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then the denominator is 2p(1 − p)(x − 1)(x2p − 1)(x2(1−p) − 1) log x ≥ 0 for x ≥ 1 when we reduce the difference right hand side minus the left hand side to a common denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Also we set the numerator as f(x, p), namely f(x, p) := (x + 1)(x2p − 1)(x2(1−p) − 1) + 2p(1 − p) � (x2p − 1)(x2(1−p) − 1) − (x + 1)2� log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since f(x, 1 − p) = f(x, p),we have only to prove f(x, p) ≥ 0 for x ≥ 1 and 0 ≤ p ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We calculate df(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dp = 4(x1−2p + 1)(log x)g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) := h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) + p(1 − p)(x − 1)(x − x2p) log x h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) := px2 − (1 − p)x2p+1 − px + x − px2p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' dh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx = 2px − (1 − p)(1 + 2p)x2p − 2p2x2p−1 + 1 − p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' d2h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx2 = 2p � −(1 − p)(1 + 2p)x2p−1 + p(1 − 2p)x2p−2 + 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' d3h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx3 = 2p(1 − p)(1 − 2p)x2p−3 {2p(x − 1) + x} ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (x ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 0 ≤ p ≤ 1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' so that we have d2h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx2 ≥ d2h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx2 = 0 =⇒ dh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx ≥ dh(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) dx = 0 =⇒ h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) ≥ h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 3 From p(1 − p)(x − 1)(x − x2p) log x ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2) with the above results, we have df(x, p) dp ≥ 0 which implies f(x, p) ≥ f(x, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' To prove the second inequality, we set k(x, p) := x + 1 2 − p(1 − p) �√x − 1 �2 − x − 1 log x , (x > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then we have k(x, p) ≥ k(x, 1/2) = 4 − 4x + (√x + 1)2 log x 4 log x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Indeed, we have x − 1 log x ≤ �√x + 1 2 �2 which implies 4−4x+(√x +1)2 log x ≥ 0 for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' This completes the proof with k(1, p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For the special case p = 1/2 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1, the inequalities in (2) are reduced to G(x, 1) ≤ L(x, 1) ≤ B1/2(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Note that the right hand side of the second inequality in (2) can not be replaced by W1/2(x, 1) which is less than or equal to A(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Secondly we consider the ratio type reverse inequality of L(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From the known wesults, we have Wp(x, 1) ≤ �√x + 1 2 �2 ≤ A(x, 1) ≤ S(x)G(x, 1) ≤ S(x)L(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (3) Where S(x) := x 1 x−1 e log x 1 x−1 is Specht ratio [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From the relation K(x) := (x + 1)2 4x ≥ S(x), Specht ratio in (3) can be replaced by Kantorovich constant K(x) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' See [2, Chapter 2] and references therein for the recent results on the inequalities with Specht ratio and Kantorovich constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Moreover we have the following inequality if we use Kantorovich constant K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Wp(x, 1) ≤ �√x + 1 2 �2 = K(√x)√x ≤ K(√x)L(x, 1), (x > 0, 0 ≤ p ≤ 1) (4) From (3) and (4), it may be expected that �√x + 1 2 �2 ≤ S(√x)√x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' However, this fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Indeed we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In this point, we see that the ordering K(x) ≥ S(x) is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For x > 0, �√x + 1 2 �2 ≥ S(√x)√x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' When x = 1, we have equality of (5) since S(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The inequality (5) is equivalent to the following inequality: (x − 1)x x x−1 e log x ≤ �x + 1 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (6) By the similar reason as we stated in the beginning of the proof in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1, it is sufficient to prove (6) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Taking a logarithm of both sides in (6) and considering its difference: f(x) := 2 log �x + 1 2 � − log(x − 1) − x x − 1 log x + 1 + log (log x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 4 Since L(x, 1) ≥ H(x, 1) and L(x, 1)−1 ≥ A(x, 1)−1 for x > 0, we have f ′(x) = 1 x(x − 1) �x − 1 log x + x log x x − 1 − 4x x + 1 � ≥ 0, (x > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have f(x) ≥ f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is notable that the inequality (5) can be also obtaind by putting v = 1/2 in [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1], taking a square the both sides and then replacing x by √x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The following result is the ratio type reverse inequality of L(x, y) ≤ Wp(x, y) for 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For x > 0, 0 ≤ p ≤ 1, we have Wp(x, 1) ≤ K(x)p(1−p)L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For x = 1, we have equality in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' So it is sufficient to prove (7) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Take a logarithm of the both sides in (7) and put the function f(x, p) as its difference, namely f(x, p) := − log(x − 1) − log (log x) + 2p(1 − p) log(x + 1) − p(1 − p) log 4x − log p − log(1 − p) + log(xp − 1) + log(x1−p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We calculate df(x, p) dx = − 1 x − 1 − p(1 − p) x + 2p(1 − p) x + 1 + pxp−1 xp − 1 + p − 1 xp − x + 1 x log x = − 1 x − 1 + p(1 − p) (x − 1) x(x + 1) + 1 x1−p lnp x + 1 xp ln1−p x + 1 x log x ≥ − 1 x − 1 + 1 x1−p lnp x + 1 xp ln1−p x =: g(x, p) and g(x, p) = h(x, p) (x − 1)(xp − 1)(x − xp) ≥ 0, (x > 1, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Indeed, h(x, p) : = −(xp − 1)(x − xp) + pxp−1(x − 1)(x − xp) + (1 − p)(x − 1)(x − xp) = (1 − p) + px − 2xp + (1 − p)x2p + px2p−1 = (1 − p)(1 + x2p) + p(x + x2p−1) − 2xp ≥ (1 − p) × 2xp + p × 2xp − 2xp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we have f(x, p) ≥ f(1, p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is natural to consider the replacement K(x) by S(x) in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' However we have not prove Wp(x, 1) ≤ S(x)p(1−p)L(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We also have not found any counter-example of the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is known that S(x) ≤ K(x), S(xr) ≤ S(x)r and K(xr) ≤ K(x)r for x > 0 and 0 ≤ r ≤ 1 [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Also it is known that both K(x) and S(x) are decreasing for 0 < x < 1 and increasing x > 1 with S(1) = K(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We are interested to find the smaller constant depending p ∈ [0, 1] than K(x)p(1−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' By the numerical computations we found the counter-example for Wp(x, 1) ≤ K(xp)L(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus the inequalities Wp(x, 1) ≤ S(xp)L(x, 1) and Wp(x, 1) ≤ K(xp(1−p))L(x, 1) do not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 5 3 Comparisons of the bounds for logarithmic mean From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3, we have K(x)−p(1−p)Wp(x, 1) ≤ L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (8) On the other hand, from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2, we have ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (9) In this section, we compare the bounds of L(x, 1) in (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The first result is the comparison on the upper bounds of L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Let x > 0 and p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (i) If p ∈ [0, 1/2], then Wp(x, 1) ≥ ˆAp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (ii) If p /∈ (0, 1/2), then Wp(x, 1) ≤ ˆAp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is sufficient to prove for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Taking account of x1−p − 1 1 − p ≥ 0 for x ≥ 1, we set fp(x) := 2(x − 1) − (xp + 1)(x1−p − 1) (1 − p) = � 2 − 1 1 − p � (x − 1) + xp − x1−p 1 − p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since we have f ′ p(x) = � 2 − 1 1 − p � + pxp−1 − (1 − p)x−p 1 − p , f ′′ p (x) = px−p−1(1 − x2p−1), we have f ′′ p (x) ≥ 0 for p ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='Thus we have f ′ p(x) ≥ f ′ p(1) = 0 which implies fp(x) ≥ fp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we obtain (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Similarly we have f ′′ p (x) ≤ 0 for p /∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have f ′ p(x) ≤ f ′ p(1) = 0 which implies fp(x) ≤ fp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we obtain (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The second result is the comparison on the lower bounds of L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' To prove it, we prepare the following lemma which is interesting itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For x > 0, we have L(x, 1)2 − G(x, 1)2 2L(x, 1)2 ≤ A(x, 1) − L(x, 1) L(x, 1) ≤ log K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is sufficient to prove (10) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We firstly prove the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' To this end, we set u(x) := 2(x − 1) − (x + 1) log x + 4(x − 1) log(x + 1) − 2(x − 1) log(4x), (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We calculate u′(x) = 4x x + 1 − 4 x + 1 + 1 x − 1 − 3 log x + 4 log(x + 1) − 4 log 2 u′′(x) = (x − 1)(x2 + 6x + 1) x2(x + 1)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have u′(x) ≥ u′(1) = 0 =⇒ u(x) ≥ u(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 6 We secondly prove the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' To this end, we set v(x) := (x2 − 1) log x + x(log x)2 − 3(x − 1)2, (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We calculate v′(x) = 2(x + 1) log x + (log x)2 − 5x + 6 − 1 x, v′′(x) = 1 x2 � 2x(x + 1) log x − 3x2 + 2x + 1 � v(3)(x) = 2 x3 w(x), w(x) := x2 − 1 − x log x, w′(x) = 2x − 1 − log x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have w(x) ≥ w(1) = 0 =⇒ v(3)(x) ≥ 0 =⇒ v′′(x) ≥ v′′(1) = 0 =⇒ v′(x) ≥ v′(1) = 0 =⇒ v(x) ≥ v(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we obtain 3L(x, 1) ≤ 2A(x, 1) + G(x, 1)2/L(x, 1), (x > 0) which is equivalent to the first inequality of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Here, the second inequality of (10) is equivalent to the inequality: 1 − (x + 1) log x 2(x − 1) + log �(x + 1)2 4x � ≥ 0 (11) Also the inequality L(x, 1)2 − G(x, 1)2 2L(x, 1)2 ≤ log K(x) is equivalent to the inequality: 1 − x(log x)2 (x − 1)2 + 2 log � 4x (x + 1)2 � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (12) The inequalities (11) and (12) will be used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Let x > 0 and p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (i) If 0 ≤ p ≤ 1 2, then ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (ii) If p ≤ 0 or p ≥ 1, then ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is sufficient to prove for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since K(x) ≥ 1, in order to prove (i), ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1) ⇐⇒ ˆGp(x, 1)K(x)p(1−p) ≥ Wp(x, 1) ⇐⇒ x p 2 �(x + 1)2 4x �p(1−p) ≥ (1 − p)(x − 1) x1−p − 1 we set fp(x) := p 2 log x + p(1 − p) {2 log(x + 1) − log(4x)} − log(1 − p) − log(x − 1) + log(x1−p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then we have f ′ p(x) = − 1 x − 1 + p 2x + p(1 − p)(x − 1) x(x + 1) + 1 − p x − xp = gp(x) 2(x − 1)(x + 1)(x − xp) 7 where gp(x) := 2(x + 1)(xp − 1 − p(x − 1)) + p(x − 1)(1 − xp−1) ((3 − 2p)x + (2p − 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' When x ≥ 1 and 0 ≤ p ≤ 1/2, we have 1 ≥ xp−1, −xp−3 ≥ −xp−2 and (1−p)(1−2p)(p−2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Also when x ≥ 1 and 0 ≤ p ≤ 1/2, we have xp−1 ≥ xp−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we calculate g′ p(x) = (p + 1)(2p2 − 3p + 2)xp − 2p(2p2 − 2p − 1)xp−1 + p(1 − p)(1 − 2p)xp−2 +2p(1 − 2p)x + 2(2p2 − 2p − 1) g′′ p(x) = p � (p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2 +(1 − p)(1 − 2p)(p − 2)xp−3 + 2(1 − 2p) � ≥ p � (p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − 2p)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2 +(1 − p)(1 − 2p)(p − 2)xp−2� = (2p3 − p2 − 5p + 4)(xp−1 − xp−2) = 2(1 − p) {(1 − p)(1 + p) + 1 − p/2} (xp−1 − xp−2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have g′ p(x) ≥ g′ p(1) = 0 so that gp(x) ≥ gp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we have f ′ p(x) ≥ 0 for x ≥ 1 and taking an accout for lim x→1 (1 − p)(x − 1) x1−p − 1 = 1, we havefp(x) ≥ fp(1) = 0 which proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is also sufficent to prove (ii) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For the special cases p = 0 or p = 1 we have equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Since ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1) ⇐⇒ x p 2 �(x + 1)2 4x �p(1−p) ≤ (1 − p)(x − 1) x1−p − 1 , we have only to prove fp(x) ≤ 0 for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (a) We consider the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We set g′′ p(x) = p · hp(x), namely hp(x) := (p+1)(2p2−3p+2)xp−1+2(1−p)(2p2−2p−1)xp−2+(1−p)(1−2p)(p−2)xp−3+2(1−2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then h′ p(x) = (p − 1)xp−4kp(x), where kp(x) := 2p3(x − 1)2 − p2(x − 1)(x − 11) − p(x2 + 6x − 17) + 2(x − 3)(x + 1) and we have k′ p(x) = 4p3(x−1)−2p2(x−6)−2p(x+3)+4(x−1), k′′ p(x) = 2(p+1)(2p2−3p+2) > 0, (p > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have k′ p(x) ≥ k′ p(1) = 10p2 − 8p > 0, (p > 1) so that kp(x) ≥ kp(1) = 10p − 8 > 0, (p > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore we have h′ p(x) ≥ 0, (p > 1) which implies hp(x) ≥ hp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have g′′ p(x) ≥ 0 so that we have g′ p(x) ≥ g′ p(1) = 0 which implies gp(x) ≥ gp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Taking account of x − xp ≤ 0 when x ≥ 1, p > 1, we have f ′ p(x) ≤ 0 Therefore we have fp(x) ≤ fp(1) = 0 which proves (ii) for the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (b) We consider the case p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We calculate dfp(x) dp = 1 1 − p + �1 2 + x xp − x � log x + (2p − 1) log(4x) + (2 − 4p) log(x + 1), d2fp(x) dp2 = −xp+1(log x)2 (x − xp)2 + 1 (p − 1)2 + 2 log(4x) − 4 log(x + 1), d3fp(x) dp3 = xp+1 (xp + x) (log x)3 (xp − x)3 − 2 (p − 1)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 8 We further calculate d dx �d3fp(x) dp3 � = −xp(log x)2 (x − xp)4 s(x, p), s(x, p) := 3(x2 − x2p) + (p − 1) � x2 + x2p + 4x1+p� log x, ds(x, p) dp = � x2 − 5x2p + 4xp+1 + 2(p − 1) � x2p + 2xp+1� log(x) � log x, d2s(x, p) dp2 = 4xp(log x)2 {2(x − xp) + (p − 1)(x + xp) log x} ≤ 0 (p ≤ 1, x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Indeed, putting a := x, b := xp in the inequality a − b log a − log b ≤ a + b 2 , (a, b > 0), we have 2(x−xp)+(p−1)(x+xp) log x ≤ 0 for p < 1 and x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (The equality holds when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=') From d2s(x, p) dp2 ≤ 0, we have ds(x, p) dp ≥ ds(x, p) dp ���� p=1 = 0 which implies s(x, p) ≤ s(x, 1) = 0 so that we have d dx �d3fp(x) dp3 � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From this, we have d3fp(x) dp3 ≥ d3fp(1) dp3 = − 2 (p − 1)3 > 0, (p < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' By the inequality (12), we have p ≤ 0 =⇒ d2fp(x) dp2 ≤ d2fp(x) dp2 ���� p=0 = 1 − x(log x)2 (x − 1)2 + 2 log � 4x (x + 1)2 � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (13) Thus we have by the inequality (11), p ≤ 0 =⇒ dfp(x) dp ≥ dfp(x) dp ���� p=0 = 1 − (x + 1) log x 2(x − 1) + log �(x + 1)2 4x � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore p ≤ 0 =⇒ fp(x) ≤ f0(x) = 0 which proves (ii) for the case p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (i) Since fp(x) = log ��(x + 1)2 4x �p(1−p) × xp/2(x1−p − 1) (1 − p)(x − 1) � appeared in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3 and comparing the maximum degree of the numerator and the denominator insides of the logarithmic function, we found that if p < 0 or p > 1/2, then we have lim x→∞ ��(x + 1)2 4x �p(1−p) × xp/2(x1−p − 1) (1 − p)(x − 1) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Thus we have lim x→∞ fp(x) = −∞ if p < 0 or p > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' On the other hand, by the muner- ical computations, we have f3/4(e) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='0063209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Therefore there is no ordering between K(x)−p(1−p)Wp(x, 1) and ˆGp(x, 1) for x > 0 and 1/2 < p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (ii) From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1 (i) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3 (i), we have for x > 0 and 0 ≤ p ≤ 1/2, K(x)−p(1−p)Wp(x, 1) ≤ ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1) ≤ Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (iii) From the proof (b) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3, we obtained d3fp(x) dp3 ≥ 0, (p < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From this with simple calculations, we have H(x, xp)3 ≤ xp+1A(x, xp) ≤ L(x, xp)3, (p ≤ 1, x > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 9 4 Conclusion As we have seen, we studied the inequalities on the relations between the Wigner–Yanase– Dyson function Wp(·, ·) and the logarithmic mean L(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' As one of main results, we obtained two kinds of the reverse inequalities for L(x, y) ≤ Wp(x, y) for x, y > 0 and 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' That is, the inequalities (2) and (7) shown in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3 are respectively equivalent to the following inequalities for x, y > 0 and 0 ≤ p ≤ 1: Wp(x, y) ≤ p(1 − p) �√x − √y �2 + L(x, y), (14) Wp(x, y) ≤ K (x/y)p(1−p) L(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (15) The inequality (14) and (15) are the difference type reverse inequality and the ratio type reverse inequality for L(x, y) ≤ Wp(x, y), (0 ≤ p ≤ 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' In addition, we compared the obtained inequality (15) with the known result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is summerized in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' The inequalities given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='4 (ii) are equivalent to the following inequalities for x, y > 0 and 0 ≤ p ≤ 1/2: K (x/y)−p(1−p) Wp(x, y) ≤ ˆGp(x, y) ≤ L(x, y) ≤ ˆAp(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' (16) We conclude this paper by giving operator inequalities baesd on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' For positive operators S, T and 0 ≤ p ≤ 1, we define the operator version of the logarithmic mean and the Wigner–Yanase–Dyson function as L(S, T) := � 1 0 S♯tTdt, Wp(S, T) := p(1 − p) 2 (S − T) (S∇T − Hzp(S, T))−1 (S − T), (S ̸= T), Wp(S, S) := S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' where S♯pT := S1/2 � S−1/2TS−1/2�p S1/2, S∇T := S + T 2 , Hzp(S, T) := 1 2 (S♯pT + S♯1−pT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' We often use the symbol S♯T := S♯1/2T for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Hzp(S, T) is often called the Heinz mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' It is notable that we have the relation for the validity in the case p := 1/2, �S − T 2 � (S∇T + S♯T)−1 �S − T 2 � = S∇T − S♯T, which can be confirmed by multiplying S−1/2 to both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Let S and T be positive operators and let 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then we have Wp(S, T) ≤ 2p(1 − p) (S∇T − S♯T) + L(S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='3, we also have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Let S and T be positive operators with αS ≤ T ≤ βS for 0 < α ≤ β and let 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' Then we have Wp(S, T) ≤ kp · L(S, T), kp := max α≤x≤β K(x)p(1−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} +page_content=' 10 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+page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'} diff --git a/0tAzT4oBgHgl3EQftv2j/content/tmp_files/2301.01681v1.pdf.txt b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/2301.01681v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..910226aaf2c2b88e19830f7f48dbe7e4eaac80bd --- /dev/null +++ b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/2301.01681v1.pdf.txt @@ -0,0 +1,2084 @@ +Tower of quantum scars in a partially many-body localized system +Michael Iversen and Anne E. B. Nielsen +Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark +Isolated quantum many-body systems are often well-described by the eigenstate thermalization +hypothesis. There are, however, mechanisms that cause different behavior: many-body localization +and quantum many-body scars. Here, we show how one can find disordered Hamiltonians hosting +a tower of scars by adapting a known method for finding parent Hamiltonians. Using this method, +we construct a spin-1/2 model which is both partially localized and contains scars. We demonstrate +that the model is partially localized by studying numerically the level spacing statistics and bipar- +tite entanglement entropy. As disorder is introduced, the adjacent gap ratio transitions from the +Gaussian orthogonal ensemble to the Poisson distribution and the entropy shifts from volume-law +to area-law scaling. We investigate the properties of scars in a partially localized background and +compare with a thermal background. At strong disorder, states initialized inside or outside the scar +subspace display different dynamical behavior but have similar entanglement entropy. We demon- +strate that localization stabilizes scar revivals of initial states with support both inside and outside +the scar subspace. Finally, we show how strong disorder introduces additional towers of approximate +scar states. +I. +INTRODUCTION +The eigenstate thermalization hypothesis (ETH) de- +scribes how isolated quantum systems reach thermal +equilibrium [1–3]. The hypothesis is a statement about +generic quantum many-body systems and has been veri- +fied for a wide variety of physical models [3–13]. Despite +the effectiveness of ETH, several phenomena are known +to cause non-thermal behavior. +One such mechanism is many-body localization (MBL) +[14–17]. +MBL appears in many-body interacting sys- +tems with local disorder. +When the disorder strength +is sufficiently strong, it causes a change in the structure +of the energy eigenstates. +An extensive set of quasi- +local integrals of motion (LIOM) emerges and the en- +ergy eigenstates localize [18, 19]. Consequently, all en- +ergy eigenstates behave non-thermally and MBL repre- +sents a strong violation of ETH. While this phenomenon +is well-established for finite systems, the stability of MBL +in the thermodynamic limit is still an open question [20]. +Another mechanism leading to non-thermal behav- +ior was discovered in experiments with kinetically con- +strained Rydberg atoms [21]. The atoms were arranged +with strong nearest neighbor interactions so the simul- +taneous excitation of neighboring atoms was prohibited. +When initializing the system in the N´eel state, observ- +ables displayed abnormal persistent oscillations – con- +trary to the predictions by ETH. Subsequent theoreti- +cal works uncovered that the revivals were caused by a +small number of non-thermal eigenstates dubbed quan- +tum many-body scars (QMBS) [22–25]. These scar states +have approximately equal energy spacing so any initial +state in the scar subspace displays revivals. +The scar +states are uncommon and represent a vanishingly small +part of an otherwise thermalizing spectrum. Therefore, +QMBS represent a weak violation of ETH. +In this work, we realize both ETH-breaking mecha- +nisms simultaneously. We study a one-dimensional dis- +ordered spin-1/2 chain hosting a tower of QMBS. As the +disorder strength is increased, the model transitions from +the thermal phase to being partially localized while pre- +serving the scar states. +In earlier works, a single scar +state was embedded in an otherwise MBL spectrum [26– +28]. Our work adds to these studies by considering a full +tower of QMBS in an MBL spectrum. The presence of +multiple scar states, enables us to study the effect of lo- +calization on the dynamical revivals characteristic of scar +states. Using this model, we demonstrate how scar states +can be distinguished from a localized background. We +also find two phenomena originating from the interplay +between QMBS and localization: disorder stabilization +of scar revivals and disorder induced approximate scars. +The paper is structured as follows. In Sec. II A, we +summarize the model by Iadecola and Schecter which is +the starting point of our analysis. In Sec. II B, we explain +how we find Hamiltonians having a set of scar states with +equal energy spacing. In Sec. II C, we use this method to +determine all local 1- and 2-body Hamiltonians for the +tower of scar states in the Iadecola and Schecter model. +In Sec. III A, we show that a subset of these Hamiltoni- +ans partially localize as disorder is introduced. We quan- +tify the partial localization as a special structure in the +energy eigenstates and compare with results from exact +diagonalization. We verify the localization by studying +the level spacing statistics in Sec. III B and the entan- +glement entropy in Sec. III C. In Sec. IV, we show that +the fidelity between initial states and the corresponding +time evolved states can be utilized to distinguish the scar +states from the partially localized background. We fur- +ther show that the bipartite entanglement entropy is an +ineffective tool for distinguishing scar states from a par- +tially localized background. In Sec. V, we demonstrate +how scar revivals are stabilized by strong disorder. In +Sec. VI, we uncover additional towers of approximate scar +states which emerge as disorder is introduced. Finally, we +summarize our results in Sec. VII. +arXiv:2301.01681v1 [cond-mat.dis-nn] 4 Jan 2023 + +2 +II. +MODEL +A. +Model by Iadecola and Schecter +We take the model by Iadecola and Schecter as our +starting point [29]. Consider a one-dimensional spin- 1 +2 +chain of even length L with periodic boundary conditions. +The local Hilbert space on each site is described by the +eigenkets |↑⟩ and |↓⟩ of the Pauli z-matrix, i.e. ˆσz |↑⟩ = |↑⟩ +and ˆσz |↓⟩ = − |↓⟩. The model by Iadecola and Schecter +is given by +ˆH0 = +L +� +i=1 +� +λ(ˆσx +i − ˆσz +i−1ˆσx +i ˆσz +i+1) + ∆ˆσz +i + J ˆσz +i ˆσz +i+1 +� +, (1) +with λ, ∆, J ∈ R. All indices are understood as modulo +L, i.e. the index i+L is identified as i. The operators ˆσx +i , +ˆσy +i and ˆσz +i are the Pauli matrices acting on site i. The +first term in Eq. (1) flips the spin si at site i if its nearest +neighbors are in different states, i.e. si−1 ̸= si+1. The +second term is a magnetic field along the z-direction with +strength ∆. The third term represents nearest neighbor +interactions with strength J. +Two adjacent spins in different states represent a do- +main wall, i.e. ↑↓ or ↓↑. The Hamiltonian conserves the +number of domain walls Ndw because only spins with dif- +ferent neighbors are allowed to change their state. Fur- +thermore, the Hamiltonian is invariant under spatial in- +version and translation, but these symmetries are broken +when disorder is introduced in section III and we will not +consider them any further. +For nonzero values of λ, ∆ and J, the energy eigen- +states are thermal except for a small number of ETH- +violating scar states grouped into two towers. Through- +out this work, we only focus on one of these towers. This +tower contains L/2+1 eigenstates and the n-th state |Sn⟩ +is constructed by acting n times with the operator ˆQ† on +the “all-spin-down” state +|Sn⟩ ∝ +� ˆQ†�n |↓↓ . . . ↓⟩ . +(2) +The operator ˆQ† is given by +ˆQ† = +L +� +i=1 +(−1)i ˆP ↓ +i−1ˆσ+ +i ˆP ↓ +i+1, +(3) +where ˆσ+ +i = (ˆσx +i +iˆσy +i )/2 is the raising operator and ˆP ↓ +i = +(ˆ1 − ˆσz +i )/2 is the local projection onto spin down. The +n-th scar state has energy En = 2(∆ − 2J)n + (J − ∆)L, +number of domain walls Ndw = 2n and generally appears +central in the spectrum after resolving all symmetries. +Since the scar states are equally spaced in energy, any +initial state in the scar subspace displays the dynami- +cal revivals characteristic of QMBS. Furthermore, it was +shown in Ref. [29] that the bipartite entanglement en- +tropy of the scar states displays logarithmic scaling with +system size. +B. +Determining Hamiltonians +All eigenstates of ˆH0 located near the middle of the +spectrum are thermal except the scar states. We wish +to extend the model so the scar states are embedded +in a MBL background instead of a thermal background. +MBL is possible in systems with quench disorder, and it +has been realized in numerous models by e.g. introduc- +ing a disordered magnetic field [17], bond-disorder [30] +or disordered nearest-neighbor interactions [31]. Unfor- +tunately, disorder cannot be introduced naively to the +Hamiltonian ˆH0. When promoting any parameter to be- +ing site-dependent λ → λi, ∆ → ∆i or J → Ji, the +scar states are no longer eigenstates. Therefore, disorder +must be introduced through new terms. In this section, +we uncover all local few-body Hamiltonians which share +the scar states as eigenstates and maintain equal energy +spacing. In the next section, we show that a subset of +these Hamiltonians are partially localized. +We search for local Hamiltonians following Refs. [32, +33]. The set of 2L×2L Hermitian operators form a vector +space. Most of these operators are long-ranged, contain +many-body interactions and are difficult to realize in ex- +periments. Therefore, we restrict ourselves to Hamiltoni- +ans containing local 1- and 2-body Hermitian operators. +This subspace is spanned by the operator basis +B2 = +� +ˆσa +i +���a ∈ {x, y, z}, i ∈ ZL +� +∪ +� +ˆσa +i ˆσb +i+1 +���a, b ∈ {x, y, z}, i ∈ ZL +� +, +(4) +where ZL = {1, 2, . . . , L} are the first L integers. This +subspace is considerably smaller than the full operator +vector space and has dimension |B2| = 12L where | · | +denotes the number of elements in a set. Any local 1- +or 2-body interacting Hamiltonian can be expressed as a +linear combination of the basis elements +ˆH = +|B2| +� +i=1 +αiˆhi, +ˆhi ∈ B2, +(5) +where αi ∈ R are free coefficients. +To simplify no- +tation, we collect the coefficients in a vector α += +(α1, α2, . . . , α|B2|)T where T is the transpose. +We search for the vector of parameters α so the re- +sulting Hamiltonian has |Sn⟩ as eigenstates for n = +0, 1, . . . , L/2. The scar state |Sn⟩ is an eigenstate of ˆH if +and only if the energy variance of |Sn⟩ is exactly zero +⟨Sn| ˆH2|Sn⟩ − ⟨Sn| ˆH|Sn⟩ +2 = 0. +(6) +Inserting Eq. (5), the expression becomes +αT Cnα = 0, +(7) +where Cn is the quantum covariance matrix +[Cn]ij = ⟨Sn|ˆhiˆhj|Sn⟩ − ⟨Sn|ˆhi|Sn⟩ ⟨Sn|ˆhj|Sn⟩ . +(8) + +3 +Equation (7) is satisfied when the vector of coefficients +lies in the null space of the quantum covariance matrix +α ∈ Null(Cn), i.e. Cnα = 0. We ensure all scar states +|Sn⟩ are simultaneously eigenstates of ˆH by demanding +the vector of coefficients α lies in the null space of ev- +ery covariance matrix α ∈ Null(C0) ∩ Null(C1) ∩ . . . ∩ +Null(CL/2). While this condition ensures all scar states +are eigenstates of ˆH, they are not necessarily equally +spaced in energy. +Equal energy spacing is established +by imposing another set of requirements +⟨Sn+2| ˆH|Sn+2⟩ − ⟨Sn+1| ˆH|Sn+1⟩ += ⟨Sn+1| ˆH|Sn+1⟩ − ⟨Sn| ˆH|Sn⟩ , +(9) +for all n = 0, 1, . . . , L/2 − 2. Inserting Eq. (5), we find +Gα = 0, +(10) +where we introduce the rectangular matrix of energy gap +differences +[G]ij = ⟨Si+2|ˆhj|Si+2⟩ − 2 ⟨Si+1|ˆhj|Si+1⟩ + ⟨Si|ˆhj|Si⟩ . +(11) +We observe that the scar states are equally spaced in en- +ergy when the coefficient vector resides in the null space +of the gap matrix. In summary, the scar states appear as +eigenstates of the Hamiltonian with equal energy spacing +when the vector of coefficients lies in the intersection +α ∈ +L/2 +� +n=0 +Null(Cn) ∩ Null(G). +(12) +It is straightforward to determine this subspace numeri- +cally since the scar states are known analytically. Note +however, that while the matrices Cn and G are com- +plex, we only search for real vectors α ∈ R|B2| (for com- +plex vectors α ∈ C|B2|, the linear combination in Eq. +(5) is not necessarily Hermitian). +We find real coeffi- +cient vectors by stacking the real and imaginary parts of +the matrices (Re(C0), Im(C0), . . . , Re(CL/2), Im(CL/2), +Re(G), Im(G))T and determining the null space of the +resulting rectangular matrix by e.g. singular value de- +composition. +The vectors αi produced by this numerical method are +typically dense, i.e. have few nonzero entries. As a con- +sequence, the corresponding operator � +i αiˆhi is difficult +to interpret. We overcome this difficulty by noting that +if {αi|i = 1, 2, . . .} lies in the null space Eq. (12), then +any linear combination of these vectors also lies in the +null space. We apply a heuristic algorithm to determine +sparse vectors in the subspace [34]. +C. +Generalized models +We apply the numerical method for system sizes L = 8, +10, 12, 14 and for all sizes find L + 4 linearly indepen- +dent vectors αi satisfying Eq. (12). The corresponding +(i) +ˆHz = �L +i=1 ˆσz +i +(ii) +ˆDi = ˆσz +i + ˆσz +i+1 + ˆσz +i ˆσz +i+1, +for i ∈ ZL +(iii) ˆHodd +zz += �L/2 +i=1 ˆσz +2i−1ˆσz +2i +(iv) +ˆHalt +xz = �L +i=1(−1)i(ˆσx +i ˆσz +i+1 + ˆσz +i ˆσx +i+1) +(v) +ˆHalt +yz = �L +i=1(−1)i(ˆσy +i ˆσz +i+1 + ˆσz +i ˆσy +i+1) +TABLE I. Local 1- and 2-body operators which have |Sn⟩ +for n = 0, 1, . . . , L/2 as energy eigenstates with equal energy +spacing. The operators are determined by applying the nu- +merical method presented in Sec. II B and Appendix A proves +the statement rigorously. +operators are summarized in Tab. I. The first operator +ˆHz was already present in the initial model Eq. (1) and +adds nothing new. The L operators ˆDi act locally on +sites i and i+1 and represent good candidates for adding +quench disorder into the model in Eq. (1). Indeed, in Sec. +III, we demonstrate the system partially localizes when +introducing sufficiently strong disorder via these opera- +tors. The third operator ˆHodd +zz +represents an interaction +between every odd site and its right neighbor with equal +interaction strength. The fourth and fifth operators ˆHalt +xz +and ˆHalt +yz flip spins with the sign of the term determined +by the nearest neighbors. +Using the numerical method, we rediscover the 1- and +2-body terms of the model in Eq. (1) by starting from +the scar states. As noted above, the operator ˆHz was +already present in the original model. Furthermore, the +third term in Eq. (1) is a linear combination of the oper- +ators in Tab. I: �L +i=1 ˆσz +i ˆσz +i+1 = �L +i=1 ˆDi − 2 ˆHz. Hence, +the operators in Tab. I only represent L + 2 non-trivial +extensions to the initial model. +The numerical method presented in Sec. II B finds all +operators in the operator subspace span(B2) hosting the +tower of scars for finite L (up to length L = 14 in our +case). However, in principle, the scar states may not be +eigenstates of these operators at larger L. Therefore, in +Appendix A we prove analytically for all even L that the +scar states remain eigenstates with equal energy spacing +for all operators in Tab. I. +The method from Sec. II B can be extended by in- +cluding all 3-body terms to the basis B3 += +B2 ∪ +{ˆσa +i ˆσb +i+1ˆσc +i+2 +���a, b, c ∈ {x, y, z}, i ∈ ZL}. +This results +in a myriad of new operators – including the first term +from Eq. (1). Hence, with a large enough operator basis, +the numerical method fully recovers the original model. +Since long-ranged many-body interactions are less rele- +vant experimentally, we will not explore this possibility +any further. +Finally, we remark that the effectiveness of this ap- +proach is highly non-trivial. For an eigenstate of a generic +local Hamiltonian, it is unlikely for another local Hamil- + +4 +tonian to exist that shares the same eigenstate [35]. Con- +trary to this, we find a large subspace of local Hamiltoni- +ans sharing a full tower of scar states. We attribute the +effectiveness of our study to the analytical structure of +the scar states, i.e. Eq. (2) and (3). Our methods are not +expected to be valuable starting from generic eigenstates +but may be equally effective in other scarred models with +similar amount of structure. +III. +MANY-BODY LOCALIZATION +In the last section, we determined a subspace of Hamil- +tonians with the scar states |Sn⟩ as eigenstates equally +spaced in energy. Now, we study a concrete Hamiltonian +from this subspace +ˆH = ˆH0 + +L +� +i=1 +di ˆDi, +(13) +with di chosen randomly from the uniform probability +distribution di ∈ [−W, W] where W > 0 is the disorder +strength. The action of ˆDi is given by +ˆDi |s1 . . . sisi+1 . . . sL⟩ += +� +3 |s1 . . . sisi+1 . . . sL⟩ , +if si = si+1 = ↑ +− |s1 . . . sisi+1 . . . sL⟩ , +otherwise +(14) +The operator ˆDi is related to the projection operators +through ˆDi = 4 ˆP ↑ +i ˆP ↑ +i+1 − ˆ1 with ˆP ↑ +i = (ˆ1 + ˆσz +i )/2. We +remark that Ref. [29] also observes that the operator +ˆP ↑ +i ˆP ↑ +i+1 preserves the scar states. +The model conserves the number of domain walls. The +dimension of the symmetry sector containing Ndw do- +main walls is given by the binomial coefficient 2( +L +Ndw ). +We generally consider the largest symmetry sector with +Ndw = 2⌊L/4⌋ domain walls where ⌊·⌋ is the function +rounding down to the nearest integer. +A. +Partial many-body localization +A physical system may transition to the MBL phase +when disorder is introduced. +MBL is usually real- +ized with the disorder term in the Hamiltonian acting +uniquely on each basis state. Consequently, a complete +set of LIOMs emerge and all energy eigenstates are fully +described by their eigenvalues of the LIOMs. +The situation is slightly different in our model because +the disorder term � +i di ˆDi treats some basis states the +same. The operator ˆDi is only sensitive to whether spins +i and i+1 are both up (it acts identically on states where +spins i and i+1 are ↓↓, ↓↑ or ↑↓). Therefore, the operator +� +i di ˆDi has the same action on product states with all +consecutive spin-ups placed identically. We do not expect +these to localize in the usual sense. Instead, we anticipate +the spectrum to separate into fully MBL eigenstates and +partially localized eigenstates. +This structure is most easily described when the prod- +uct states |s1s2 . . . sL⟩ are relabeled to reflect the ac- +tion of � +i di ˆDi. +In this spirit, we define |Ndw, D, n⟩ +as a simultaneous eigenstate of the ˆDi’s with eigenvalues +D = (D1, D2, . . . DL) where Di ∈ {−1, 3}. We will refer +to D as the disorder indices. As discussed above, the +state |s1s2 . . . sL⟩ is not fully described by D since mul- +tiple states can have the same eigenvalues. Therefore, we +further label the states by their number of domain walls +Ndw and introduce a dummy index n = 1, 2, . . . , N (Ndw) +D +to distinguish states with identical Ndw and D. For in- +stance, if two states |s1s2 . . . sL⟩ and |s′ +1s′ +2 . . . s′ +L⟩ have the +same number of domain walls Ndw and disorder indices +D, then they are relabeled as |Ndw, D, n⟩ for n = 1, 2. +Note that some labelings are invalid. Consider the vector +of eigenvalues D = (3, −1, 3, 3) for a small system L = 4. +The “3”s imply all spins are up, while the “−1” entail at +least one spin is down. In the following, we study a single +symmetry sector and hence omit the Ndw index for clar- +ity but reintroduce it in Secs. V and VI when studying +multiple symmetry sectors at once. +Upon introducing strong disorder, we expect LIOMs to +emerge which are localized on the operators ˆDi and en- +ergy eigenstates are characterized by their eigenvalues of +the LIOMs. Therefore, we expect the energy eigenstates +to be close to linear combinations of product states with +the same disorder indices +|ED,m⟩ ≈ +ND +� +n=1 +αmn |D, n⟩ . +(15) +with αmn ∈ R and m = 1, 2, . . . , ND. This expression +is an approximation rather than an equality due to an +exponentially small overlap with states |D′, n⟩ with dif- +ferent disorder indices D′ ̸= D. The special case ND = 1 +corresponds to the disorder term acting uniquely on the +basis state |D, 1⟩. We expect the corresponding energy +eigenstate |ED,1⟩ ≈ |D, 1⟩ to be MBL. For ND > 1, +the states {|ED,m⟩ |m = 1, 2, . . . , ND} are only partially +MBL since the LIOMs do not fully describe each state +and all additional structure is captured by the extra in- +dex m. +The above considerations are verified in numerical sim- +ulations by considering a system of size L = 8 at strong +disorder W = 10. Figure 1 illustrates the norm squared +overlap of all energy eigenstates |ED,m⟩ with the prod- +uct states |D, n⟩. The (i, j)-th pixel displays the norm +squared overlap between the i-th product state and j- +th energy eigenstate. The product states on the second +axis are sorted according to ND. The energy eigenstates +are reordered to allow the diagonal shape in Fig. 1. In +the upper left corner of Fig. 1, each eigenstate has high +overlap with a single product state. Numerical analysis +reveals that these product states exactly coincide with +those being fully described by their disorder indices, i.e. +ND = 1. These results support the claim that such eigen- + +5 +|ED,m⟩ +|D, n⟩ +1 +2 +3 +4 +20 +ND +(a) +(b) +(c) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +|⟨D, n|ED,m⟩|2 +FIG. 1. +The norm squared overlap of the energy eigenstates +with the product states | ⟨D, n|ED,m⟩ |2 for system size L = 8, +disorder strength W = 10 and parameters λ = ∆ = J = 1. +The color of pixel (i, j) displays the overlap between the i’th +product state and the j’th eigenstate. The product states are +sorted into ascending order according to ND. The second axis +on the right hand side groups the product states according to +ND. The insets show eigenstates with significant weight on +(a) two, (b) three and (c) four product states. +The figure +verifies that all energy eigenstates are approximately linear +combinations of product states with the same disorder indices. +states fully localize. The next eigenstates shown in Fig. +1(a) each has significant overlap with exactly two product +states of the same disorder indices. The pattern contin- +ues: we find eigenstates that are linear combinations of +Fig. 1(b) three, Fig. 1(c) four, and (bottom right corner) +twenty product states. In each case, the product states +have the same disorder indices and hence correspond to +{|D, n⟩ |n = 1, 2, . . . , ND} for ND = 3, 4, 20. These ob- +servations are not restricted to L = 8, but seem universal +at all system sizes. For larger system sizes, the number +and sizes of the blocks increase. Finally, we note that +the scar state within the considered symmetry sector is +located in the block ND = 20 in Fig. 1. The scar state +is generally an equal weight linear combination of prod- +uct states with the maximum ND. This fact will play an +important role when we explore the system dynamics in +Sec. V. +Next, we discuss how the eigenstates are distributed +in energy. The magnetization MD = � +i σz +i of a prod- +uct state |D, n⟩ is fixed by the symmetry sector Ndw +E +Thermal +|Sn⟩ +Partial MBL +|ED1,m1⟩ +|ED2,1⟩ +|ED2,2⟩ +|ED2,3⟩ +|ED3,m3⟩ +|ED4,m4⟩ +|ED5,m5⟩ +FIG. 2. +Sketch of the spectrum in the thermal phase (left) +and in the partially localized phase (right). In the thermal +phase, the energy levels follow the Wigner-Dyson surmise. +As disorder is introduced, the spectrum experiences partial +localization. Eigenstates with similar indices D are near de- +generate and the spectrum forms clusters of such eigenstates. +The scar state lies in the largest of these clusters. +and disorder indices D. Likewise, the number N (↑↑,↓↓) +D +of adjacent spins pointing in the same direction (↑↑ or +↓↓) and the number N (↑↓,↓↑) +D +of adjacent spins point- +ing in opposite directions (↑↓ or ↓↑) are also fully de- +termined. +Therefore, the terms ∆ � +i ˆσz +i , J � +i ˆσz +i ˆσz +i+1 +and � +i di ˆDi have the same action on all product states +with the same number of domain walls and disorder in- +dices: {|D, n⟩ |n = 1, 2, . . . , ND}. +At strong disorder, +the energy of an eigenstate is approximately ED,m ≈ +∆MD + J(N (↑↑,↓↓) +D +− N (↑↓,↓↑) +D +) + � +i diDi with a small +correction that depends on the value of the m index. The +slight additional contribution originates from the term +� +i λ(ˆσx +i − ˆσz +i−1ˆσx +i ˆσz +i+1) and scales with λ. Consequently, +at large disorder, the set of eigenstates {|ED,m⟩ |m = +1, 2, . . . , ND} are near degenerate and form clusters. A +scar state resides in the largest of these clusters in all +symmetry sectors. Figure 2 illustrates the spectral struc- +ture. Note that Fig. 2 is highly idealized to highlight the +structure described above. In practice, it is highly likely +for two or more clusters to overlap making the structure +less apparent. +B. +Spectral statistics +The distribution of energy gaps distinguishes the ther- +mal and MBL phases. +Let Ei be the energies of the +Hamiltonian in ascending order and δi = Ei+1 − Ei ≥ 0 +the i-th energy gap. In the thermal phase, the number of +energy levels in an interval [E, E+∆E] is known to follow +the Wigner-surmise [36, 37]. In particular, it follows the +Gaussian orthogonal ensemble (GOE) since the model in +Eq. (13) is time-reversal invariant. On the other hand, +the number of energy levels in an interval follows the Pois- +son distribution in the MBL phase. Since our model only +partially localizes, we review how the Poisson distribu- +tion accurately describes the MBL phase and investigate + +T6 +the validity of these arguments in our model. Consider +two adjacent eigenstates with energies Ei and Ei+1. At +large disorder, the energy of these states are dominated +by the disorder term � +i di ˆDi. If the states have differ- +ent disorder indices |ED,m⟩ and |ED′,m′⟩, then their en- +ergies originate from different linear combinations of the +random numbers di: � +i diDi ≈ � +i diD′ +i with Di ̸= D′ +i +for some i’s. Consequently, the eigenstates “arrive” at +this energy independently of each other and hence fol- +low the Poisson distribution. +These arguments are no +longer valid when two adjacent eigenstates have the same +disorder indices and different m indices. +In this case, +we expect the level spacing distribution to follow GOE. +Thus, the distribution of energy levels still identifies the +transition to partial localization if we only consider level +spacings between eigenstates of different disorder indices. +Instead of working directly with the level spacing dis- +tribution, it is convenient to analyze the adjacent gap +ratio since it removes the need for unfolding the spec- +trum [37, 38]. The adjacent gap ratio is defined by [16] +ri = min(δi, δi+1) +max(δi, δi+1). +(16) +This quantity is bounded by the interval ri ∈ [0, 1] and +follows the distributions below in the thermal and MBL +phases respectively [39] +PGOE(r) = 27 +4 +r(1 + r) +(1 + r + r2)5/2 , +(17a) +PPoisson(r) = +2 +(1 + r)2 . +(17b) +The mean values of the distributions in Eq. (17) are given +by ⟨r⟩GOE = 2(2 − +√ +3) ≈ 0.536 and ⟨r⟩Poisson = 2 ln 2 − +1 ≈ 0.386. +Figure 3(a) illustrates the mean adjacent gap ratio as +a function of disorder strength for different system sizes. +We average the adjacent gap ratio over 2 × 103 disor- +der realizations for L = 8, 103 disorder realizations for +L = 10, 12, 14 and 500 disorder realizations for L = 16. +For each disorder realization, we average over all energies +in the interval Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the +q-th quantile of the energy distribution for the current +disorder realization. For system size L = 16, we average +over the 103 energies closest to (Emin + Emax)/2 where +Emin and Emax are the smallest and largest energies in +the spectrum. The errorbars indicate two standard devi- +ations of the average when assuming a Gaussian distri- +bution. As discussed above, the distribution of adjacent +gap ratios only converges to Eq. (17b) if the analysis is +restricted to adjacent energy levels with different disor- +der indices. In practice, however, it is unlikely for two +neighboring eigenstates to have the same disorder indices. +Furthermore, the likelihood of neighboring eigenstates +having the same disorder indices decreases rapidly with +system size. With this in mind, we study the mean adja- +cent gap ratio using all eigenstates in the central third of +the spectrum. We verify the considerations above by also +computing the mean adjacent gap ratio using only adja- +cent eigenstates with different disorder indices at large +disorder. For each energy gap δi = Ei+1 −Ei, we inspect +the eigenstates |ED,m⟩ and |ED′,m′⟩ corresponding to the +energies Ei and Ei+1. At large disorder, the disorder in- +dices D are accurately determined by computing which +D yields �ND +m=1 | ⟨D, m|ED,m⟩ |2 ≈ 1. The mean of the +adjacent gap ratio is then restricted to energy gaps with +D ̸= D′. For small system sizes, there is a large differ- +ence between the two methods, but the difference is seen +to be small for large systems. +The mean adjacent gap ratio agrees well with the GOE +value at weak disorder 0 <∼ W <∼ 1. +As the disorder +strength is increased, the mean adjacent gap ratio de- +creases and ultimately approaches the Poisson value at +5 <∼ W. The agreement of data with the GOE and Pois- +son values improves with increasing system size and the +transition between the thermal and localized phase be- +comes steeper for larger systems. +Figures 3(b)-(d) illustrate the adjacent gap ratio dis- +tribution at (b) weak disorder W = 0.46, (c) intermedi- +ate disorder strength W = 2.27 and (d) strong disorder +W = 6. The figures display the distributions in Eq. (17) +for comparison. As expected, the data agrees with Eq. +(17a) at weak disorder and (17b) at strong disorder. Fig- +ure 3 indicates the system transitions from the thermal +phase to being partially localized as disorder is intro- +duced. +C. +Bipartite entanglement entropy +In this section, we further verify the transition from +the thermal phase to partial localization by studying the +bipartite entanglement entropy. We separate the system +into a left part L containing the first L/2 sites and a +right part R containing the remaining sites. The reduced +density matrix of the left part is obtained by tracing out +the right part +ρL = TrR(ρ) +(18) +where ρ is the density matrix of the full system and +TrR(·) is the partial trace over R. +The entanglement +entropy between the left and right halves is given by, +S = − TrL +� +ρL ln(ρL) +� +. +(19) +In the thermal phase, we expect eigenstates near the +center of the spectrum to display volume-law scaling with +system size. Specifically, the entropy is approximately +described by the Page value SPage = [L ln(2) − 1]/2 [40]. +On the other hand, the entanglement entropy displays +area-law scaling for MBL eigenstates [41]. While some +eigenstates in our model are fully MBL, others are only +partially localized. Hence, the precise scaling behavior +of the entanglement entropy is not clear. Nonetheless, +we expect the entropy of partially localized eigenstates +to grow slower with system size than thermal eigenstates + +7 +P(r) +(b) +Poisson +GOE +P(r) +(c) +0.0 +0.5 +1.0 +r +P(r) +(d) +0 +1 +2 +3 +4 +5 +6 +W +0.40 +0.45 +0.50 +⟨r⟩ +(a) +GOE +Poisson +L = 8 +10 +12 +14 +16 +FIG. 3. +(a) Mean adjacent gap ratio ⟨r⟩ (solid line) as a function of disorder strength W for different system sizes L with +parameters λ = ∆ = J = 1. The shaded areas display two standard deviations on the estimate of ⟨r⟩ when assuming a Gaussian +distribution of data. For L = 8, the adjacent gap ratio is averaged over 2 × 103 disorder realizations, for L = 10, 12, 14 we +use 103 disorder realizations and for L = 16 we use 500 disorder realizations. For system sizes L = 8, 10, 12, 14, we average +over all energies Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile. For system size L = 16, we average over the 103 +energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum. +At low +disorder 0 <∼ W <∼ 1, the system is thermal and ⟨r⟩ coincides with the Gaussian orthogonal ensemble ⟨r⟩GOE ≈ 0.536 (upper +dashed line). At strong disorder 5 <∼ W, the mean adjacent gap ratio agrees with the Poisson distribution ⟨r⟩Poisson ≈ 0.386 +(lower dotted line). The agreement between data and the GOE and Poisson values improves with system size. Additionally, +the transition from the thermal phase to partial localization happens more rapidly as a function of disorder strength for larger +system sizes. The figure also illustrates the mean adjacent gap ratio when only averaging over neighboring energy eigenstates +with different disorder indices (dots). The errorbars show two standard deviations on the estimate of the mean. This average +coincides with the naive calculation at large system sizes. The figure also shows the adjacent gap ratio distribution for L = 16 +at (b) weak disorder W = 0.46, (c) intermediate disorder strength W = 2.27 and (d) strong disorder W = 6. These plots +include the distributions Eq. (17a) (dashed curve) and Eq. (17b) (dotted curve). The data agrees with Eq. (17a) at weak +disorder and transitions to the distribution (17b) at strong disorder. +and we use the entropy to identify the onset of partial +localization. +Figure 4(a) shows the entropy of the eigenstate with +energy closest to (Emin + Emax)/2 as a function of disor- +der strength W for different system sizes L. Each data +point represents the average entropy over 103 disorder +realizations with errorbars displaying two standard devi- +ations of the mean when assuming a Gaussian distribu- +tion. For low disorder, the entanglement entropy scales +linearly with the system size and hence agrees with the +expected volume-law scaling in the thermal phase. Ad- +ditionally, the entropy approaches the Page value with +increasing system size. +At large disorder, the entropy +seems to be roughly independent of system size. Thus, +the scaling of entropy is consistent with area-law for par- +tially localized eigenstates. +The sudden shift in scaling behavior of the entropy +verifies the transition from the thermal phase to partial +localization at strong disorder. The transition point is +identified by analyzing the variance of entanglement en- +tropy. Figure 4(b) illustrates the sample variance of the +entropy over 103 disorder realizations. The variance dis- +plays a peak when the system transitions from volume- +law to area-law scaling. +IV. +DISTINGUISHABLE FEATURES OF SCAR +STATES IN A PARTIALLY LOCALIZED +BACKGROUND +Scar states are commonly distinguished from a thermal +background by their low entanglement and oscillatory dy- +namics. In this section, we show that oscillatory dynam- +ics can also be utilized to distinguish scar states from +a partially localized background, while entanglement en- +tropy turns out not to be an effective tool to identify the +scar states. +A. +Entanglement entropy +The entanglement entropy of the scar states scales log- +arithmically with system size [29], while thermal states +display volume-law scaling. Therefore, the entanglement + +8 +0 +1 +2 +3 +4 +5 +⟨S⟩ +(a) +L = 8 +10 +12 +14 +16 +0 +2 +4 +6 +W +0 +1 +Var(S) +(b) +FIG. 4. +(a) Average bipartite entanglement entropy of the +eigenstate closest to the center of the spectrum ⟨S⟩ as a func- +tion of disorder strength W for different system sizes L. The +entropy is averaged over 103 disorder realizations with system +parameters λ = ∆ = J = 1. +Errorbars display two stan- +dard deviations on the estimate of average entropy assuming +a Gaussian distribution. At low disorder, the entropy displays +volume-law scaling with system size and approaches the Page +value (dashed lines) as expected in the thermal phase. +At +large disorder, the entropy follows area-law scaling with sys- +tem size. (b) Variance of bipartite entanglement entropy of +the eigenstate closest to the center of the spectrum. +The +variance is computed from 103 disorder realizations. As the +disorder strength is increased, the variance displays a sudden +peak. This indicates a transition from the thermal phase to +partial localization. The peak becomes higher at larger sys- +tem sizes. +entropy provides a way to identify the scar states in a +thermal background. Figure 5(a) illustrates the entropy +as a function of energy of a thermal system with size +L = 14 and disorder strength W = 0.5. The thermal +states form a narrow arc with maximum in the middle +of the spectrum while the scar state appears as an out- +lier at much lower entropy. The situation is different in +a partially localized background. Figure 5(b) illustrates +the entropy as a function of energy at strong disorder +W = 6. As discussed above, partially localized eigen- +states are weakly entangled making it difficult to identify +the scar state. We conclude that entanglement entropy +is an ineffective tool for distinguishing scar states from a +partially localized background. +0.0 +0.5 +1.0 +ϵ +0 +2 +4 +S +(a) +0.0 +0.5 +1.0 +ϵ +(b) +FIG. 5. +The entanglement entropy S as a function of nor- +malized energy ϵ = (E − Emin)/(Emax − Emin) where Emin +and Emax are the smallest and largest energies in the spec- +trum. +Lighter (darker) colors indicate lower (higher) den- +sity of points. +(a) We consider a thermal system of size +L = 14, disorder strength W = 0.5 and system parameters +λ = ∆ = J = 1. In the thermal phase, the energy eigenstates +form a narrow band with maximum at the center of the spec- +trum. The scar state (inside the green ring) is easily identified +since it appears isolated below the curve. (b) We consider a +partially localized system at strong disorder W = 6. The en- +ergy eigenstates are spread out at low entropy with the scar +state embedded among them. The entanglement entropy is +hence not an effective tool to distinguish the scar state from +a partially localized background. +B. +Fidelity +States initialized in the scar subspace distinguish them- +selves from a thermal background by displaying persis- +tent dynamic revivals. We now show that this behavior +also enables the identification of scar states from a par- +tially localized background. We quantify the dynamics of +quantum systems by the fidelity F(t). Let |ψ(0)⟩ be the +initial state and |ψ(t)⟩ = e−i ˆ +Ht |ψ(0)⟩ the time evolved +state. The fidelity is given by +F(t) = | ⟨ψ(0)|ψ(t)⟩ |2. +(20) +The time evolution of fidelity is most clearly understood +by considering the overlap of the initial state with all +energy eigenstates. Let |φi⟩ be the i-th energy eigenstate +with corresponding energy Ei and let ci be the inner +product between the i-th energy eigenstate and the initial +state ci = ⟨φi|ψ(0)⟩. The relation between fidelity and +the expansion coefficients ci is highlighted by rewriting +the fidelity according to +F(t) = +� +i +|ci|4 + +� +i̸=j +|ci|2|cj|2ei(Ei−Ej)t +(21) +It is clear from this expression that the dynamics of fi- +delity is sensitive to the distribution of |ci|2. We generally +display this distribution along with the fidelity for clarity. +We demonstrate the different dynamical behavior of +the thermal and partial MBL phases by initializing a sys- +tem of size L = 14 in a product state. First, we consider + +9 +0.0 +0.5 +1.0 +F +(a) +⟨F T⟩ +⟨F MBL⟩ +Fscar +0.0 +0.5 +1.0 +F +(b) +0 +1 +2 +3 +4 +5 +t/Tscar +0.0 +0.5 +1.0 +F +(c) +−25 +0 +25 +Ei +0.0 +0.1 +|ci|2 +(d) +−50 +0 +50 +Ei +0.0 +0.5 +(e) +−20 +0 +Ei +0.0 +0.1 +(f) +FIG. 6. +(a) The average fidelity of a random product state in +a thermal system at disorder strength W = 0.5. (b) The aver- +age fidelity in a partially localized system at disorder strength +W = 10. The system is initialized in a product state which +fully localizes (solid line). For comparison, the system is ini- +tialized in a random product state which only partially local- +izes |ψ(0))⟩ = |D, n⟩ with ND = 5 (dashed line), 10 (dashed +dotted line) and 35 (dotted line). (c) The system is initialized +in the scar subspace at any disorder strength. The average +fidelity is in all cases calculated over 103 disorder realizations. +The bottom panel displays the distribution of expansion coef- +ficients |ci|2 across energy in a single disorder realization. (d) +For the thermal phase W = 0.5. (e) For partial MBL W = 10 +with initial state |ψ(0)⟩ = |D, n⟩ for ND = 5. (f) For the +initial state being an equal weight linear combination of the +scar states. +a thermal system at disorder strength W = 0.5. +The +initial state is chosen as a random product state with +all product states having the same probability of being +drawn. +We ensure the initial state resides outside the +scar subspace by drawing a new product state if the first +has non-zero overlap with a scar state. We consider 103 +disorder realizations and draw a random product state +in each realization. In the i-th realization, the fidelity is +computed as a function of time Fi(t) and Fig. 6(a) shows +the average fidelity ⟨F(t)⟩ = 10−3 �103 +i=1 Fi(t) over all re- +alizations. Figure 6(d) shows the expansion coefficients +|ci|2 of a single disorder realization following the Gaus- +sian distribution as expected [42, 43]. Since the initial +state has large overlap with many different eigenstates, +the second sum in Eq. (21) rapidly vanishes due to can- +cellation between terms with different phase factors. As +a consequence, the fidelity quickly decreases and satu- +rates at Fi(t) ≈ � +i |ci|4 ≈ 0 at long times Tscar ≪ t for +all disorder realizations. These considerations agree with +the observed time evolution of the average fidelity in Fig. +6(a) which rapidly decreases to a value near zero. +Next, we consider the same setup when the system is +partially localized at large disorder W = 10. As discussed +in Sec. III A, the spectrum separates into fully MBL +eigenstates and partially localized eigenstates. +Conse- +quently, the dynamics depend greatly on the initial state. +The solid blue line in Fig. 6(b) is the average fidelity +over 103 disorder realizations when initialing the sys- +tem in a random product state which fully localizes, i.e. +|ψ(0)⟩ = |D, n⟩ with ND = 1. Fully MBL eigenstates +have significant overlap with only one product state, and +the average fidelity remains far from zero at all times as +observed in Fig. 6(b). +We note that a stronger disor- +der strength is needed to achieve MBL in larger systems. +Therefore, the average fidelity saturates significantly be- +low unity in Fig. 6(b) even though all product states with +ND = 1 in Fig. 1 are near identical to an energy eigen- +state. The average fidelity saturates closer to unity at +larger disorder strengths. +When the initial state is chosen as a product state that +only partially localizes, it has significant overlap with +multiple eigenstates. Consequently, the average fidelity +drops closer to zero as illustrated by the dashed and dot- +ted curves in Fig. 6(b). For these curves, we choose the +initial state randomly as |ψ(0)⟩ = |D, n⟩ with ND = 5, 10 +and 35. These initial states have significant support on +up to ND eigenstates causing the average fidelity to de- +crease with increasing ND. +Figure 6(e) illustrates the +distribution of |ci|2 for a single disorder realization for a +random initial state |ψ(0)⟩ = |D, n⟩ with ND = 5. The +distribution is more sparse than the thermal case. +Finally, we consider the initial state being a linear com- +bination of scar states +|ψscar⟩ = +1 +� +L +2 + 1 +L/2 +� +n=0 +|Sn⟩ . +(22) +When the initial state is chosen within the scar subspace, +the equal energy spacing causes the fidelity to display +persistent periodic revivals. In particular, for the equal +weight linear combination in Eq. (22), the fidelity is given +by +Fscar(t) = +1 +L +2 + 1 +� +1 + 2 +L/2 +� +n=1 +� +1 − +n +L +2 + 1 +� +cos(n∆Et) +� +. +(23) +Revivals occur at times tℓ = Tscarℓ = +2πℓ +∆Escar where ℓ ∈ N +and ∆Escar is the energy spacing between consecutive +scar states. +Figure 6(c) illustrates the fidelity of this +initial state and Fig. 6(f) shows the distribution of the +expansion coefficients. + +10 +In the thermal phase, states initialized respectively in- +side and outside the scar subspace behave differently. +The fidelity of states outside the scar subspace quickly +drops to zero, while any linear combination of scar states +display persistent revivals. In our analysis, we specifi- +cally initialized the system as a product state, but the +same conclusions hold for generic linear combinations of +product states. In a partially localized background, the +average fidelity distinguishes between states with sup- +port inside and outside the scar subspace. The average +fidelity of partially localized states saturates while scar +states display revivals. Again, our analysis concerns the +special case of initializing the system as a random prod- +uct state. If instead the initial state is a generic linear +combination of a large number of product states, the sec- +ond term of Eq. (21) will generally vanish due to phase +cancellation, and the average fidelity saturates near zero. +While this is true for generic linear combinations, there +exists particular states where the phase cancellation hap- +pens exceptionally slowly. We discuss these special initial +states in section VI and how to distinguish them from the +scar states. Summing up, the average fidelity represents +an effective tool for identifying scar states in both a ther- +mal and localized background. +Finally, we remark that the fidelity of individual dis- +order realizations are enough to distinguish initial states +with support inside and outside the scar subspace. This +statement is simple in the thermal phase where initial +states outside the scar subspace rapidly converges to zero. +At large disorder, the fidelity of individual disorder re- +alizations may oscillate rapidly contrary to the average +fidelity. However, these oscillations are generally com- +posed of frequencies different from the scar revivals. The +amplitude of the oscillations are also typically different +from the scar revivals. Thus, the scar states can be dis- +tinguished from a partially localized background. +V. +DISORDER STABILIZATION OF SCAR +REVIVALS +We study the dynamics of initial states with support +both inside and outside the scar subspace across all sym- +metry sectors. +In this case, we generally expect the +scar revivals to diminish. +The scar revivals are stabi- +lized when the initial state only has support on product +states with the same disorder indices as the scar states +D0 = (−1, −1, . . . , −1). We demonstrate this behavior +by initializing the system in a generic state only having +support on product states with disorder indices D0 +|ψstable⟩ = +1 +Nstable +� +|ψscar⟩ + +� +Ndw,n +β(Ndw) +n +|Ndw, D0, n⟩ +� +, +(24) +where Nstable is a normalization constant and β(Ndw) +n +are +drawn +randomly +from +the +interval +β(Ndw) +n +∈ +[0, 1.5/ +� +N (Ndw) +D0 +]. We reintroduce the index Ndw to de- +scribe product states with the same disorder indices in +different symmetry sectors. The time evolution of fidelity +is investigated at weak and strong disorder in 103 real- +izations. The coefficients β(Ndw) +n +are redrawn in each dis- +order realization. Figure 7(a) displays the disorder aver- +aged fidelity for a thermal system and a partially local- +ized system. In both cases, the average fidelity displays +persistent revivals with the revival amplitude decaying +and eventually saturating at a value around 0.5. +The fidelity amplitude quickly decays for a thermal +system. The explanation can be found by studying the +expansion coefficients |ci|2 as illustrated in Fig. 7(b). Be- +cause the system is thermal, the initial state has support +on many energy eigenstates. Consequently, terms with +different phases quickly cancel causing the fidelity ampli- +tude to saturate almost immediately. +At large disorder, the fidelity amplitude decays at a +much slower rate and only saturates alongside the ther- +mal graph after many revivals t ∼ 7Tscar. +We under- +stand this behavior by recalling the spectral structure at +large disorder. First, recall that the energy eigenstates +{|ED0,m⟩ |m = 1, 2, . . . , ND0} are near degenerate and +only have significant overlap with product states of the +same disorder indices as described in Eq. (15). There- +fore, the second term in Eq. (24) can be rewritten as a +sum of near degenerate eigenstates, +ND0 +� +n=1 +β(Ndw) +n +|Ndw, D0, n⟩ ≈ +ND0 +� +m=1 +γ(Ndw) +m +|ENdw,D0,m⟩ , +(25) +with γ(Ndw) +m += � +n β(Ndw) +n +⟨ENdw,D0,m|Ndw, D0, n⟩. Fur- +thermore, the scar states themselves are described by +the disorder indices D0, so the eigenstates |ENdw,D0,m⟩ +are close in energy to a scar state. +Consequently, the +eigenstates outside the scar subspace having large over- +lap with |ψstab⟩ are always close in energy to a scar state. +We sketch this structure in Fig. 8 where the eigenstates +|ENdw,D0,m⟩ have similar energy to the scar states for +all Ndw. These considerations agree with the observed +distribution of |ci|2 for a single disorder realization illus- +trated in Fig. 7(c). The expansion coefficients are sharply +peaked around the scar states and consequently the can- +cellation of terms with different phases takes place at +much larger times. +In this way, the partially localized background stabi- +lizes the scar revivals by rearranging the support outside +the scar subspace. The stabilization takes place whenever +the initial state is predominantly a linear combination of +product states with the same disorder indices as the scar +states D0. If product states with other disorder indices +D′ ̸= D0 are included, the stabilization will be less pro- +nounced. + +11 +10−4 +10−2 +100 +|ci|2 +(b) +−50 +0 +50 +E +10−4 +10−2 +100 +|ci|2 +(c) +0 +2 +4 +6 +8 +t/Tscar +0.00 +0.25 +0.50 +0.75 +1.00 +F +(a) +W = 10 +W = 0.5 +FIG. 7. +A system of size L = 14 with parameters ∆ = 1, J = 5, λ = 1 is initialized according to Eq. (24) in the thermal phase +at disorder strength W = 0.5 and the partial MBL phase at disorder strength W = 10. (a) The average fidelity over 103 disorder +realizations when the system is thermal and partially MBL. The disorder protects the scar revivals and the fidelity amplitude +decays much slower compared to the thermal case. The right panels illustrate the distribution of expansion coefficients |ci|2 +over energy Ei for a single disorder realization at disorder strength (b) W = 0.5 and (c) W = 10. The distribution of the +expansion coefficients is wide in the thermal phase and consists of narrow peaks near the scar states in the localized phase. +E +Symmetry sectors +∆Escar +∆Escar +∆Escar +Ndw0 +Ndw1 +Ndw2 +Ndw3 +FIG. 8. +At large disorder, the initial state Eq. (24) has +significant overlap with a small number of energy eigenstates +(black lines) as sketched in the figure. These eigenstates ap- +pear in clusters around the energy of the scar states (green +lines). A single cluster exists in every symmetry sector and +the energy gap between two adjacent clusters equals the en- +ergy gap between scar states ∆Escar. +VI. +DISORDER INDUCED APPROXIMATE +SCARS +Additional approximate scar states emerge as disorder +is introduced. These approximate scars appear because +some symmetry sectors contain energy eigenstates with +the same disorder indices. For instance, the eigenstates +|E2,D,1⟩ ≈ |↑↑↓↓↓↓⟩ and |E4,D,m⟩ ≈ αm1 |↑↑↓↑↓↓⟩ + +αm2 |↑↑↓↓↑↓⟩ for m = 1, 2 have the same disorder in- +dices D = (3, −1, −1, −1, −1, −1) but different number +of domain walls Ndw. Recall from Sec. III A that the en- +ergy of an eigenstate at large disorder is approximately +given by, +ENdw,D,m ≈ ∆MNdw,D + J +� +N (↑↑,↓↓) +Ndw,D − N (↑↓,↓↑) +Ndw,D +� ++ +� +i +diDi, +(26) +If an eigenstate |ENdw,D,m⟩ is described by the values +MNdw,D, N (↑↑,↓↓) +Ndw,D and N (↑↓,↓↑) +Ndw,D , then another eigenstate +|ENdw+2,D,m⟩ with Ndw + 2 domain walls and identical +disorder indices D is described by +MNdw+2,D = MNdw,D + 2, +(27a) +N (↑↑,↓↓) +Ndw+2,D = N (↑↑,↓↓) +Ndw,D − 2, +(27b) +N (↑↓,↓↑) +Ndw+2,D = N (↑↓,↓↑) +Ndw,D + 2. +(27c) +Using Eq. (26) and (27), one can show the energy dif- +ference between two eigenstates with the same disorder +indices D and number of domain walls ND and ND + 2 +is approximately +ENdw+2,D,m − ENdw,D,m ≈ ∆Escar, +(28) +where ∆Escar = 2(∆−2J) is the energy gap between the +scar states. This calculation demonstrates that towers +of approximate scar states appear in the spectrum as +disorder is introduced. +We demonstrate how the appearance of approximate +scars generates non-trivial dynamics. The system is ini- +tialized in a generic linear combination of product states +with disorder indices D1 = (3, −1, −1, . . . , −1) +|ψinduced +D1 +⟩ = +1 +Ninduced +� +Ndw,n +ζ(Ndw) +n +|Ndw, D1, n⟩ . +(29) +The coefficients are chosen randomly from the interval +ζ(Ndw) +n +∈ [0, 1] and Ninduced is a normalization constant. + +12 +0 +1 +2 +3 +4 +5 +t/Tscar +0.0 +0.5 +1.0 +F +(a) +0 +1 +2 +3 +4 +5 +t/Tscar +0.0 +0.5 +1.0 +(b) +0 +1 +2 +3 +4 +5 +t/Tscar +0.0 +0.5 +1.0 +(c) +−50 +0 +50 +E +0.000 +0.025 +|ci|2 +(d) +−50 +0 +50 +E +0.0 +0.1 +(e) +−50 +0 +50 +E +0.0 +0.2 +(f) +FIG. 9. +The average fidelity of the initial state Eq. (29) over 103 disorder realizations for system size L = 14 with parameters +λ = ∆ = 1, J = 5 at disorder strength (a) W = 0.5, (b) W = 5 and (c) W = 10. The shaded areas show the interquartile range +(middle 50%) of the disorder realizations. The corresponding distribution of expansion coefficients |ci|2 of a single disorder +realization at disorder strength (d) W = 0.5, (e) W = 5 and (f) W = 10. At weak disorder, the initial state has significant +overlap with many energy eigenstates and the average fidelity quickly decays to zero. As the disorder strength is increased, +the initial state has significant overlap with a small number of energy eigenstates with equal energy spacing. Consequently, the +average fidelity shows persistent revivals. +We study this initial state because, at large disorder, it +is a linear combination of an approximate scar tower. +We consider 103 disorder realizations at different disor- +der strengths and the fidelity is computed for each re- +alization. Figure 9(a) displays the average fidelity of a +thermal system at weak disorder W = 0.5. In this case, +there is nothing special about the initial state in Eq. (29) +and it quickly decays to zero similar to Fig. 6(a). The +dynamical behavior changes remarkably as the disorder +strength is increased as illustrated in Fig. 9(b)-(c). At +stronger disorder, the initial state Eq. (29) has large over- +lap with eigenstates that are approximately equidistant +in energy. +Consequently, the average fidelity oscillates +with a period given by the energy gap Tscar = +2π +∆Escar . +The revival amplitude increases with disorder strength. +The shaded area in Fig. 9(a)-(c) displays the interquar- +tile range of disorder realizations. Figures 9(d)-(f) shows +the expansion of the initial state in energy eigenstates at +(d) weak disorder W = 0.5, (e) strong disorder W = 5 +and (f) very strong disorder W = 10. As expected, the +initial state is distributed over a wide range of eigenstates +in the thermal phase similar to Fig. 6(d). As the disor- +der strength increases, the initial state has higher and +higher overlap with eigenstates in an approximate tower +of equidistant states. +Figure 9 demonstrates that it is possible to observe +revivals from generic linear combinations of the states +{|Ndw, D, n⟩ |Ndw = 0, 2, . . . ; n = 1, 2, . . .} at large dis- +order. However, the effects may be enhanced by choosing +the initial state more carefully. The initial state in Eq. +(29) is, in some sense, the worst case scenario. When all +product states with disorder indices D are included in +the sum, the initial state generally has significant overlap +with all relevant energy eigenstates {|ENdw,D,m⟩ |Ndw = +0, 2, . . . ; m = 1, 2, . . .}. This causes a large spread in the +distribution of |ci|2 resulting in a faster decay of the av- +erage fidelity. If instead, we consider an initial state with +exactly one product state from each symmetry sector, the +spread of |ci|2 is smaller +| ˜ψinduced +D1 +⟩ = +1 +� +L +2 − 1 +� +|↑↑↓↓↓↓↓ . . . ↓⟩ + |↑↑↓↑↓↓↓ . . . ↓⟩ ++ |↑↑↓↑↓↑↓ . . . ↓⟩ + . . . + |↑↑↓↑↓↑ . . . ↓↑↓↓⟩ +� +. +(30) +Figure 10(a) shows the average fidelity of this initial state +over 103 disorder realizations at strong disorder W = +10 and Fig. 10(b) displays the distribution of |ci|2 for a +single realization. As expected, the distribution of |ci|2 +is narrower and the revival amplitude larger compared to +Fig. 9. +The initial states Eq. (29) and (30) display revivals +similar to the scar states. However, one may distinguish +these initial states from the scar subspace by noting that +the average fidelity in Fig. 9 and 10 decays to zero, while +the amplitude in Fig. 6(c) and 7 remain strictly larger +than zero. The different dynamical behavior is caused by +Eq. (29) and (30) being composed of approximate scar +towers while the original scars |Sn⟩ are exactly equally +spaced in energy. + +13 +0 +2 +4 +t/Tscar +0.0 +0.5 +1.0 +F +(a) +−50 +0 +50 +E +0.0 +0.1 +|ci|2 +(b) +FIG. 10. +(a) Average fidelity of the initial state Eq. (30) +over 103 disorder realizations with system size L = 14 and +parameters λ = ∆ = 1, J = 5 and W = 10. The shaded +area displays the interquartile range of the disorder realiza- +tions. The average fidelity displays persistent revivals with +larger amplitude compared to Fig. 7. (b) Expansion of the +initial state across energy eigenstates. The coefficients |ci|2 +are sharply peaked around certain energies which are approx- +imately equally spaced. +VII. +CONCLUSION +Building on a known method to find parent Hamil- +tonians, we proposed a way to determine Hamiltonians +hosting a tower of QMBS. Starting from the model by +Iadecola and Schecter, we used this method to identify +all local 1- and 2-body Hamiltonians of the scar tower +|Sn⟩. Among these Hamiltonians, we found operators fa- +cilitating the implementation of local disorder while pre- +serving the scar states. When introducing disorder, the +mean level spacing statistics shifts from the GOE to the +Poisson distribution and the entanglement entropy goes +from volume-law to area-law scaling with system size. We +conclude the system transitions from the thermal phase +to being partially localized. A theory describing the par- +tially localized eigenstates was developed and verified nu- +merically. +In total, we determined a system hosting a +tower of scar states with the remaining spectrum being +either thermal or partially localized depending on the +disorder strength. +We studied the properties of scar states embedded in a +localized spectrum and compared with the corresponding +features in a thermal spectrum. In contrast to thermal +systems, the bipartite entanglement entropy does not en- +able the identification of scar states in a localized back- +ground. The average fidelity, on the other hand, effec- +tively identifies the scar subspace. +We investigated the effect of localization on initial +states with support both inside and outside the scar sub- +space. For a thermal system, the fidelity displays persis- +tent revivals with rapidly decreasing amplitude. In con- +trast, the revival amplitude decays slower for a partially +localized system. +Hence, partial localization stabilizes +the persistent revivals of states initialized partly outside +the scar subspace. +Finally, we demonstrated how additional approximate +scar states emerge as disorder is introduced. When ini- +tializing the system as a superposition of these states, the +average fidelity displays revivals with the same period as +the true scar states. While this effect does not rely on +fine-tuning the initial state, the revivals are amplified by +choosing the initial state appropriately. +ACKNOWLEDGMENTS +This work has been supported by the Carlsberg Foun- +dation under grant number CF20-0658. +Appendix A: Proof that |Sn⟩ are eigenstates of all +operators in Tab. I with equal energy spacing +In section II C, we found L + 4 operators having the +scar states as eigenstates equidistantly spaced in energy. +Since this analysis was carried out for finite system sizes +L = 8, 10, 12, 14, the validity of this statement is not +guaranteed for larger system sizes. In this appendix, we +rigorously prove the scar states |Sn⟩ are equally spaced +eigenstates of all operators in Tab. I. Since the scar states +are constructed iteratively by applying the operator Q†, +we generally prove this statement using proof by induc- +tion. +First, we consider the operator ˆHz = � +i ˆσz +i . +The +lowest scar state |S0⟩ = |↓↓ . . . ↓⟩ is trivially an eigen- +state of ˆHz. +A straightforward calculation shows that +[ ˆHz, ˆQ†] = 2 ˆQ† and by induction all other scar states are +eigenstates because +ˆHz |Sn+1⟩ ∝ ˆHz ˆQ† |Sn⟩ += +� +Ez,n ˆQ† + 2 ˆQ†� +|Sn⟩ += +� +Ez,n + 2 +� +|Sn+1⟩ , +(A1) +where ˆHz |Sn⟩ = Ez,n |Sn⟩. +The scar states are also +equally spaced in energy En+1,z − En,z = 2. +A simi- +lar argument holds for ˆHodd +zz +since [ ˆHodd +zz , ˆQ†] = −4 ˆQ† +where the energy gap between scar states is −4. +Next, we consider the operators ˆDi = ˆσz +i + ˆσz +i+1 + +ˆσz +i ˆσz +i+1. Recall that ˆDi is related to the projection oper- +ators through ˆDi = 4 ˆP ↑ +i ˆP ↑ +i+1 − ˆ1 where ˆP ↑ +i = (ˆ1 + ˆσz +i )/2 +projects site i onto spin-up. First note that ˆDi |S0⟩ = +(4 ˆP ↑ +i ˆP ↑ +i+1 − ˆ1) |↓↓ . . . ↓⟩ = − |↓↓ . . . ↓⟩. A simple calcu- +lation shows that ˆDi commutes with ˆQ† by noting that + +14 +ˆP ↑ +i ˆP ↓ +i = 0 +[ ˆDi, ˆQ†] = 4 +L +� +j=1 +(−1)j� +ˆP ↓ +j−1[ ˆP ↑ +i , ˆσ+ +j ] ˆP ↓ +j+1 ˆP ↑ +i+1 ++ ˆP ↑ +i ˆP ↓ +j−1[ ˆP ↑ +i+1, ˆσ+ +j ] ˆP ↓ +j+1 +� += 4(−1)i� +ˆP ↓ +i−1ˆσ+ +i ˆP ↓ +i+1 ˆP ↑ +i+1 − ˆP ↑ +i ˆP ↓ +i ˆσ+ +i+1 ˆP ↓ +i+2 +� += 0. +(A2) +Thus, for all scar states we have ˆDi |Sn⟩ = − |Sn⟩. Alter- +natively, one may note that |Sn⟩ by construction does not +contain adjacent sites being spin-up. Therefore, ˆP ↑ +i ˆP ↑ +i+1 +naturally annihilates the state. +Next, we consider the operator ˆHalt +xz . Before studying +the action of ˆHalt +xz on the scar states, we prove by in- +duction that the commutator [ ˆHalt +xz , ˆQ†] annihilates |Sn⟩. +The commutator is given by +[ ˆHalt +xz , ˆQ†] = +L +� +i=1 +� +2 +� ˆP ↓ +i ˆσ+ +i+1ˆσ− +i+2 − ˆσ+ +i ˆσ+ +i+1 ˆP ↓ +i+2 +� ++ i +� ˆP ↓ +i ˆσ+ +i+1ˆσy +i+2 + ˆσy +i ˆσ+ +i+1 ˆP ↓ +i+2 ++ ˆσz +i ˆσy +i+1ˆσ+ +i+2 ˆP ↓ +i+3 − ˆP ↓ +i ˆσ+ +i+1ˆσy +i+2ˆσz +i+3 +�� +, +(A3) +where ˆP ↓ +i = (ˆ1 − ˆσz +i )/2 is the local projection onto spin- +down. By direct calculation, one can show the lowest scar +state is annihilated by this expression [ ˆHalt +xz , ˆQ†] |S0⟩ = 0. +A lengthy, yet straightforward, calculation also shows the +nested commutator vanishes +� +[ ˆHalt +xz , ˆQ†], ˆQ†� += 0. +We +now prove by induction that the commutator annihilates +all scar states. Assume [ ˆHalt +xz , ˆQ†] |Sn⟩ = 0 and consider, +[ ˆHalt +xz , ˆQ†] |Sn+1⟩ ∝ [ ˆHalt +xz , ˆQ†] ˆQ† |Sn⟩ += +� +ˆQ†[ ˆHalt +xz , ˆQ†] + +� +[ ˆHalt +xz , ˆQ†], ˆQ†�� +|Sn⟩ += 0. +(A4) +Having shown this intermediate result, we prove by in- +duction that the operator ˆHalt +xz annihilates the scar states. +First we show the operator ˆHalt +xz annihilates |S0⟩ +ˆHalt +xz |S0⟩ = +L +� +i=1 +(−1)i(ˆσx +i ˆσz +i+1 + ˆσz +i ˆσx +i+1) |↓↓ . . . ↓⟩ += +L +� +i=1 +(−1)i+1(ˆσx +i + ˆσx +i+1) |↓↓ . . . ↓⟩ += 0, +(A5) +where the second term cancels the first after changing +summation index i + 1 → i. Next, we show by induction +that the n-th scar state is annihilated by ˆHalt +xy . Assume +ˆHalt +xz annihilates |Sn⟩ and consider +ˆHalt +xz |Sn+1⟩ ∝ ˆHalt +xz ˆQ† |Sn⟩ += ( ˆQ† ˆHalt +xz + [ ˆHalt +xz , ˆQ†]) |Sn⟩ += 0. +(A6) +The first term vanishes by assumption and the second +term is exactly what we considered in Eq. (A4). In total, +we conclude ˆHalt +xy has |Sn⟩ as eigenstates equidistantly +separated in energy (with zero energy spacing). +Finally we consider the operator ˆHalt +yz . One can prove +this operator annihilates the scar states using similar ar- +guments to above. The commutator is given by +[ ˆHalt +yz , ˆQ†] =i +L +� +i=1 +� +2 +� ˆP ↓ +i ˆσ+ +i+1ˆσ− +i+2 + ˆσ+ +i ˆσ+ +i+1 ˆP ↓ +i+2 +� +− ˆσx +i ˆσ+ +i+1 ˆP ↓ +i+2 − ˆP ↓ +i ˆσ+ +i+1ˆσx +i+2 ++ ˆP ↓ +i ˆσ+ +i+1ˆσx +i+2ˆσz +i+3 − ˆσz +i ˆσx +i+1ˆσ+ +i+2 ˆP ↓ +i+3 +� +. +(A7) +Using induction, one can prove the commutator annihi- +lates all scar states [ ˆHalt +yz , ˆQ†] |Sn⟩ = 0 and the operator +annihilates the lowest scar state ˆHalt +yz |S0⟩ = 0. Retracing +the steps in Eq. (A6), we find that ˆHalt +yz annihilates all +scar states. +[1] J. M. Deutsch, Quantum statistical mechanics in a closed +system, Phys. Rev. A 43, 2046 (1991). +[2] M. Srednicki, Chaos and quantum thermalization, Phys. +Rev. E 50, 888 (1994). +[3] M. Rigol, V. 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E 85, 036209 +(2012). + diff --git a/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7d4ed081deed526927194479f7c9335bf3d5e95 --- /dev/null +++ b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt @@ -0,0 +1,1181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf,len=1180 +page_content='Tower of quantum scars in a partially many-body localized system Michael Iversen and Anne E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Nielsen Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark Isolated quantum many-body systems are often well-described by the eigenstate thermalization hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' There are, however, mechanisms that cause different behavior: many-body localization and quantum many-body scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Here, we show how one can find disordered Hamiltonians hosting a tower of scars by adapting a known method for finding parent Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using this method, we construct a spin-1/2 model which is both partially localized and contains scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate that the model is partially localized by studying numerically the level spacing statistics and bipar- tite entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As disorder is introduced, the adjacent gap ratio transitions from the Gaussian orthogonal ensemble to the Poisson distribution and the entropy shifts from volume-law to area-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We investigate the properties of scars in a partially localized background and compare with a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder, states initialized inside or outside the scar subspace display different dynamical behavior but have similar entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demon- strate that localization stabilizes scar revivals of initial states with support both inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we show how strong disorder introduces additional towers of approximate scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' INTRODUCTION The eigenstate thermalization hypothesis (ETH) de- scribes how isolated quantum systems reach thermal equilibrium [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The hypothesis is a statement about generic quantum many-body systems and has been veri- fied for a wide variety of physical models [3–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Despite the effectiveness of ETH, several phenomena are known to cause non-thermal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' One such mechanism is many-body localization (MBL) [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL appears in many-body interacting sys- tems with local disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When the disorder strength is sufficiently strong, it causes a change in the structure of the energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' An extensive set of quasi- local integrals of motion (LIOM) emerges and the en- ergy eigenstates localize [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, all en- ergy eigenstates behave non-thermally and MBL repre- sents a strong violation of ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this phenomenon is well-established for finite systems, the stability of MBL in the thermodynamic limit is still an open question [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Another mechanism leading to non-thermal behav- ior was discovered in experiments with kinetically con- strained Rydberg atoms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The atoms were arranged with strong nearest neighbor interactions so the simul- taneous excitation of neighboring atoms was prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When initializing the system in the N´eel state, observ- ables displayed abnormal persistent oscillations – con- trary to the predictions by ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Subsequent theoreti- cal works uncovered that the revivals were caused by a small number of non-thermal eigenstates dubbed quan- tum many-body scars (QMBS) [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These scar states have approximately equal energy spacing so any initial state in the scar subspace displays revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar states are uncommon and represent a vanishingly small part of an otherwise thermalizing spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, QMBS represent a weak violation of ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this work, we realize both ETH-breaking mecha- nisms simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We study a one-dimensional dis- ordered spin-1/2 chain hosting a tower of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the model transitions from the thermal phase to being partially localized while pre- serving the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In earlier works, a single scar state was embedded in an otherwise MBL spectrum [26– 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Our work adds to these studies by considering a full tower of QMBS in an MBL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The presence of multiple scar states, enables us to study the effect of lo- calization on the dynamical revivals characteristic of scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using this model, we demonstrate how scar states can be distinguished from a localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We also find two phenomena originating from the interplay between QMBS and localization: disorder stabilization of scar revivals and disorder induced approximate scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II A, we summarize the model by Iadecola and Schecter which is the starting point of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B, we explain how we find Hamiltonians having a set of scar states with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II C, we use this method to determine all local 1- and 2-body Hamiltonians for the tower of scar states in the Iadecola and Schecter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A, we show that a subset of these Hamiltoni- ans partially localize as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We quan- tify the partial localization as a special structure in the energy eigenstates and compare with results from exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We verify the localization by studying the level spacing statistics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III B and the entan- glement entropy in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' IV, we show that the fidelity between initial states and the corresponding time evolved states can be utilized to distinguish the scar states from the partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We fur- ther show that the bipartite entanglement entropy is an ineffective tool for distinguishing scar states from a par- tially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V, we demonstrate how scar revivals are stabilized by strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VI, we uncover additional towers of approximate scar states which emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='01681v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='dis-nn] 4 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Model by Iadecola and Schecter We take the model by Iadecola and Schecter as our starting point [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider a one-dimensional spin- 1 2 chain of even length L with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The local Hilbert space on each site is described by the eigenkets |↑⟩ and |↓⟩ of the Pauli z-matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ˆσz |↑⟩ = |↑⟩ and ˆσz |↓⟩ = − |↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The model by Iadecola and Schecter is given by ˆH0 = L � i=1 � λ(ˆσx i − ˆσz i−1ˆσx i ˆσz i+1) + ∆ˆσz i + J ˆσz i ˆσz i+1 � , (1) with λ, ∆, J ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' All indices are understood as modulo L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' the index i+L is identified as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operators ˆσx i , ˆσy i and ˆσz i are the Pauli matrices acting on site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) flips the spin si at site i if its nearest neighbors are in different states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' si−1 ̸= si+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The second term is a magnetic field along the z-direction with strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The third term represents nearest neighbor interactions with strength J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Two adjacent spins in different states represent a do- main wall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↑↓ or ↓↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The Hamiltonian conserves the number of domain walls Ndw because only spins with dif- ferent neighbors are allowed to change their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fur- thermore, the Hamiltonian is invariant under spatial in- version and translation, but these symmetries are broken when disorder is introduced in section III and we will not consider them any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For nonzero values of λ, ∆ and J, the energy eigen- states are thermal except for a small number of ETH- violating scar states grouped into two towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Through- out this work, we only focus on one of these towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This tower contains L/2+1 eigenstates and the n-th state |Sn⟩ is constructed by acting n times with the operator ˆQ† on the “all-spin-down” state |Sn⟩ ∝ � ˆQ†�n |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (2) The operator ˆQ† is given by ˆQ† = L � i=1 (−1)i ˆP ↓ i−1ˆσ+ i ˆP ↓ i+1, (3) where ˆσ+ i = (ˆσx i +iˆσy i )/2 is the raising operator and ˆP ↓ i = (ˆ1 − ˆσz i )/2 is the local projection onto spin down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The n-th scar state has energy En = 2(∆ − 2J)n + (J − ∆)L, number of domain walls Ndw = 2n and generally appears central in the spectrum after resolving all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the scar states are equally spaced in energy, any initial state in the scar subspace displays the dynami- cal revivals characteristic of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, it was shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [29] that the bipartite entanglement en- tropy of the scar states displays logarithmic scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Determining Hamiltonians All eigenstates of ˆH0 located near the middle of the spectrum are thermal except the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We wish to extend the model so the scar states are embedded in a MBL background instead of a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL is possible in systems with quench disorder, and it has been realized in numerous models by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' introduc- ing a disordered magnetic field [17], bond-disorder [30] or disordered nearest-neighbor interactions [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Unfor- tunately, disorder cannot be introduced naively to the Hamiltonian ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When promoting any parameter to be- ing site-dependent λ → λi, ∆ → ∆i or J → Ji, the scar states are no longer eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, disorder must be introduced through new terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this section, we uncover all local few-body Hamiltonians which share the scar states as eigenstates and maintain equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the next section, we show that a subset of these Hamiltonians are partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We search for local Hamiltonians following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The set of 2L×2L Hermitian operators form a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Most of these operators are long-ranged, contain many-body interactions and are difficult to realize in ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we restrict ourselves to Hamiltoni- ans containing local 1- and 2-body Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This subspace is spanned by the operator basis B2 = � ˆσa i ���a ∈ {x, y, z}, i ∈ ZL � ∪ � ˆσa i ˆσb i+1 ���a, b ∈ {x, y, z}, i ∈ ZL � , (4) where ZL = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L} are the first L integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This subspace is considerably smaller than the full operator vector space and has dimension |B2| = 12L where | · | denotes the number of elements in a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Any local 1- or 2-body interacting Hamiltonian can be expressed as a linear combination of the basis elements ˆH = |B2| � i=1 αiˆhi, ˆhi ∈ B2, (5) where αi ∈ R are free coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' To simplify no- tation, we collect the coefficients in a vector α = (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , α|B2|)T where T is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We search for the vector of parameters α so the re- sulting Hamiltonian has |Sn⟩ as eigenstates for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state |Sn⟩ is an eigenstate of ˆH if and only if the energy variance of |Sn⟩ is exactly zero ⟨Sn| ˆH2|Sn⟩ − ⟨Sn| ˆH|Sn⟩ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (6) Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5), the expression becomes αT Cnα = 0, (7) where Cn is the quantum covariance matrix [Cn]ij = ⟨Sn|ˆhiˆhj|Sn⟩ − ⟨Sn|ˆhi|Sn⟩ ⟨Sn|ˆhj|Sn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (8) 3 Equation (7) is satisfied when the vector of coefficients lies in the null space of the quantum covariance matrix α ∈ Null(Cn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Cnα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We ensure all scar states |Sn⟩ are simultaneously eigenstates of ˆH by demanding the vector of coefficients α lies in the null space of ev- ery covariance matrix α ∈ Null(C0) ∩ Null(C1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ∩ Null(CL/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this condition ensures all scar states are eigenstates of ˆH, they are not necessarily equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Equal energy spacing is established by imposing another set of requirements ⟨Sn+2| ˆH|Sn+2⟩ − ⟨Sn+1| ˆH|Sn+1⟩ = ⟨Sn+1| ˆH|Sn+1⟩ − ⟨Sn| ˆH|Sn⟩ , (9) for all n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5), we find Gα = 0, (10) where we introduce the rectangular matrix of energy gap differences [G]ij = ⟨Si+2|ˆhj|Si+2⟩ − 2 ⟨Si+1|ˆhj|Si+1⟩ + ⟨Si|ˆhj|Si⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (11) We observe that the scar states are equally spaced in en- ergy when the coefficient vector resides in the null space of the gap matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In summary, the scar states appear as eigenstates of the Hamiltonian with equal energy spacing when the vector of coefficients lies in the intersection α ∈ L/2 � n=0 Null(Cn) ∩ Null(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12) It is straightforward to determine this subspace numeri- cally since the scar states are known analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note however, that while the matrices Cn and G are com- plex, we only search for real vectors α ∈ R|B2| (for com- plex vectors α ∈ C|B2|, the linear combination in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5) is not necessarily Hermitian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We find real coeffi- cient vectors by stacking the real and imaginary parts of the matrices (Re(C0), Im(C0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , Re(CL/2), Im(CL/2), Re(G), Im(G))T and determining the null space of the resulting rectangular matrix by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' singular value de- composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The vectors αi produced by this numerical method are typically dense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' have few nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As a con- sequence, the corresponding operator � i αiˆhi is difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We overcome this difficulty by noting that if {αi|i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='} lies in the null space Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12), then any linear combination of these vectors also lies in the null space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We apply a heuristic algorithm to determine sparse vectors in the subspace [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Generalized models We apply the numerical method for system sizes L = 8, 10, 12, 14 and for all sizes find L + 4 linearly indepen- dent vectors αi satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The corresponding (i) ˆHz = �L i=1 ˆσz i (ii) ˆDi = ˆσz i + ˆσz i+1 + ˆσz i ˆσz i+1, for i ∈ ZL (iii) ˆHodd zz = �L/2 i=1 ˆσz 2i−1ˆσz 2i (iv) ˆHalt xz = �L i=1(−1)i(ˆσx i ˆσz i+1 + ˆσz i ˆσx i+1) (v) ˆHalt yz = �L i=1(−1)i(ˆσy i ˆσz i+1 + ˆσz i ˆσy i+1) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Local 1- and 2-body operators which have |Sn⟩ for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2 as energy eigenstates with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operators are determined by applying the nu- merical method presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B and Appendix A proves the statement rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' operators are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The first operator ˆHz was already present in the initial model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) and adds nothing new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The L operators ˆDi act locally on sites i and i+1 and represent good candidates for adding quench disorder into the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Indeed, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III, we demonstrate the system partially localizes when introducing sufficiently strong disorder via these opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The third operator ˆHodd zz represents an interaction between every odd site and its right neighbor with equal interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fourth and fifth operators ˆHalt xz and ˆHalt yz flip spins with the sign of the term determined by the nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using the numerical method, we rediscover the 1- and 2-body terms of the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) by starting from the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As noted above, the operator ˆHz was already present in the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, the third term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) is a linear combination of the oper- ators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I: �L i=1 ˆσz i ˆσz i+1 = �L i=1 ˆDi − 2 ˆHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, the operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I only represent L + 2 non-trivial extensions to the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The numerical method presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B finds all operators in the operator subspace span(B2) hosting the tower of scars for finite L (up to length L = 14 in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, in principle, the scar states may not be eigenstates of these operators at larger L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, in Appendix A we prove analytically for all even L that the scar states remain eigenstates with equal energy spacing for all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The method from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B can be extended by in- cluding all 3-body terms to the basis B3 = B2 ∪ {ˆσa i ˆσb i+1ˆσc i+2 ���a, b, c ∈ {x, y, z}, i ∈ ZL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This results in a myriad of new operators – including the first term from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, with a large enough operator basis, the numerical method fully recovers the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since long-ranged many-body interactions are less rele- vant experimentally, we will not explore this possibility any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we remark that the effectiveness of this ap- proach is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For an eigenstate of a generic local Hamiltonian, it is unlikely for another local Hamil- 4 tonian to exist that shares the same eigenstate [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Con- trary to this, we find a large subspace of local Hamiltoni- ans sharing a full tower of scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We attribute the effectiveness of our study to the analytical structure of the scar states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Our methods are not expected to be valuable starting from generic eigenstates but may be equally effective in other scarred models with similar amount of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MANY-BODY LOCALIZATION In the last section, we determined a subspace of Hamil- tonians with the scar states |Sn⟩ as eigenstates equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Now, we study a concrete Hamiltonian from this subspace ˆH = ˆH0 + L � i=1 di ˆDi, (13) with di chosen randomly from the uniform probability distribution di ∈ [−W, W] where W > 0 is the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The action of ˆDi is given by ˆDi |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ = � 3 |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ , if si = si+1 = ↑ − |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ , otherwise (14) The operator ˆDi is related to the projection operators through ˆDi = 4 ˆP ↑ i ˆP ↑ i+1 − ˆ1 with ˆP ↑ i = (ˆ1 + ˆσz i )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We remark that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [29] also observes that the operator ˆP ↑ i ˆP ↑ i+1 preserves the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The model conserves the number of domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The dimension of the symmetry sector containing Ndw do- main walls is given by the binomial coefficient 2( L Ndw ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We generally consider the largest symmetry sector with Ndw = 2⌊L/4⌋ domain walls where ⌊·⌋ is the function rounding down to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Partial many-body localization A physical system may transition to the MBL phase when disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL is usually real- ized with the disorder term in the Hamiltonian acting uniquely on each basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, a complete set of LIOMs emerge and all energy eigenstates are fully described by their eigenvalues of the LIOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The situation is slightly different in our model because the disorder term � i di ˆDi treats some basis states the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operator ˆDi is only sensitive to whether spins i and i+1 are both up (it acts identically on states where spins i and i+1 are ↓↓, ↓↑ or ↑↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the operator � i di ˆDi has the same action on product states with all consecutive spin-ups placed identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We do not expect these to localize in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Instead, we anticipate the spectrum to separate into fully MBL eigenstates and partially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This structure is most easily described when the prod- uct states |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ are relabeled to reflect the ac- tion of � i di ˆDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this spirit, we define |Ndw, D, n⟩ as a simultaneous eigenstate of the ˆDi’s with eigenvalues D = (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DL) where Di ∈ {−1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We will refer to D as the disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, the state |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ is not fully described by D since mul- tiple states can have the same eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we further label the states by their number of domain walls Ndw and introduce a dummy index n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , N (Ndw) D to distinguish states with identical Ndw and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For in- stance, if two states |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ and |s′ 1s′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' s′ L⟩ have the same number of domain walls Ndw and disorder indices D, then they are relabeled as |Ndw, D, n⟩ for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note that some labelings are invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider the vector of eigenvalues D = (3, −1, 3, 3) for a small system L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The “3”s imply all spins are up, while the “−1” entail at least one spin is down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the following, we study a single symmetry sector and hence omit the Ndw index for clar- ity but reintroduce it in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V and VI when studying multiple symmetry sectors at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Upon introducing strong disorder, we expect LIOMs to emerge which are localized on the operators ˆDi and en- ergy eigenstates are characterized by their eigenvalues of the LIOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we expect the energy eigenstates to be close to linear combinations of product states with the same disorder indices |ED,m⟩ ≈ ND � n=1 αmn |D, n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (15) with αmn ∈ R and m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This expression is an approximation rather than an equality due to an exponentially small overlap with states |D′, n⟩ with dif- ferent disorder indices D′ ̸= D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The special case ND = 1 corresponds to the disorder term acting uniquely on the basis state |D, 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We expect the corresponding energy eigenstate |ED,1⟩ ≈ |D, 1⟩ to be MBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For ND > 1, the states {|ED,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} are only partially MBL since the LIOMs do not fully describe each state and all additional structure is captured by the extra in- dex m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The above considerations are verified in numerical sim- ulations by considering a system of size L = 8 at strong disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 1 illustrates the norm squared overlap of all energy eigenstates |ED,m⟩ with the prod- uct states |D, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The (i, j)-th pixel displays the norm squared overlap between the i-th product state and j- th energy eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The product states on the second axis are sorted according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The energy eigenstates are reordered to allow the diagonal shape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the upper left corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1, each eigenstate has high overlap with a single product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Numerical analysis reveals that these product states exactly coincide with those being fully described by their disorder indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ND = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These results support the claim that such eigen- 5 |ED,m⟩ |D, n⟩ 1 2 3 4 20 ND (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 |⟨D, n|ED,m⟩|2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The norm squared overlap of the energy eigenstates with the product states | ⟨D, n|ED,m⟩ |2 for system size L = 8, disorder strength W = 10 and parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The color of pixel (i, j) displays the overlap between the i’th product state and the j’th eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The product states are sorted into ascending order according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The second axis on the right hand side groups the product states according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The insets show eigenstates with significant weight on (a) two, (b) three and (c) four product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure verifies that all energy eigenstates are approximately linear combinations of product states with the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' states fully localize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The next eigenstates shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(a) each has significant overlap with exactly two product states of the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The pattern contin- ues: we find eigenstates that are linear combinations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(b) three, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(c) four, and (bottom right corner) twenty product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In each case, the product states have the same disorder indices and hence correspond to {|D, n⟩ |n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} for ND = 3, 4, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These ob- servations are not restricted to L = 8, but seem universal at all system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For larger system sizes, the number and sizes of the blocks increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we note that the scar state within the considered symmetry sector is located in the block ND = 20 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state is generally an equal weight linear combination of prod- uct states with the maximum ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This fact will play an important role when we explore the system dynamics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we discuss how the eigenstates are distributed in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The magnetization MD = � i σz i of a prod- uct state |D, n⟩ is fixed by the symmetry sector Ndw E Thermal |Sn⟩ Partial MBL |ED1,m1⟩ |ED2,1⟩ |ED2,2⟩ |ED2,3⟩ |ED3,m3⟩ |ED4,m4⟩ |ED5,m5⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Sketch of the spectrum in the thermal phase (left) and in the partially localized phase (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the energy levels follow the Wigner-Dyson surmise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As disorder is introduced, the spectrum experiences partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Eigenstates with similar indices D are near de- generate and the spectrum forms clusters of such eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state lies in the largest of these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' and disorder indices D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Likewise, the number N (↑↑,↓↓) D of adjacent spins pointing in the same direction (↑↑ or ↓↓) and the number N (↑↓,↓↑) D of adjacent spins point- ing in opposite directions (↑↓ or ↓↑) are also fully de- termined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the terms ∆ � i ˆσz i , J � i ˆσz i ˆσz i+1 and � i di ˆDi have the same action on all product states with the same number of domain walls and disorder in- dices: {|D, n⟩ |n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder, the energy of an eigenstate is approximately ED,m ≈ ∆MD + J(N (↑↑,↓↓) D − N (↑↓,↓↑) D ) + � i diDi with a small correction that depends on the value of the m index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The slight additional contribution originates from the term � i λ(ˆσx i − ˆσz i−1ˆσx i ˆσz i+1) and scales with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, at large disorder, the set of eigenstates {|ED,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} are near degenerate and form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A scar state resides in the largest of these clusters in all symmetry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 2 illustrates the spectral struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 2 is highly idealized to highlight the structure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In practice, it is highly likely for two or more clusters to overlap making the structure less apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Spectral statistics The distribution of energy gaps distinguishes the ther- mal and MBL phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let Ei be the energies of the Hamiltonian in ascending order and δi = Ei+1 − Ei ≥ 0 the i-th energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the number of energy levels in an interval [E, E+∆E] is known to follow the Wigner-surmise [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In particular, it follows the Gaussian orthogonal ensemble (GOE) since the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (13) is time-reversal invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' On the other hand, the number of energy levels in an interval follows the Pois- son distribution in the MBL phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since our model only partially localizes, we review how the Poisson distribu- tion accurately describes the MBL phase and investigate T6 the validity of these arguments in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider two adjacent eigenstates with energies Ei and Ei+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the energy of these states are dominated by the disorder term � i di ˆDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If the states have differ- ent disorder indices |ED,m⟩ and |ED′,m′⟩, then their en- ergies originate from different linear combinations of the random numbers di: � i diDi ≈ � i diD′ i with Di ̸= D′ i for some i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the eigenstates “arrive” at this energy independently of each other and hence fol- low the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These arguments are no longer valid when two adjacent eigenstates have the same disorder indices and different m indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, we expect the level spacing distribution to follow GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the distribution of energy levels still identifies the transition to partial localization if we only consider level spacings between eigenstates of different disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Instead of working directly with the level spacing dis- tribution, it is convenient to analyze the adjacent gap ratio since it removes the need for unfolding the spec- trum [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The adjacent gap ratio is defined by [16] ri = min(δi, δi+1) max(δi, δi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (16) This quantity is bounded by the interval ri ∈ [0, 1] and follows the distributions below in the thermal and MBL phases respectively [39] PGOE(r) = 27 4 r(1 + r) (1 + r + r2)5/2 , (17a) PPoisson(r) = 2 (1 + r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) The mean values of the distributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17) are given by ⟨r⟩GOE = 2(2 − √ 3) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='536 and ⟨r⟩Poisson = 2 ln 2 − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 3(a) illustrates the mean adjacent gap ratio as a function of disorder strength for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We average the adjacent gap ratio over 2 × 103 disor- der realizations for L = 8, 103 disorder realizations for L = 10, 12, 14 and 500 disorder realizations for L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For each disorder realization, we average over all energies in the interval Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile of the energy distribution for the current disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system size L = 16, we average over the 103 energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The errorbars indicate two standard devi- ations of the average when assuming a Gaussian distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, the distribution of adjacent gap ratios only converges to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) if the analysis is restricted to adjacent energy levels with different disor- der indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In practice, however, it is unlikely for two neighboring eigenstates to have the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, the likelihood of neighboring eigenstates having the same disorder indices decreases rapidly with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' With this in mind, we study the mean adja- cent gap ratio using all eigenstates in the central third of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We verify the considerations above by also computing the mean adjacent gap ratio using only adja- cent eigenstates with different disorder indices at large disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For each energy gap δi = Ei+1 −Ei, we inspect the eigenstates |ED,m⟩ and |ED′,m′⟩ corresponding to the energies Ei and Ei+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the disorder in- dices D are accurately determined by computing which D yields �ND m=1 | ⟨D, m|ED,m⟩ |2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The mean of the adjacent gap ratio is then restricted to energy gaps with D ̸= D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For small system sizes, there is a large differ- ence between the two methods, but the difference is seen to be small for large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The mean adjacent gap ratio agrees well with the GOE value at weak disorder 0 <∼ W <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the mean adjacent gap ratio de- creases and ultimately approaches the Poisson value at 5 <∼ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The agreement of data with the GOE and Pois- son values improves with increasing system size and the transition between the thermal and localized phase be- comes steeper for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figures 3(b)-(d) illustrate the adjacent gap ratio dis- tribution at (b) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='46, (c) intermedi- ate disorder strength W = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='27 and (d) strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figures display the distributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the data agrees with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) at weak disorder and (17b) at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fig- ure 3 indicates the system transitions from the thermal phase to being partially localized as disorder is intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Bipartite entanglement entropy In this section, we further verify the transition from the thermal phase to partial localization by studying the bipartite entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We separate the system into a left part L containing the first L/2 sites and a right part R containing the remaining sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The reduced density matrix of the left part is obtained by tracing out the right part ρL = TrR(ρ) (18) where ρ is the density matrix of the full system and TrR(·) is the partial trace over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy between the left and right halves is given by, S = − TrL � ρL ln(ρL) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (19) In the thermal phase, we expect eigenstates near the center of the spectrum to display volume-law scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Specifically, the entropy is approximately described by the Page value SPage = [L ln(2) − 1]/2 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' On the other hand, the entanglement entropy displays area-law scaling for MBL eigenstates [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While some eigenstates in our model are fully MBL, others are only partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, the precise scaling behavior of the entanglement entropy is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Nonetheless, we expect the entropy of partially localized eigenstates to grow slower with system size than thermal eigenstates 7 P(r) (b) Poisson GOE P(r) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 r P(r) (d) 0 1 2 3 4 5 6 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='50 ⟨r⟩ (a) GOE Poisson L = 8 10 12 14 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Mean adjacent gap ratio ⟨r⟩ (solid line) as a function of disorder strength W for different system sizes L with parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded areas display two standard deviations on the estimate of ⟨r⟩ when assuming a Gaussian distribution of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For L = 8, the adjacent gap ratio is averaged over 2 × 103 disorder realizations, for L = 10, 12, 14 we use 103 disorder realizations and for L = 16 we use 500 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system sizes L = 8, 10, 12, 14, we average over all energies Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system size L = 16, we average over the 103 energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At low disorder 0 <∼ W <∼ 1, the system is thermal and ⟨r⟩ coincides with the Gaussian orthogonal ensemble ⟨r⟩GOE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='536 (upper dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder 5 <∼ W, the mean adjacent gap ratio agrees with the Poisson distribution ⟨r⟩Poisson ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='386 (lower dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The agreement between data and the GOE and Poisson values improves with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Additionally, the transition from the thermal phase to partial localization happens more rapidly as a function of disorder strength for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure also illustrates the mean adjacent gap ratio when only averaging over neighboring energy eigenstates with different disorder indices (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The errorbars show two standard deviations on the estimate of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This average coincides with the naive calculation at large system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure also shows the adjacent gap ratio distribution for L = 16 at (b) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='46, (c) intermediate disorder strength W = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='27 and (d) strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These plots include the distributions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) (dashed curve) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) (dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The data agrees with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) at weak disorder and transitions to the distribution (17b) at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' and we use the entropy to identify the onset of partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 4(a) shows the entropy of the eigenstate with energy closest to (Emin + Emax)/2 as a function of disor- der strength W for different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Each data point represents the average entropy over 103 disorder realizations with errorbars displaying two standard devi- ations of the mean when assuming a Gaussian distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For low disorder, the entanglement entropy scales linearly with the system size and hence agrees with the expected volume-law scaling in the thermal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Ad- ditionally, the entropy approaches the Page value with increasing system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the entropy seems to be roughly independent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the scaling of entropy is consistent with area-law for par- tially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The sudden shift in scaling behavior of the entropy verifies the transition from the thermal phase to partial localization at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The transition point is identified by analyzing the variance of entanglement en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 4(b) illustrates the sample variance of the entropy over 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The variance dis- plays a peak when the system transitions from volume- law to area-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISTINGUISHABLE FEATURES OF SCAR STATES IN A PARTIALLY LOCALIZED BACKGROUND Scar states are commonly distinguished from a thermal background by their low entanglement and oscillatory dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this section, we show that oscillatory dynam- ics can also be utilized to distinguish scar states from a partially localized background, while entanglement en- tropy turns out not to be an effective tool to identify the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Entanglement entropy The entanglement entropy of the scar states scales log- arithmically with system size [29], while thermal states display volume-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the entanglement 8 0 1 2 3 4 5 ⟨S⟩ (a) L = 8 10 12 14 16 0 2 4 6 W 0 1 Var(S) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Average bipartite entanglement entropy of the eigenstate closest to the center of the spectrum ⟨S⟩ as a func- tion of disorder strength W for different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entropy is averaged over 103 disorder realizations with system parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Errorbars display two stan- dard deviations on the estimate of average entropy assuming a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At low disorder, the entropy displays volume-law scaling with system size and approaches the Page value (dashed lines) as expected in the thermal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the entropy follows area-law scaling with sys- tem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) Variance of bipartite entanglement entropy of the eigenstate closest to the center of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The variance is computed from 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the variance displays a sudden peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This indicates a transition from the thermal phase to partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The peak becomes higher at larger sys- tem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' entropy provides a way to identify the scar states in a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 5(a) illustrates the entropy as a function of energy of a thermal system with size L = 14 and disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The thermal states form a narrow arc with maximum in the middle of the spectrum while the scar state appears as an out- lier at much lower entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The situation is different in a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 5(b) illustrates the entropy as a function of energy at strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, partially localized eigen- states are weakly entangled making it difficult to identify the scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We conclude that entanglement entropy is an ineffective tool for distinguishing scar states from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 ϵ 0 2 4 S (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 ϵ (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy S as a function of nor- malized energy ϵ = (E − Emin)/(Emax − Emin) where Emin and Emax are the smallest and largest energies in the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Lighter (darker) colors indicate lower (higher) den- sity of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) We consider a thermal system of size L = 14, disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and system parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the energy eigenstates form a narrow band with maximum at the center of the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state (inside the green ring) is easily identified since it appears isolated below the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) We consider a partially localized system at strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The en- ergy eigenstates are spread out at low entropy with the scar state embedded among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy is hence not an effective tool to distinguish the scar state from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fidelity States initialized in the scar subspace distinguish them- selves from a thermal background by displaying persis- tent dynamic revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We now show that this behavior also enables the identification of scar states from a par- tially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We quantify the dynamics of quantum systems by the fidelity F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let |ψ(0)⟩ be the initial state and |ψ(t)⟩ = e−i ˆ Ht |ψ(0)⟩ the time evolved state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity is given by F(t) = | ⟨ψ(0)|ψ(t)⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (20) The time evolution of fidelity is most clearly understood by considering the overlap of the initial state with all energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let |φi⟩ be the i-th energy eigenstate with corresponding energy Ei and let ci be the inner product between the i-th energy eigenstate and the initial state ci = ⟨φi|ψ(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The relation between fidelity and the expansion coefficients ci is highlighted by rewriting the fidelity according to F(t) = � i |ci|4 + � i̸=j |ci|2|cj|2ei(Ei−Ej)t (21) It is clear from this expression that the dynamics of fi- delity is sensitive to the distribution of |ci|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We generally display this distribution along with the fidelity for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate the different dynamical behavior of the thermal and partial MBL phases by initializing a sys- tem of size L = 14 in a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, we consider 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) ⟨F T⟩ ⟨F MBL⟩ Fscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (b) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (c) −25 0 25 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 |ci|2 (d) −50 0 50 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 (e) −20 0 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) The average fidelity of a random product state in a thermal system at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) The aver- age fidelity in a partially localized system at disorder strength W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The system is initialized in a product state which fully localizes (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For comparison, the system is ini- tialized in a random product state which only partially local- izes |ψ(0))⟩ = |D, n⟩ with ND = 5 (dashed line), 10 (dashed dotted line) and 35 (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (c) The system is initialized in the scar subspace at any disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity is in all cases calculated over 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The bottom panel displays the distribution of expansion coef- ficients |ci|2 across energy in a single disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (d) For the thermal phase W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (e) For partial MBL W = 10 with initial state |ψ(0)⟩ = |D, n⟩ for ND = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (f) For the initial state being an equal weight linear combination of the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' a thermal system at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial state is chosen as a random product state with all product states having the same probability of being drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We ensure the initial state resides outside the scar subspace by drawing a new product state if the first has non-zero overlap with a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We consider 103 disorder realizations and draw a random product state in each realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the i-th realization, the fidelity is computed as a function of time Fi(t) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a) shows the average fidelity ⟨F(t)⟩ = 10−3 �103 i=1 Fi(t) over all re- alizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(d) shows the expansion coefficients |ci|2 of a single disorder realization following the Gaus- sian distribution as expected [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the initial state has large overlap with many different eigenstates, the second sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (21) rapidly vanishes due to can- cellation between terms with different phase factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As a consequence, the fidelity quickly decreases and satu- rates at Fi(t) ≈ � i |ci|4 ≈ 0 at long times Tscar ≪ t for all disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These considerations agree with the observed time evolution of the average fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a) which rapidly decreases to a value near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the same setup when the system is partially localized at large disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A, the spectrum separates into fully MBL eigenstates and partially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Conse- quently, the dynamics depend greatly on the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The solid blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b) is the average fidelity over 103 disorder realizations when initialing the sys- tem in a random product state which fully localizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' |ψ(0)⟩ = |D, n⟩ with ND = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fully MBL eigenstates have significant overlap with only one product state, and the average fidelity remains far from zero at all times as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We note that a stronger disor- der strength is needed to achieve MBL in larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the average fidelity saturates significantly be- low unity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b) even though all product states with ND = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1 are near identical to an energy eigen- state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity saturates closer to unity at larger disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When the initial state is chosen as a product state that only partially localizes, it has significant overlap with multiple eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity drops closer to zero as illustrated by the dashed and dot- ted curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For these curves, we choose the initial state randomly as |ψ(0)⟩ = |D, n⟩ with ND = 5, 10 and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These initial states have significant support on up to ND eigenstates causing the average fidelity to de- crease with increasing ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(e) illustrates the distribution of |ci|2 for a single disorder realization for a random initial state |ψ(0)⟩ = |D, n⟩ with ND = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The distribution is more sparse than the thermal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we consider the initial state being a linear com- bination of scar states |ψscar⟩ = 1 � L 2 + 1 L/2 � n=0 |Sn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (22) When the initial state is chosen within the scar subspace, the equal energy spacing causes the fidelity to display persistent periodic revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In particular, for the equal weight linear combination in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (22), the fidelity is given by Fscar(t) = 1 L 2 + 1 � 1 + 2 L/2 � n=1 � 1 − n L 2 + 1 � cos(n∆Et) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (23) Revivals occur at times tℓ = Tscarℓ = 2πℓ ∆Escar where ℓ ∈ N and ∆Escar is the energy spacing between consecutive scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(c) illustrates the fidelity of this initial state and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(f) shows the distribution of the expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10 In the thermal phase, states initialized respectively in- side and outside the scar subspace behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity of states outside the scar subspace quickly drops to zero, while any linear combination of scar states display persistent revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In our analysis, we specifi- cally initialized the system as a product state, but the same conclusions hold for generic linear combinations of product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In a partially localized background, the average fidelity distinguishes between states with sup- port inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity of partially localized states saturates while scar states display revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Again, our analysis concerns the special case of initializing the system as a random prod- uct state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If instead the initial state is a generic linear combination of a large number of product states, the sec- ond term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (21) will generally vanish due to phase cancellation, and the average fidelity saturates near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this is true for generic linear combinations, there exists particular states where the phase cancellation hap- pens exceptionally slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We discuss these special initial states in section VI and how to distinguish them from the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Summing up, the average fidelity represents an effective tool for identifying scar states in both a ther- mal and localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we remark that the fidelity of individual dis- order realizations are enough to distinguish initial states with support inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This statement is simple in the thermal phase where initial states outside the scar subspace rapidly converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the fidelity of individual disorder re- alizations may oscillate rapidly contrary to the average fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, these oscillations are generally com- posed of frequencies different from the scar revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The amplitude of the oscillations are also typically different from the scar revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the scar states can be dis- tinguished from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISORDER STABILIZATION OF SCAR REVIVALS We study the dynamics of initial states with support both inside and outside the scar subspace across all sym- metry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, we generally expect the scar revivals to diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar revivals are stabi- lized when the initial state only has support on product states with the same disorder indices as the scar states D0 = (−1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate this behavior by initializing the system in a generic state only having support on product states with disorder indices D0 |ψstable⟩ = 1 Nstable � |ψscar⟩ + � Ndw,n β(Ndw) n |Ndw, D0, n⟩ � , (24) where Nstable is a normalization constant and β(Ndw) n are drawn randomly from the interval β(Ndw) n ∈ [0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5/ � N (Ndw) D0 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We reintroduce the index Ndw to de- scribe product states with the same disorder indices in different symmetry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The time evolution of fidelity is investigated at weak and strong disorder in 103 real- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The coefficients β(Ndw) n are redrawn in each dis- order realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 7(a) displays the disorder aver- aged fidelity for a thermal system and a partially local- ized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In both cases, the average fidelity displays persistent revivals with the revival amplitude decaying and eventually saturating at a value around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity amplitude quickly decays for a thermal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The explanation can be found by studying the expansion coefficients |ci|2 as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Be- cause the system is thermal, the initial state has support on many energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, terms with different phases quickly cancel causing the fidelity ampli- tude to saturate almost immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the fidelity amplitude decays at a much slower rate and only saturates alongside the ther- mal graph after many revivals t ∼ 7Tscar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We under- stand this behavior by recalling the spectral structure at large disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, recall that the energy eigenstates {|ED0,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND0} are near degenerate and only have significant overlap with product states of the same disorder indices as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' There- fore, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) can be rewritten as a sum of near degenerate eigenstates, ND0 � n=1 β(Ndw) n |Ndw, D0, n⟩ ≈ ND0 � m=1 γ(Ndw) m |ENdw,D0,m⟩ , (25) with γ(Ndw) m = � n β(Ndw) n ⟨ENdw,D0,m|Ndw, D0, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fur- thermore, the scar states themselves are described by the disorder indices D0, so the eigenstates |ENdw,D0,m⟩ are close in energy to a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the eigenstates outside the scar subspace having large over- lap with |ψstab⟩ are always close in energy to a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We sketch this structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 8 where the eigenstates |ENdw,D0,m⟩ have similar energy to the scar states for all Ndw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These considerations agree with the observed distribution of |ci|2 for a single disorder realization illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The expansion coefficients are sharply peaked around the scar states and consequently the can- cellation of terms with different phases takes place at much larger times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this way, the partially localized background stabi- lizes the scar revivals by rearranging the support outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The stabilization takes place whenever the initial state is predominantly a linear combination of product states with the same disorder indices as the scar states D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If product states with other disorder indices D′ ̸= D0 are included, the stabilization will be less pro- nounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 11 10−4 10−2 100 |ci|2 (b) −50 0 50 E 10−4 10−2 100 |ci|2 (c) 0 2 4 6 8 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='00 F (a) W = 10 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A system of size L = 14 with parameters ∆ = 1, J = 5, λ = 1 is initialized according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) in the thermal phase at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and the partial MBL phase at disorder strength W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) The average fidelity over 103 disorder realizations when the system is thermal and partially MBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The disorder protects the scar revivals and the fidelity amplitude decays much slower compared to the thermal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The right panels illustrate the distribution of expansion coefficients |ci|2 over energy Ei for a single disorder realization at disorder strength (b) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and (c) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The distribution of the expansion coefficients is wide in the thermal phase and consists of narrow peaks near the scar states in the localized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' E Symmetry sectors ∆Escar ∆Escar ∆Escar Ndw0 Ndw1 Ndw2 Ndw3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) has significant overlap with a small number of energy eigenstates (black lines) as sketched in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These eigenstates ap- pear in clusters around the energy of the scar states (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A single cluster exists in every symmetry sector and the energy gap between two adjacent clusters equals the en- ergy gap between scar states ∆Escar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISORDER INDUCED APPROXIMATE SCARS Additional approximate scar states emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These approximate scars appear because some symmetry sectors contain energy eigenstates with the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For instance, the eigenstates |E2,D,1⟩ ≈ |↑↑↓↓↓↓⟩ and |E4,D,m⟩ ≈ αm1 |↑↑↓↑↓↓⟩ + αm2 |↑↑↓↓↑↓⟩ for m = 1, 2 have the same disorder in- dices D = (3, −1, −1, −1, −1, −1) but different number of domain walls Ndw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Recall from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A that the en- ergy of an eigenstate at large disorder is approximately given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ENdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m ≈ ∆MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + J � N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D − N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D � + � i diDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (26) If an eigenstate |ENdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m⟩ is described by the values MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D and N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' then another eigenstate |ENdw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m⟩ with Ndw + 2 domain walls and identical disorder indices D is described by MNdw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27a) N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27b) N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27c) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (26) and (27), one can show the energy dif- ference between two eigenstates with the same disorder indices D and number of domain walls ND and ND + 2 is approximately ENdw+2,D,m − ENdw,D,m ≈ ∆Escar, (28) where ∆Escar = 2(∆−2J) is the energy gap between the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This calculation demonstrates that towers of approximate scar states appear in the spectrum as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate how the appearance of approximate scars generates non-trivial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The system is ini- tialized in a generic linear combination of product states with disorder indices D1 = (3, −1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , −1) |ψinduced D1 ⟩ = 1 Ninduced � Ndw,n ζ(Ndw) n |Ndw, D1, n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) The coefficients are chosen randomly from the interval ζ(Ndw) n ∈ [0, 1] and Ninduced is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 12 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 (b) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 (c) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='025 |ci|2 (d) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 (e) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='2 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity of the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) over 103 disorder realizations for system size L = 14 with parameters λ = ∆ = 1, J = 5 at disorder strength (a) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (b) W = 5 and (c) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded areas show the interquartile range (middle 50%) of the disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The corresponding distribution of expansion coefficients |ci|2 of a single disorder realization at disorder strength (d) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (e) W = 5 and (f) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At weak disorder, the initial state has significant overlap with many energy eigenstates and the average fidelity quickly decays to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the initial state has significant overlap with a small number of energy eigenstates with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity shows persistent revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We study this initial state because, at large disorder, it is a linear combination of an approximate scar tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We consider 103 disorder realizations at different disor- der strengths and the fidelity is computed for each re- alization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 9(a) displays the average fidelity of a thermal system at weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, there is nothing special about the initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and it quickly decays to zero similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The dynamical behavior changes remarkably as the disorder strength is increased as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9(b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At stronger disorder, the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) has large over- lap with eigenstates that are approximately equidistant in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity oscillates with a period given by the energy gap Tscar = 2π ∆Escar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The revival amplitude increases with disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9(a)-(c) displays the interquar- tile range of disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figures 9(d)-(f) shows the expansion of the initial state in energy eigenstates at (d) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (e) strong disorder W = 5 and (f) very strong disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the initial state is distributed over a wide range of eigenstates in the thermal phase similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disor- der strength increases, the initial state has higher and higher overlap with eigenstates in an approximate tower of equidistant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 9 demonstrates that it is possible to observe revivals from generic linear combinations of the states {|Ndw, D, n⟩ |Ndw = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='} at large dis- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, the effects may be enhanced by choosing the initial state more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) is, in some sense, the worst case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When all product states with disorder indices D are included in the sum, the initial state generally has significant overlap with all relevant energy eigenstates {|ENdw,D,m⟩ |Ndw = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This causes a large spread in the distribution of |ci|2 resulting in a faster decay of the av- erage fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If instead, we consider an initial state with exactly one product state from each symmetry sector, the spread of |ci|2 is smaller | ˜ψinduced D1 ⟩ = 1 � L 2 − 1 � |↑↑↓↓↓↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + |↑↑↓↑↓↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + |↑↑↓↑↓↑↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' + |↑↑↓↑↓↑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓↑↓↓⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (30) Figure 10(a) shows the average fidelity of this initial state over 103 disorder realizations at strong disorder W = 10 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10(b) displays the distribution of |ci|2 for a single realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the distribution of |ci|2 is narrower and the revival amplitude larger compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial states Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and (30) display revivals similar to the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, one may distinguish these initial states from the scar subspace by noting that the average fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9 and 10 decays to zero, while the amplitude in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(c) and 7 remain strictly larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The different dynamical behavior is caused by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and (30) being composed of approximate scar towers while the original scars |Sn⟩ are exactly equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 13 0 2 4 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 |ci|2 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Average fidelity of the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (30) over 103 disorder realizations with system size L = 14 and parameters λ = ∆ = 1, J = 5 and W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded area displays the interquartile range of the disorder realiza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity displays persistent revivals with larger amplitude compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) Expansion of the initial state across energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The coefficients |ci|2 are sharply peaked around certain energies which are approx- imately equally spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' CONCLUSION Building on a known method to find parent Hamil- tonians, we proposed a way to determine Hamiltonians hosting a tower of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Starting from the model by Iadecola and Schecter, we used this method to identify all local 1- and 2-body Hamiltonians of the scar tower |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Among these Hamiltonians, we found operators fa- cilitating the implementation of local disorder while pre- serving the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When introducing disorder, the mean level spacing statistics shifts from the GOE to the Poisson distribution and the entanglement entropy goes from volume-law to area-law scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We conclude the system transitions from the thermal phase to being partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A theory describing the par- tially localized eigenstates was developed and verified nu- merically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In total, we determined a system hosting a tower of scar states with the remaining spectrum being either thermal or partially localized depending on the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We studied the properties of scar states embedded in a localized spectrum and compared with the corresponding features in a thermal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In contrast to thermal systems, the bipartite entanglement entropy does not en- able the identification of scar states in a localized back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity, on the other hand, effec- tively identifies the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We investigated the effect of localization on initial states with support both inside and outside the scar sub- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For a thermal system, the fidelity displays persis- tent revivals with rapidly decreasing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In con- trast, the revival amplitude decays slower for a partially localized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, partial localization stabilizes the persistent revivals of states initialized partly outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we demonstrated how additional approximate scar states emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When ini- tializing the system as a superposition of these states, the average fidelity displays revivals with the same period as the true scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this effect does not rely on fine-tuning the initial state, the revivals are amplified by choosing the initial state appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has been supported by the Carlsberg Foun- dation under grant number CF20-0658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Appendix A: Proof that |Sn⟩ are eigenstates of all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I with equal energy spacing In section II C, we found L + 4 operators having the scar states as eigenstates equidistantly spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since this analysis was carried out for finite system sizes L = 8, 10, 12, 14, the validity of this statement is not guaranteed for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this appendix, we rigorously prove the scar states |Sn⟩ are equally spaced eigenstates of all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the scar states are constructed iteratively by applying the operator Q†, we generally prove this statement using proof by induc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, we consider the operator ˆHz = � i ˆσz i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The lowest scar state |S0⟩ = |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ is trivially an eigen- state of ˆHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A straightforward calculation shows that [ ˆHz, ˆQ†] = 2 ˆQ† and by induction all other scar states are eigenstates because ˆHz |Sn+1⟩ ∝ ˆHz ˆQ† |Sn⟩ = � Ez,n ˆQ† + 2 ˆQ†� |Sn⟩ = � Ez,n + 2 � |Sn+1⟩ , (A1) where ˆHz |Sn⟩ = Ez,n |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar states are also equally spaced in energy En+1,z − En,z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A simi- lar argument holds for ˆHodd zz since [ ˆHodd zz , ˆQ†] = −4 ˆQ† where the energy gap between scar states is −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the operators ˆDi = ˆσz i + ˆσz i+1 + ˆσz i ˆσz i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Recall that ˆDi is related to the projection oper- ators through ˆDi = 4 ˆP ↑ i ˆP ↑ i+1 − ˆ1 where ˆP ↑ i = (ˆ1 + ˆσz i )/2 projects site i onto spin-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First note that ˆDi |S0⟩ = (4 ˆP ↑ i ˆP ↑ i+1 − ˆ1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = − |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A simple calcu- lation shows that ˆDi commutes with ˆQ† by noting that 14 ˆP ↑ i ˆP ↓ i = 0 [ ˆDi, ˆQ†] = 4 L � j=1 (−1)j� ˆP ↓ j−1[ ˆP ↑ i , ˆσ+ j ] ˆP ↓ j+1 ˆP ↑ i+1 + ˆP ↑ i ˆP ↓ j−1[ ˆP ↑ i+1, ˆσ+ j ] ˆP ↓ j+1 � = 4(−1)i� ˆP ↓ i−1ˆσ+ i ˆP ↓ i+1 ˆP ↑ i+1 − ˆP ↑ i ˆP ↓ i ˆσ+ i+1 ˆP ↓ i+2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A2) Thus, for all scar states we have ˆDi |Sn⟩ = − |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Alter- natively, one may note that |Sn⟩ by construction does not contain adjacent sites being spin-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, ˆP ↑ i ˆP ↑ i+1 naturally annihilates the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the operator ˆHalt xz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Before studying the action of ˆHalt xz on the scar states, we prove by in- duction that the commutator [ ˆHalt xz , ˆQ†] annihilates |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The commutator is given by [ ˆHalt xz , ˆQ†] = L � i=1 � 2 � ˆP ↓ i ˆσ+ i+1ˆσ− i+2 − ˆσ+ i ˆσ+ i+1 ˆP ↓ i+2 � + i � ˆP ↓ i ˆσ+ i+1ˆσy i+2 + ˆσy i ˆσ+ i+1 ˆP ↓ i+2 + ˆσz i ˆσy i+1ˆσ+ i+2 ˆP ↓ i+3 − ˆP ↓ i ˆσ+ i+1ˆσy i+2ˆσz i+3 �� , (A3) where ˆP ↓ i = (ˆ1 − ˆσz i )/2 is the local projection onto spin- down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' By direct calculation, one can show the lowest scar state is annihilated by this expression [ ˆHalt xz , ˆQ†] |S0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A lengthy, yet straightforward, calculation also shows the nested commutator vanishes � [ ˆHalt xz , ˆQ†], ˆQ†� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We now prove by induction that the commutator annihilates all scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Assume [ ˆHalt xz , ˆQ†] |Sn⟩ = 0 and consider, [ ˆHalt xz , ˆQ†] |Sn+1⟩ ∝ [ ˆHalt xz , ˆQ†] ˆQ† |Sn⟩ = � ˆQ†[ ˆHalt xz , ˆQ†] + � [ ˆHalt xz , ˆQ†], ˆQ†�� |Sn⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A4) Having shown this intermediate result, we prove by in- duction that the operator ˆHalt xz annihilates the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First we show the operator ˆHalt xz annihilates |S0⟩ ˆHalt xz |S0⟩ = L � i=1 (−1)i(ˆσx i ˆσz i+1 + ˆσz i ˆσx i+1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = L � i=1 (−1)i+1(ˆσx i + ˆσx i+1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = 0, (A5) where the second term cancels the first after changing summation index i + 1 → i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we show by induction that the n-th scar state is annihilated by ˆHalt xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Assume ˆHalt xz annihilates |Sn⟩ and consider ˆHalt xz |Sn+1⟩ ∝ ˆHalt xz ˆQ† |Sn⟩ = ( ˆQ† ˆHalt xz + [ ˆHalt xz , ˆQ†]) |Sn⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A6) The first term vanishes by assumption and the second term is exactly what we considered in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In total, we conclude ˆHalt xy has |Sn⟩ as eigenstates equidistantly separated in energy (with zero energy spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally we consider the operator ˆHalt yz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' One can prove this operator annihilates the scar states using similar ar- guments to above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The commutator is given by [ ˆHalt yz , ˆQ†] =i L � i=1 � 2 � ˆP ↓ i ˆσ+ i+1ˆσ− i+2 + ˆσ+ i ˆσ+ i+1 ˆP ↓ i+2 � − ˆσx i ˆσ+ i+1 ˆP ↓ i+2 − ˆP ↓ i ˆσ+ i+1ˆσx i+2 + ˆP ↓ i ˆσ+ i+1ˆσx i+2ˆσz i+3 − ˆσz i ˆσx i+1ˆσ+ i+2 ˆP ↓ i+3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A7) Using induction, one can prove the commutator annihi- lates all scar states [ ˆHalt yz , ˆQ†] |Sn⟩ = 0 and the operator annihilates the lowest scar state ˆHalt yz |S0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Retracing the steps in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A6), we find that ˆHalt 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Izrailev, Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' E 85, 036209 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} diff --git a/1dAzT4oBgHgl3EQfDfo-/content/tmp_files/2301.00976v1.pdf.txt b/1dAzT4oBgHgl3EQfDfo-/content/tmp_files/2301.00976v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7570a13ecd1a9ba95588006d55a7ab326cc9b079 --- /dev/null +++ b/1dAzT4oBgHgl3EQfDfo-/content/tmp_files/2301.00976v1.pdf.txt @@ -0,0 +1,1251 @@ +arXiv:2301.00976v1 [hep-ph] 3 Jan 2023 +The Σ and Ξ electromagnetic form factors in the extended vector meson dominance model +Bing Yan,1,2, ∗ Cheng Chen,2,3, † and Ju-Jun Xie2, 3, 4, ‡ +1College of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China +2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +3School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China +4Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, +Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China +We propose a phenomenological extended vector meson dominance model for the baryon electromagnetic +structure, and it is found that the current experimental data on the Σ and Ξ electromagnetic form factors in the +time-like region can be well described. Meanwhile, we can also reproduce the ratios of the total cross sections +of reactions e+e− → Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+, which are 9.7 ± 1.3 : 3.3 ± 0.7 : 1 at center-of-mass +energies from 2.3864 to 3.02 GeV. We also analytically continue the expression of the form factors to space- +like region and estimate the charge radii of the Σ and Ξ hyperons. The result for the Σ− is in agreement with +the experimental date. +I. +INTRODUCTION +The electromagnetic structure information of hadrons is +characterized by the electromagnetic form factors (EMFFs), +which are functions of the four-momentum transfer squared +q2, with q the four-momentum carried by the exchanged vir- +tual photon. Study of these EMFFs can lead to a better under- +standing of fundamental structure of hadrons. On the experi- +mental side, most commonly the baryon EMFFs in the space- +like region (q2 < 0) were measured in the electron-baryon +scattering [1–4]. While for these unstable hadrons, for ex- +ample, these hyperons, their EMFFs in the space-like region +are very difficult to be experimentally measured. However, in +the time-like region (q2 > 0), their EMFFs can be measured +through the electron-positron annihilation reactions by the +BESIII and Belle collaborations [5–12]. Meanwhile, the ef- +fective form factor Geff(q2) of hyperons can be extracted from +the high-precision measured Born cross sections of the reac- +tions e+e− → Y ¯Y (Y stands for hyperon; ¯Y is anti-hyperon). +It was pointed out that these baryon EMFFs in the time-like +region can be associated with the time evolution of the charge +and magnetic distributions inside the baryon [13, 14]. +The hyperon effective form factors Geff(q2) are the func- +tions that parametrize the γY ¯Y vertex generated by the strong +interaction. While, the production vertex γY ¯Y is very poorly +understood so far [15, 16]. +The vector meson dominance +(VMD) model is a very successful tool for studying the nu- +cleon electromagnetic form factors, in both the space-like and +time-like regions [17–19]. Within a modified VMD model, +the EMFFs of Λ hyperon were investigated in Refs. [20, 21]. +By considering the Y ¯Y final sate interactions, the EMFFs +of hyperons in the time-like region have been studied in +Ref. [22]. It is worth to mention that the enhancement of the +∗Electronic address: yanbing@impcas.ac.cn +†Electronic address: chencheng22@mails.ucas.ac.cn +‡Electronic address: xiejujun@impcas.ac.cn +effective form factor of the Λ hyperon seen in the e+e− → +Λ¯Λ reaction, was reproduced within the two above different +calculations in Refs. [20, 21] and Ref. [22], respectively. In +the vector meson dominance model for studying the electro- +magnetic form factors of baryons, there is a phenomenological +intrinsic form factor g(q2). From these studies of the nucleon +and hyperon EMFFs [17–29], it is found that a better choice +of g(q2) is the dipole form +g(q2) = +1 +(1 − γq2)2 , +(1) +with γ a free parameter. In the space-like region, the dipole +form is consistent with the results obtained from perturbative +quantum chromodynamics calculations [30, 31]. In the time- +like region, it should be noticed that γ is a positive parameter, +thus g(q2) will have a pole in the position γ = 1/q2, such +pole could be restricted in the unphysical region, if γ satisfy +γ > 1/(4m2 +Y ) for hyperon Y . +For a long time, the simple dipole form paremetrization is +very useful for the discussion of different baryons. For exam- +ple, the dipole form of g(q2) can well describe the effective +form factors of Λ [20, 21], Σ [32], and Ξ [27]. While for the +nucleon, a good general review is given in Refs. [33–36], both +from the theoretical and from the experimental points of view. +However, these determined values of γ for different ground +state octet baryons with spin 1/2 are much different, even for +the triplet Σ+, Σ− and Σ0 [32]. The determined values of γ, +from previous works, for nucleon, Λ, Σ, and Ξ0 baryons are +collected in Table I. Nevertheless, the VMD model and the +parametrization of g(q2) can give a reasonable description of +the experimental data on the baryon EMFFs at the considered +energy region. +Various experimental and theoretical efforts have been con- +tributed to the electromagnetic form factors. Very recently, the +EMFFs of Σ+, Σ−, and Σ0 hyperons in the time-like region, +have been measured with high-precision by the BESIII collab- +oration through e+e− → Σ+ ¯Σ− [9], Σ− ¯Σ+ [9], and Σ0 ¯Σ0 +reactions [10] at center-of-mass energies from 2.3864 to 3.02 +GeV. The resulting ratios of total cross sections of these above + +2 +TABLE I: Values of γ (in GeV−2) for octet baryons used in previous +works. +Proton ([17–19]) Neutron ([24–27]) +Λ ([20]) +Λ ([21]) +γ +1.408 +1.408 +0.336 +0.48 ± 0.08 +Σ+ ([32]) +Σ− ([32]) +Σ0 ([27]) +Ξ0 ([27]) +γ +0.46 ± 0.01 +1.18 ± 0.13 +0.26 ± 0.01 0.21 ± 0.02 +three reactions are 9.7 ± 1.3 : 1 : 3.3 ± 0.7 [9, 37, 38], which +disagree with various theoretical model predictions [39, 40]. +After the experimental measurements of e+e− → Σ+ ¯Σ− and +Σ− ¯Σ+ [9], the effective form factors of Σ+ and Σ− were in- +vestigated by using the VMD model [32], where the param- +eter γ were taken with different values for Σ+ and Σ−. In +Ref. [22], by considering the final state interactions of Y ¯Y , +the energy dependence of the three reactions e+e− → Σ+ ¯Σ−, +Σ0 ¯Σ0, and Σ− ¯Σ+ at low energies can be roughly reproduced, +and it was found that there is a strong interplay between +Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+ channel in the near-threshold re- +gion, caused by the Σ¯Σ final state interactions. In the present +work, we revisit the EMFFs at the time-like region of Σ and Ξ +hyperons within an extended vector meson dominance model, +where the affects of the isospin combinations from isovector +ρ0 and isoscalar ω and φ mesons are taken into account. Fur- +thermore, we assume that the values of model parameter γ +are same for Σ and Ξ hyperons. In addition, a vector meson +with mass around 2.7 GeV was considered for the sake of bet- +ter fitting the EMFFs of the Ξ0 and Ξ− hyperons. We then +progress to an analysis of the electromagnetic form factors in +the space-like region and evaluate the electromagnetic radius +of Σ hyperons. The theoretical result for the Σ− hyperon is in +agreement with the experimental measurements. This article +is organized as follows: in next section we will show the theo- +retical formalism of the Σ and Ξ electromagnetic form factors +in the VMD model. Numerical results about the effective form +factors of Σ and Ξ and total cross sections of e+e− → Σ¯Σ and +Ξ¯Ξ are shown in Sec. III, and a short summary is given in the +final section. +II. +FORMALISM +As already pointed out, as fixed-energy e+e− colliders, +the EMFFs of hyperons in the time-like region was extracted +from the data on the differential cross section of the process +e+e− → Y ¯Y . For analysis the data, the BESIII Collaboration +use the energy scan method [41–43], while the initial state +radiation method was used by Belle Collaboration [12] and +BABAR collaboration [44, 45]. Besides, the effective form +factors Geff can be easily obtained from the data of the total +cross sections. The module squared of effective form factor +|Geff|2 is a linear combination of |GE|2 and |GM|2, and pro- +portional to the total cross section of e+e− → Y ¯Y reaction. +In this work, we study the EMFFs of Σ and Ξ baryons in the +time-like region with the experimental measurements on the +e+e− → Y ¯Y reactions. Based on parity conservation and +Lorentz invariant, the electromagnetic current of the baryons +with a spin of 1/2 characterize two independent scalar func- +tions F1(q2) and F2(q2) depending on q2, which are the Dirac +and Pauli form factors, respectively. While the corresponding +electrical and magnetic form factors GE(q2) and GM(q2) are +written as [38, 46, 47], +GE(q2) = F1(q2) + τF2(q2), +(2) +GM(q2) = F1(q2) + F2(q2), +(3) +where M is the baryon mass and τ = q2/(4M 2). +With +GE(q2) and GM(q2), the magnitude of the effective form fac- +tor |Geff(q2)| is defined as +|Geff(q2)| = +� +2τ|GM(q2)|2 + |GE(q2)|2 +1 + 2τ +. +(4) +In the time-like region, the effective form factors of hyper- +ons are experimentally studied via the electron-positron anni- +hilation processes. Under the one photon exchange approx- +imation, the total cross section of the annihilation reaction +e+e− → ¯Y Y can be expressed in terms of the effective form +factor Geff as [44, 48, 49] +σe+e−→ ¯Y Y = 4πα2βCY +3s +(1 + 1 +2τ ) | Geff(s) |2, +(5) +with α = e2/(4π) = 1/137.036 the fine-structure con- +stant, and β = +� +1 − 4M 2 +Y /s is a phase-space factor. Here, +s = q2 is the invariant mass square of the e+e− system. The +coulomb enhancement factor CY accounts for the electromag- +netic interaction of charged point-like fermion pairs in the fi- +nal state [50], which is given by +CY = +� +y +1−e−y +for Σ+, Σ−, and Ξ−, +1 +for Σ0 and Ξ0, +(6) +with y = απ +β +2MY +√s . Considering the CY factor, it is expected +that the cross section of process e+e− → Y ¯Y is nonzero at +the reaction threshold for charged hyperons pairs. As plotted +in Fig. 1 for the case of Ξ− 1, the factor CY affects only at +the energy region very close to the reaction threshold, and it +decreases very quickly as the reaction energy growing and it +follows that few MeV above the reaction threshold it is CY ∼ +1, then its effect can be neglected [50–53]. +A. +The EMFFs of Σ hyperon +In the VMD model, the virtual photon couples to Σ and ¯Σ +through isovector ρ0 meson and isoscalar ω and φ mesons. +Since both the ω and φ are far from the mass threshold of Σ¯Σ, +the behavior of the contributions from them are similar, thus +we combine their contributions. In this way, one can param- +eterize Dirac and Pauli form factors for Σ+ and Σ− in the +1 The numerical results for Σ+ and Σ− are similar. + +3 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +√ +s +− +2M +Ξ +− +(MeV) +0 +2 +4 +6 +8 +10 +12 +C +Ξ +− +FIG. 1: The Coulomb factor for Ξ−. +time-like region as follows [17, 19], 2 +F Σ+ +1 += g(q2)(f Σ+ +1 ++ βρ +√ +2Bρ − βωφ +√ +3 Bωφ), +(8) +F Σ+ +2 += g(q2)(f Σ+ +2 +Bρ − αωφ +√ +3 Bωφ), +(9) +F Σ− +1 += g(q2)(f Σ− +1 +− βρ +√ +2Bρ − βωφ +√ +3 Bωφ), +(10) +F Σ− +2 += g(q2)(f Σ− +2 +Bρ − αωφ +√ +3 Bωφ), +(11) +F Σ0 +1 += g(q2)(βωφ +√ +3 +− βωφ +√ +3 +Bωφ), +(12) +F Σ0 +2 += g(q2)µΣ0Bωφ, +(13) +with +Bρ = +m2 +ρ +m2ρ − q2 − imρΓρ +, +(14) +Bωφ = +m2 +ωφ +m2 +ωφ − q2 − imωφΓωφ +, +(15) +where the widths of ρ, ω and φ are taken into account. In this +work, we take mρ = 0.775 MeV, Γρ = 149.1 MeV, Γωφ = +2 We have followed: +|Σ+ ¯Σ−⟩ = +1 +√ +2 +|1, 0⟩ + +1 +√ +3 +|0, 0⟩ + +1 +√ +6 +|2, 0⟩ , +|Σ− ¯Σ+⟩ = − 1 +√ +2 +|1, 0⟩ + +1 +√ +3 +|0, 0⟩ + +1 +√ +6 +|2, 0⟩ , +|Σ0 ¯Σ0⟩ = − 1 +√ +3 +|0, 0⟩ + +� +2 +3 |2, 0⟩ , +(7) +with the basis of |IΣ¯Σ, IZ +Σ¯Σ⟩. In the one photon exchange approximation, +there is no contributions from the isospin tensor terms. +(Γω + Γφ)/2 = 6.4645 MeV, and mωφ = (mω + mφ)/2 = +0.9005 GeV, which are quoted in the review of particle +physics book [54]. +Besides, we take µΣ+ = 3.112ˆµΣ+, +µΣ− = −1.479ˆµΣ−, µΣ0 = 2.044ˆµΣ0 in natural unit [54], +i.e., ˆµ = +e +2MΣ . In addition, at q2 = 0, with the constraints +GΣ+ +E += 1 and GΣ+ +M = µΣ+, GΣ− +E += −1 and GΣ− +M = µΣ−, the +coefficients f Σ+ +1 +and f Σ+ +2 +, f Σ− +1 +and f Σ− +2 +can be calculated, +f Σ+ +1 += 1 − βρ +√ +2 + βωφ +√ +3 , +f Σ+ +2 += 2.112 + αωφ +√ +3 , +(16) +f Σ− +1 += −1 + βρ +√ +2 + βωφ +√ +3 , +f Σ− +2 += −0.479 + αωφ +√ +3 .(17) +Finally, the model parameters γ, the coefficients βρ, βωφ, and +αωφ will be determined by fitting them to the experimental +data on the time-like effective form factors of Σ+, Σ0, and +Σ−, which will be discussed in following. +B. +The EMFFs of Ξ hyperon +For the case of e+e− → Ξ−¯Ξ+ and Ξ0¯Ξ0 reactions, since +Ξ− and Ξ0 are isospin doublets, we express the Ξ−¯Ξ+ and +Ξ0¯Ξ0 states in terms of isospin 0 and 1 components. The mix- +tures of isoscalar and isovector for Ξ−¯Ξ+ and Ξ0¯Ξ0 of equal +relative wight but different sign are imposed by the isospin +symmetry as introduced by the underlying Clebsch-Gorden +coefficients [54]. Then, the Dirac and Pauli form factors F1 +and F2 for Ξ− and Ξ0 can be easily obtained as before for the +Σ hyperon, +F Ξ− +1 += g(q2)(f Ξ− +1 +− βρ +√ +2 +Bρ − βV1 +√ +2 +BV1 +−βV2 +√ +2 BV2 + βωφ +√ +2 Bωφ), +(18) +F Ξ− +2 += g(q2)(f Ξ− +2 +Bρ − αV1 +√ +2 BV1 +−αV2 +√ +2 BV2 + αωφ +√ +2 Bωφ), +(19) +F Ξ0 +1 += g(q2)(f Ξ0 +1 ++ βρ +√ +2 +Bρ + βV1 +√ +2 +BV1 ++βV2 +√ +2 BV2 + βωφ +√ +2 Bωφ), +(20) +F Ξ0 +2 += g(q2)(f Ξ0 +2 Bρ + αV1 +√ +2 BV1 ++αV2 +√ +2 +BV2 + αωφ +√ +2 +Bωφ), +(21) +with +BV 1 = +M 2 +V1 +M 2 +V1 − q2 − iMV1ΓV1 +, +(22) +BV 2 = +M 2 +V2 +M 2 +V2 − q2 − iMV2ΓV2 +, +(23) +where we have considered contributions from two more ex- +cited vector mesons, V1 and V2, in addition the contribu- +tions from ground states ρ, ω and φ. Their mass and width + +4 +are MV1 (MV2) and ΓV1 (ΓV2), respectively. The mass MV2 +and width ΓV2 are taken as used in Ref. [7], which are: +MV2 = 2.993 GeV and ΓV2 = 88 MeV. +Besides, we +take µΞ− = −0.915ˆµΞ−, and µΞ0 = −1.749ˆµΞ0 in natural +unit [54]. Then the coefficients f Ξ− +1 +, f Ξ− +2 +, f Ξ0 +1 , and f Ξ0 +2 +can +be calculated as +f Ξ− +1 += −1 + βρ +√ +2 + βV1 +√ +2 + βV2 +√ +2 − βωφ +√ +2 , +(24) +f Ξ− +2 += 0.085 + αV1 +√ +2 + αV2 +√ +2 − αωφ +√ +2 , +(25) +f Ξ0 +1 += − βρ +√ +2 − βV1 +√ +2 − βV2 +√ +2 − βωφ +√ +2 , +(26) +f Ξ0 +2 += −1.749 − αV1 +√ +2 − αV2 +√ +2 − αωφ +√ +2 . +(27) +The parameter γ will be fixed as the one determined from the +case of Σ, while the other free parameters βωφ, βρ, βV1, βV2, +αωφ, αV1, αV2, ΓV1, and MV1 are determined by fitting them +to experimental data on the time-like effective form factors of +Ξ− and Ξ0. +III. +NUMERICAL RESULTS +Under the above formulations, we perform a four-parameter +(γ, βρ, βωφ, αωφ)-χ2 fit to the experimental data on the ef- +fective form factors Geff of Σ+, Σ0, and Σ− hyperons. There +are 33 data points in total, which are extracted at the center- +of-mass energies from 2.3864 to 3.0200 GeV. The fitted pa- +rameters are: γ = 0.527 ± 0.024 GeV−2, βρ = 1.63 ± 0.07, +βωφ = −0.08 ± 0.06, and αωφ = −3.18 ± 0.77. And the +obtained χ2/dof is 1.69, where dof is the number of dimen- +sion of the freedom. Note that the obtained χ2/dof is larger +than 1, since we have fitted all the experimental data from +BESIII [9, 10], Belle [12], and BABAR [45] Collaborations, +by considering these contributions from only ground state of +vector mesons. If we considered only these data of BESIII +Collaboration [9, 10], the obtained χ2/dof is 1.17. In Fig. 2 +we show the theoretical results of the effective form factors of +the Σ+, Σ0, and Σ−. The red, blue, and green curves stand +for the results for Σ+, Σ0, and Σ−, respectively. The exper- +imental data from BESIII [9, 10], Belle [12], and BABAR +Collaboration [45] are also shown for comparing. One can +see that, with same model parameters, we can describe these +data on the effective form factors of Σ+, Σ0 and Σ− quite +well, especially for the precise data measured by the BESIII +Collaboration [9, 10]. The total cross sections of e+e− → Σ¯Σ +are also calculated with these fitted parameters. The numeri- +cal results are shown in Fig. 3, compared with the experimen- +tal data. Since the effective form factors of Σ hyperons can +be well reproduced with our model, the total cross sections +of e+e− → Σ+ ¯Σ−, e+e− → Σ0 ¯Σ0 and e+e− → Σ− ¯Σ+ +reactions can be also well described. +For the case of Ξ− and Ξ0 effective form factors, γ is taken +as the result of fitting to Σ hyperon, i.e., γ = 0.527, we per- +form nine-parameter (βωφ, βρ, βV1, βV2, αωφ, αV1, αV2, ΓV1, +MV1)-χ2 fit to the experimental data on. There are totally 18 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +√ +s +(GeV� +10 +−3 +10 +−2 +10 +−1 +10 +0 +10 +1 +|Geff| +Σ ++ +BESIII +Σ +0 +BESIII +Σ +− +BESIII +Σ ++ +Belle +Σ +0 +Belle +Σ +0 +BABAR +FIG. 2: The obtained effective form factors of Σ+, Σ0, and Σ−, +compared with the experimental data. +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +√ +s +(GeV� +10 +−1 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +σ(pb) +Σ ++ +BESIII +Σ +0 +BESIII +Σ +− +BESIII +Σ ++ +Belle +Σ +0 +Belle +Σ +0 +BABAR +FIG. 3: The total cross section of Σ+, Σ0 and Σ− hyperons com- +pared with experimental data. +data points, and these data correspond to the center-of-mass +energies from 2.644 to 3.080 GeV. The fitted parameters are +listed in Table II, with a reasonably small χ2/dof = 0.29. +Since we have more free parameters and the experimental data +points is limited, we did not get the uncertainties of these pa- +rameters from the χ2 fit. In Fig. 4, we depict the effective form +factor of the Ξ− and Ξ0 using the fitted parameters shown in +Table II. The red curve stands for the results of Ξ0, while the +green curve is the fitted results for Ξ−. Again, one can see that +the experimental data on the effective form factors of Ξ− and +Ξ0 can be well reproduced. It is worth to mention that the two +resonances V1 and V2 are crucial to describe the experimental +data, and without their contributions, we cannot get a good fit +to the experimental data. In addition, the total cross section of +e+e− → Ξ−¯Ξ+ and e+e− → Ξ0¯Ξ0 are also calculated with + +5 +TABLE II: Fitted model parameters for the effective form factors of +Ξ− and Ξ0. +Parameter +Value +Parameter +Value +βωφ +−0.774 +αωφ +9.346 +βρ +0.616 +αV1 +−0.039 +βV1 +0.099 +αV2 +−0.113 +βV2 +0.115 +ΓV1 (MeV) +71 +MV1 (GeV) +2.742 +the fitted parameters shown in Table II, and the numerical re- +sults are shown Fig. 5. The two peaks of V1 and V2 can be +clear seen, and more precise data around 2744 and 2993 MeV +are needed to further study their properties. +We next pay +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +√ +s +(GeV� +10 +−2 +10 +−1 +10 +0 +|Geff| +Ξ +− +BESIII +Ξ +0 +BESIII +FIG. 4: The obtained effective form factors of Ξ− and Ξ0 compared +with the experimental data. +attention to the EMFFs at the space-like region, which can +be straightforwardly obtained with the these parameters de- +termined from the experimental data in the time-like region. +Since the EMFFs in the space-like region are real, thus we +have to ignore the widths of the vector mesons. Then one can +calculate the mean squared charge radius, which is defined by +the relation [1, 40, 55] +� +r2 +ch +� += + + + + + + + +−6 +GE(0) +dGE(Q2) +dQ2 +���� +Q2=0 +, +for Σ+, Σ− and Ξ−, +−6 dGE(Q2) +dQ2 +���� +Q2=0 +, +for Σ0 and Ξ0, +(28) +with Q2 = −q2. With the parameters fitted above, the cal- +culated results of +� +r2 +ch +� +of Σ and Ξ hyperons are shown in +Table III. Our result for Σ− is agreement with the experi- +mental data within uncertainties: +� +r2 +ch +� +Σ− = 0.61 ± 0.12 ± +0.09 [1], +� +r2 +ch +� +Σ− = 0.91 ± 0.32 ± 0.4 [2]. In Ref. [1] the +Σ− charge radius was measured in the space-like Q2 range +0.035 − 0.105 GeV2 by elastic scattering of a Σ− beam +off atomic electrons. The measurement was performed with +the SELEX (E781) spectrometer using the Fermilab hyperon +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +√ +s +(GeV� +10 +−1 +10 +0 +10 +1 +10 +2 +10 +3 +σ(pb) +Ξ +− +BESIII +Ξ +0 +BESIII +FIG. 5: The total cross sections of e+e− → Ξ−¯Ξ+ and e+e− → +Ξ0¯Ξ0 reactions compared with experimental data. +beam at a mean energy of 610GeV. In Ref. [2] it was at- +tracted from the elastic scattering of high energy Σ− off elec- +trons from carbon and copper targets using the CERN hy- +peron beam, where these events are identified using a maxi- +mum likelihood technique exploring the kinematical relations +of the scattered particles. Theoretical calculations with chi- +ral perturbation theory (ChPT) [40, 56] and the nonlocal chi- +ral effective theory (ChET) [57], and chiral constituent quark +model (ChCQM) [58] are also listed for comparison. On can +see that the orderings of the most charge radii calculated by +other works are in agreement with our results. Moreover, our +results are consistent with these calculations in Refs. [56–58] +that +� +r2 +ch +� +Σ+ > +� +r2 +ch +� +Σ−. On the contrary, the results ob- +tained with chiral perturbation theory predictions in Ref. [40] +indicate that the charge radius of Σ− is larger than the one of +Σ+. In addition, the charge radius of Ξ0 calculated here is +small and negative, which is in agreement with the nonlocal +chiral effective theory calculation in Ref. [57]. It is expected +that these results can be tested by future experimental mea- +surements. +IV. +SUMMARY +In this work, we study the effective form factor of Σ and +Ξ hyperons in time-like region within the vector meson dom- +inance model, and we take a common model parameter γ. In +addition, the effect of the isospin combination is taken into +account. For the case of Σ hyperon, the contributions from ρ, +ω and φ mesons are considered. Within same model parame- +ters, we can simultaneously describe the current experimental +data on the effective form factors of Σ+, Σ0 and Σ−. While +for the case of Ξ+ and Ξ−, in addition to the contributions of +the ground states ρ, ω and φ, it is found that one needs also +contributions from two new vector states, and their masses and +widths are: MV1 = 2.742 GeV, ΓV1 = 71 MeV, MV2 = 2.993 + +6 +TABLE III: The obtained results for mean squared electromagnetic radii +� +r2 +ch +� +(fm2) for Σ and Ξ. The results from two ChPT calculations, +ChET and, ChCQM as well as the experimental data are also listed. +Baryon +Ξ0 +Ξ− +Σ+ +Σ0 +Σ− +This work +−0.07 +0.43 +0.78 +0.12 +0.65 +ChPT [40] +0.13 ± 0.03 +0.49 ± 0.05 +0.60 ± 0.02 +−0.03 ± 0.01 +0.67 ± 0.03 +ChPT [56] +0.36 ± 0.02 +0.61 ± 0.01 +0.99 ± 0.03 +0.10 ± 0.02 +0.780 +ChET [57] +−0.015 ± 0.007 0.601 ± 0.127 0.719 ± 0.116 0.010 ± 0.004 0.700 ± 0.124 +ChCQM [58] +0.091 +0.587 +0.825 +0.089 +0.643 +GeV, and ΓV2 = 88 MeV. It is expected that new precise ex- +perimental data at BESIII [59] can be used to further study +their properties. Finally, we would like to stress that thanks +to the effects of the isospin combinations, the effective form +factors of Σ+, Σ0 and Σ− can be simultaneously reproduced +within the same model parameters by using the vector meson +dominance model. Again, the theoretical results obtained here +also indicate that the vecor meson dominance model is a valid +tool for studying the baryonic electromagnetic form factors at +the time-like region. More precise data on the e+e− → Y ¯Y +reactions can be used to improve our knowledge of hyperon +effective form factors. +Acknowledgements +We warmly thank Profs. Xiong-Fei Wang and Xiao-Rong +Zhou for useful comments and discussions. +This work is +partly supported by the National Natural Science Founda- +tion of China under Grant Nos. 12075288, 11735003, and +11961141012. It is also supported by the Youth Innovation +Promotion Association CAS. +[1] I. M. Gough Eschrich et al. [SELEX], Phys. Lett. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' ‡ 1College of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Chengdu University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Chengdu 610059,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' China 2Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' China 3School of Nuclear Sciences and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Beijing 101408,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' China 4Southern Center for Nuclear-Science Theory (SCNT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Huizhou 516000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Guangdong Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' China We propose a phenomenological extended vector meson dominance model for the baryon electromagnetic structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' and it is found that the current experimental data on the Σ and Ξ electromagnetic form factors in the time-like region can be well described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Meanwhile, we can also reproduce the ratios of the total cross sections of reactions e+e− → Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+, which are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 : 1 at center-of-mass energies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3864 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' We also analytically continue the expression of the form factors to space- like region and estimate the charge radii of the Σ and Ξ hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The result for the Σ− is in agreement with the experimental date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' INTRODUCTION The electromagnetic structure information of hadrons is characterized by the electromagnetic form factors (EMFFs), which are functions of the four-momentum transfer squared q2, with q the four-momentum carried by the exchanged vir- tual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Study of these EMFFs can lead to a better under- standing of fundamental structure of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' On the experi- mental side, most commonly the baryon EMFFs in the space- like region (q2 < 0) were measured in the electron-baryon scattering [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' While for these unstable hadrons, for ex- ample, these hyperons, their EMFFs in the space-like region are very difficult to be experimentally measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' However, in the time-like region (q2 > 0), their EMFFs can be measured through the electron-positron annihilation reactions by the BESIII and Belle collaborations [5–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Meanwhile, the ef- fective form factor Geff(q2) of hyperons can be extracted from the high-precision measured Born cross sections of the reac- tions e+e− → Y ¯Y (Y stands for hyperon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' ¯Y is anti-hyperon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It was pointed out that these baryon EMFFs in the time-like region can be associated with the time evolution of the charge and magnetic distributions inside the baryon [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The hyperon effective form factors Geff(q2) are the func- tions that parametrize the γY ¯Y vertex generated by the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' While, the production vertex γY ¯Y is very poorly understood so far [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The vector meson dominance (VMD) model is a very successful tool for studying the nu- cleon electromagnetic form factors, in both the space-like and time-like regions [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Within a modified VMD model, the EMFFs of Λ hyperon were investigated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' By considering the Y ¯Y final sate interactions, the EMFFs of hyperons in the time-like region have been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It is worth to mention that the enhancement of the ∗Electronic address: yanbing@impcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='cn †Electronic address: chencheng22@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='cn ‡Electronic address: xiejujun@impcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='cn effective form factor of the Λ hyperon seen in the e+e− → Λ¯Λ reaction, was reproduced within the two above different calculations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [20, 21] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [22], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In the vector meson dominance model for studying the electro- magnetic form factors of baryons, there is a phenomenological intrinsic form factor g(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' From these studies of the nucleon and hyperon EMFFs [17–29], it is found that a better choice of g(q2) is the dipole form g(q2) = 1 (1 − γq2)2 , (1) with γ a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In the space-like region, the dipole form is consistent with the results obtained from perturbative quantum chromodynamics calculations [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In the time- like region, it should be noticed that γ is a positive parameter, thus g(q2) will have a pole in the position γ = 1/q2, such pole could be restricted in the unphysical region, if γ satisfy γ > 1/(4m2 Y ) for hyperon Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' For a long time, the simple dipole form paremetrization is very useful for the discussion of different baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' For exam- ple, the dipole form of g(q2) can well describe the effective form factors of Λ [20, 21], Σ [32], and Ξ [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' While for the nucleon, a good general review is given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [33–36], both from the theoretical and from the experimental points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' However, these determined values of γ for different ground state octet baryons with spin 1/2 are much different, even for the triplet Σ+, Σ− and Σ0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The determined values of γ, from previous works, for nucleon, Λ, Σ, and Ξ0 baryons are collected in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Nevertheless, the VMD model and the parametrization of g(q2) can give a reasonable description of the experimental data on the baryon EMFFs at the considered energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Various experimental and theoretical efforts have been con- tributed to the electromagnetic form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Very recently, the EMFFs of Σ+, Σ−, and Σ0 hyperons in the time-like region, have been measured with high-precision by the BESIII collab- oration through e+e− → Σ+ ¯Σ− [9], Σ− ¯Σ+ [9], and Σ0 ¯Σ0 reactions [10] at center-of-mass energies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3864 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The resulting ratios of total cross sections of these above 2 TABLE I: Values of γ (in GeV−2) for octet baryons used in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Proton ([17–19]) Neutron ([24–27]) Λ ([20]) Λ ([21]) γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='408 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='08 Σ+ ([32]) Σ− ([32]) Σ0 ([27]) Ξ0 ([27]) γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 three reactions are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 : 1 : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 [9, 37, 38], which disagree with various theoretical model predictions [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' After the experimental measurements of e+e− → Σ+ ¯Σ− and Σ− ¯Σ+ [9], the effective form factors of Σ+ and Σ− were in- vestigated by using the VMD model [32], where the param- eter γ were taken with different values for Σ+ and Σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [22], by considering the final state interactions of Y ¯Y , the energy dependence of the three reactions e+e− → Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+ at low energies can be roughly reproduced, and it was found that there is a strong interplay between Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+ channel in the near-threshold re- gion, caused by the Σ¯Σ final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In the present work, we revisit the EMFFs at the time-like region of Σ and Ξ hyperons within an extended vector meson dominance model, where the affects of the isospin combinations from isovector ρ0 and isoscalar ω and φ mesons are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Fur- thermore, we assume that the values of model parameter γ are same for Σ and Ξ hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In addition, a vector meson with mass around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 GeV was considered for the sake of bet- ter fitting the EMFFs of the Ξ0 and Ξ− hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' We then progress to an analysis of the electromagnetic form factors in the space-like region and evaluate the electromagnetic radius of Σ hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The theoretical result for the Σ− hyperon is in agreement with the experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' This article is organized as follows: in next section we will show the theo- retical formalism of the Σ and Ξ electromagnetic form factors in the VMD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Numerical results about the effective form factors of Σ and Ξ and total cross sections of e+e− → Σ¯Σ and Ξ¯Ξ are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' III, and a short summary is given in the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' FORMALISM As already pointed out, as fixed-energy e+e− colliders, the EMFFs of hyperons in the time-like region was extracted from the data on the differential cross section of the process e+e− → Y ¯Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' For analysis the data, the BESIII Collaboration use the energy scan method [41–43], while the initial state radiation method was used by Belle Collaboration [12] and BABAR collaboration [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Besides, the effective form factors Geff can be easily obtained from the data of the total cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The module squared of effective form factor |Geff|2 is a linear combination of |GE|2 and |GM|2, and pro- portional to the total cross section of e+e− → Y ¯Y reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In this work, we study the EMFFs of Σ and Ξ baryons in the time-like region with the experimental measurements on the e+e− → Y ¯Y reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Based on parity conservation and Lorentz invariant, the electromagnetic current of the baryons with a spin of 1/2 characterize two independent scalar func- tions F1(q2) and F2(q2) depending on q2, which are the Dirac and Pauli form factors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' While the corresponding electrical and magnetic form factors GE(q2) and GM(q2) are written as [38, 46, 47], GE(q2) = F1(q2) + τF2(q2), (2) GM(q2) = F1(q2) + F2(q2), (3) where M is the baryon mass and τ = q2/(4M 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' With GE(q2) and GM(q2), the magnitude of the effective form fac- tor |Geff(q2)| is defined as |Geff(q2)| = � 2τ|GM(q2)|2 + |GE(q2)|2 1 + 2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (4) In the time-like region, the effective form factors of hyper- ons are experimentally studied via the electron-positron anni- hilation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Under the one photon exchange approx- imation, the total cross section of the annihilation reaction e+e− → ¯Y Y can be expressed in terms of the effective form factor Geff as [44, 48, 49] σe+e−→ ¯Y Y = 4πα2βCY 3s (1 + 1 2τ ) | Geff(s) |2, (5) with α = e2/(4π) = 1/137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='036 the fine-structure con- stant, and β = � 1 − 4M 2 Y /s is a phase-space factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Here, s = q2 is the invariant mass square of the e+e− system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The coulomb enhancement factor CY accounts for the electromag- netic interaction of charged point-like fermion pairs in the fi- nal state [50], which is given by CY = � y 1−e−y for Σ+, Σ−, and Ξ−, 1 for Σ0 and Ξ0, (6) with y = απ β 2MY √s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Considering the CY factor, it is expected that the cross section of process e+e− → Y ¯Y is nonzero at the reaction threshold for charged hyperons pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' As plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 1 for the case of Ξ− 1, the factor CY affects only at the energy region very close to the reaction threshold, and it decreases very quickly as the reaction energy growing and it follows that few MeV above the reaction threshold it is CY ∼ 1, then its effect can be neglected [50–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The EMFFs of Σ hyperon In the VMD model, the virtual photon couples to Σ and ¯Σ through isovector ρ0 meson and isoscalar ω and φ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Since both the ω and φ are far from the mass threshold of Σ¯Σ, the behavior of the contributions from them are similar, thus we combine their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In this way, one can param- eterize Dirac and Pauli form factors for Σ+ and Σ− in the 1 The numerical results for Σ+ and Σ− are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 3 0 1 2 3 4 5 6 7 8 9 10 √ s − 2M Ξ − (MeV) 0 2 4 6 8 10 12 C Ξ − FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 1: The Coulomb factor for Ξ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' time-like region as follows [17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 2 F Σ+ 1 = g(q2)(f Σ+ 1 + βρ √ 2Bρ − βωφ √ 3 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (8) F Σ+ 2 = g(q2)(f Σ+ 2 Bρ − αωφ √ 3 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (9) F Σ− 1 = g(q2)(f Σ− 1 − βρ √ 2Bρ − βωφ √ 3 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (10) F Σ− 2 = g(q2)(f Σ− 2 Bρ − αωφ √ 3 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (11) F Σ0 1 = g(q2)(βωφ √ 3 − βωφ √ 3 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (12) F Σ0 2 = g(q2)µΣ0Bωφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (13) with Bρ = m2 ρ m2ρ − q2 − imρΓρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (14) Bωφ = m2 ωφ m2 ωφ − q2 − imωφΓωφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (15) where the widths of ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' ω and φ are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In this work, we take mρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='775 MeV, Γρ = 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1 MeV, Γωφ = 2 We have followed: |Σ+ ¯Σ−⟩ = 1 √ 2 |1, 0⟩ + 1 √ 3 |0, 0⟩ + 1 √ 6 |2, 0⟩ , |Σ− ¯Σ+⟩ = − 1 √ 2 |1, 0⟩ + 1 √ 3 |0, 0⟩ + 1 √ 6 |2, 0⟩ , |Σ0 ¯Σ0⟩ = − 1 √ 3 |0, 0⟩ + � 2 3 |2, 0⟩ , (7) with the basis of |IΣ¯Σ, IZ Σ¯Σ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In the one photon exchange approximation, there is no contributions from the isospin tensor terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (Γω + Γφ)/2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='4645 MeV, and mωφ = (mω + mφ)/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='9005 GeV, which are quoted in the review of particle physics book [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Besides, we take µΣ+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='112ˆµΣ+, µΣ− = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='479ˆµΣ−, µΣ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='044ˆµΣ0 in natural unit [54], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=', ˆµ = e 2MΣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In addition, at q2 = 0, with the constraints GΣ+ E = 1 and GΣ+ M = µΣ+, GΣ− E = −1 and GΣ− M = µΣ−, the coefficients f Σ+ 1 and f Σ+ 2 , f Σ− 1 and f Σ− 2 can be calculated, f Σ+ 1 = 1 − βρ √ 2 + βωφ √ 3 , f Σ+ 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='112 + αωφ √ 3 , (16) f Σ− 1 = −1 + βρ √ 2 + βωφ √ 3 , f Σ− 2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='479 + αωφ √ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (17) Finally, the model parameters γ, the coefficients βρ, βωφ, and αωφ will be determined by fitting them to the experimental data on the time-like effective form factors of Σ+, Σ0, and Σ−, which will be discussed in following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The EMFFs of Ξ hyperon For the case of e+e− → Ξ−¯Ξ+ and Ξ0¯Ξ0 reactions, since Ξ− and Ξ0 are isospin doublets, we express the Ξ−¯Ξ+ and Ξ0¯Ξ0 states in terms of isospin 0 and 1 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The mix- tures of isoscalar and isovector for Ξ−¯Ξ+ and Ξ0¯Ξ0 of equal relative wight but different sign are imposed by the isospin symmetry as introduced by the underlying Clebsch-Gorden coefficients [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' the Dirac and Pauli form factors F1 and F2 for Ξ− and Ξ0 can be easily obtained as before for the Σ hyperon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' F Ξ− 1 = g(q2)(f Ξ− 1 − βρ √ 2 Bρ − βV1 √ 2 BV1 −βV2 √ 2 BV2 + βωφ √ 2 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (18) F Ξ− 2 = g(q2)(f Ξ− 2 Bρ − αV1 √ 2 BV1 −αV2 √ 2 BV2 + αωφ √ 2 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (19) F Ξ0 1 = g(q2)(f Ξ0 1 + βρ √ 2 Bρ + βV1 √ 2 BV1 +βV2 √ 2 BV2 + βωφ √ 2 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (20) F Ξ0 2 = g(q2)(f Ξ0 2 Bρ + αV1 √ 2 BV1 +αV2 √ 2 BV2 + αωφ √ 2 Bωφ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (21) with BV 1 = M 2 V1 M 2 V1 − q2 − iMV1ΓV1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (22) BV 2 = M 2 V2 M 2 V2 − q2 − iMV2ΓV2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (23) where we have considered contributions from two more ex- cited vector mesons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' V1 and V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' in addition the contribu- tions from ground states ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' ω and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Their mass and width 4 are MV1 (MV2) and ΓV1 (ΓV2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The mass MV2 and width ΓV2 are taken as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [7], which are: MV2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='993 GeV and ΓV2 = 88 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Besides, we take µΞ− = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='915ˆµΞ−, and µΞ0 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='749ˆµΞ0 in natural unit [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Then the coefficients f Ξ− 1 , f Ξ− 2 , f Ξ0 1 , and f Ξ0 2 can be calculated as f Ξ− 1 = −1 + βρ √ 2 + βV1 √ 2 + βV2 √ 2 − βωφ √ 2 , (24) f Ξ− 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='085 + αV1 √ 2 + αV2 √ 2 − αωφ √ 2 , (25) f Ξ0 1 = − βρ √ 2 − βV1 √ 2 − βV2 √ 2 − βωφ √ 2 , (26) f Ξ0 2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='749 − αV1 √ 2 − αV2 √ 2 − αωφ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' (27) The parameter γ will be fixed as the one determined from the case of Σ, while the other free parameters βωφ, βρ, βV1, βV2, αωφ, αV1, αV2, ΓV1, and MV1 are determined by fitting them to experimental data on the time-like effective form factors of Ξ− and Ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' NUMERICAL RESULTS Under the above formulations, we perform a four-parameter (γ, βρ, βωφ, αωφ)-χ2 fit to the experimental data on the ef- fective form factors Geff of Σ+, Σ0, and Σ− hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' There are 33 data points in total, which are extracted at the center- of-mass energies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3864 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='0200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The fitted pa- rameters are: γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='527 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='024 GeV−2, βρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='07, βωφ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='06, and αωφ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' And the obtained χ2/dof is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='69, where dof is the number of dimen- sion of the freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Note that the obtained χ2/dof is larger than 1, since we have fitted all the experimental data from BESIII [9, 10], Belle [12], and BABAR [45] Collaborations, by considering these contributions from only ground state of vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' If we considered only these data of BESIII Collaboration [9, 10], the obtained χ2/dof is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 2 we show the theoretical results of the effective form factors of the Σ+, Σ0, and Σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The red, blue, and green curves stand for the results for Σ+, Σ0, and Σ−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The exper- imental data from BESIII [9, 10], Belle [12], and BABAR Collaboration [45] are also shown for comparing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' One can see that, with same model parameters, we can describe these data on the effective form factors of Σ+, Σ0 and Σ− quite well, especially for the precise data measured by the BESIII Collaboration [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The total cross sections of e+e− → Σ¯Σ are also calculated with these fitted parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The numeri- cal results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 3, compared with the experimen- tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Since the effective form factors of Σ hyperons can be well reproduced with our model, the total cross sections of e+e− → Σ+ ¯Σ−, e+e− → Σ0 ¯Σ0 and e+e− → Σ− ¯Σ+ reactions can be also well described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' For the case of Ξ− and Ξ0 effective form factors, γ is taken as the result of fitting to Σ hyperon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=', γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='527, we per- form nine-parameter (βωφ, βρ, βV1, βV2, αωφ, αV1, αV2, ΓV1, MV1)-χ2 fit to the experimental data on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' There are totally 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1 √ s (GeV� 10 −3 10 −2 10 −1 10 0 10 1 |Geff| Σ + BESIII Σ 0 BESIII Σ − BESIII Σ + Belle Σ 0 Belle Σ 0 BABAR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 2: The obtained effective form factors of Σ+, Σ0, and Σ−, compared with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1 √ s (GeV� 10 −1 10 0 10 1 10 2 10 3 10 4 σ(pb) Σ + BESIII Σ 0 BESIII Σ − BESIII Σ + Belle Σ 0 Belle Σ 0 BABAR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 3: The total cross section of Σ+, Σ0 and Σ− hyperons com- pared with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' data points, and these data correspond to the center-of-mass energies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='644 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='080 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The fitted parameters are listed in Table II, with a reasonably small χ2/dof = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Since we have more free parameters and the experimental data points is limited, we did not get the uncertainties of these pa- rameters from the χ2 fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 4, we depict the effective form factor of the Ξ− and Ξ0 using the fitted parameters shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The red curve stands for the results of Ξ0, while the green curve is the fitted results for Ξ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Again, one can see that the experimental data on the effective form factors of Ξ− and Ξ0 can be well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It is worth to mention that the two resonances V1 and V2 are crucial to describe the experimental data, and without their contributions, we cannot get a good fit to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In addition, the total cross section of e+e− → Ξ−¯Ξ+ and e+e− → Ξ0¯Ξ0 are also calculated with 5 TABLE II: Fitted model parameters for the effective form factors of Ξ− and Ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Parameter Value Parameter Value βωφ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='774 αωφ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='346 βρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='616 αV1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='039 βV1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='099 αV2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='113 βV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='115 ΓV1 (MeV) 71 MV1 (GeV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='742 the fitted parameters shown in Table II, and the numerical re- sults are shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The two peaks of V1 and V2 can be clear seen, and more precise data around 2744 and 2993 MeV are needed to further study their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' We next pay 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1 √ s (GeV� 10 −2 10 −1 10 0 |Geff| Ξ − BESIII Ξ 0 BESIII FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 4: The obtained effective form factors of Ξ− and Ξ0 compared with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' attention to the EMFFs at the space-like region, which can be straightforwardly obtained with the these parameters de- termined from the experimental data in the time-like region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Since the EMFFs in the space-like region are real, thus we have to ignore the widths of the vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Then one can calculate the mean squared charge radius, which is defined by the relation [1, 40, 55] � r2 ch � = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −6 GE(0) dGE(Q2) dQ2 ���� Q2=0 , for Σ+, Σ− and Ξ−, −6 dGE(Q2) dQ2 ���� Q2=0 , for Σ0 and Ξ0, (28) with Q2 = −q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' With the parameters fitted above, the cal- culated results of � r2 ch � of Σ and Ξ hyperons are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Our result for Σ− is agreement with the experi- mental data within uncertainties: � r2 ch � Σ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='09 [1], � r2 ch � Σ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='4 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [1] the Σ− charge radius was measured in the space-like Q2 range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='035 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='105 GeV2 by elastic scattering of a Σ− beam off atomic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The measurement was performed with the SELEX (E781) spectrometer using the Fermilab hyperon 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1 √ s (GeV� 10 −1 10 0 10 1 10 2 10 3 σ(pb) Ξ − BESIII Ξ 0 BESIII FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 5: The total cross sections of e+e− → Ξ−¯Ξ+ and e+e− → Ξ0¯Ξ0 reactions compared with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' beam at a mean energy of 610GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [2] it was at- tracted from the elastic scattering of high energy Σ− off elec- trons from carbon and copper targets using the CERN hy- peron beam, where these events are identified using a maxi- mum likelihood technique exploring the kinematical relations of the scattered particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Theoretical calculations with chi- ral perturbation theory (ChPT) [40, 56] and the nonlocal chi- ral effective theory (ChET) [57], and chiral constituent quark model (ChCQM) [58] are also listed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' On can see that the orderings of the most charge radii calculated by other works are in agreement with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Moreover, our results are consistent with these calculations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [56–58] that � r2 ch � Σ+ > � r2 ch � Σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' On the contrary, the results ob- tained with chiral perturbation theory predictions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [40] indicate that the charge radius of Σ− is larger than the one of Σ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In addition, the charge radius of Ξ0 calculated here is small and negative, which is in agreement with the nonlocal chiral effective theory calculation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It is expected that these results can be tested by future experimental mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' SUMMARY In this work, we study the effective form factor of Σ and Ξ hyperons in time-like region within the vector meson dom- inance model, and we take a common model parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' In addition, the effect of the isospin combination is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' For the case of Σ hyperon, the contributions from ρ, ω and φ mesons are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Within same model parame- ters, we can simultaneously describe the current experimental data on the effective form factors of Σ+, Σ0 and Σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' While for the case of Ξ+ and Ξ−, in addition to the contributions of the ground states ρ, ω and φ, it is found that one needs also contributions from two new vector states, and their masses and widths are: MV1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='742 GeV, ΓV1 = 71 MeV, MV2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='993 6 TABLE III: The obtained results for mean squared electromagnetic radii � r2 ch � (fm2) for Σ and Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' The results from two ChPT calculations, ChET and, ChCQM as well as the experimental data are also listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Baryon Ξ0 Ξ− Σ+ Σ0 Σ− This work −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='65 ChPT [40] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='03 ChPT [56] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='780 ChET [57] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='015 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='601 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='719 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='700 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='124 ChCQM [58] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='643 GeV, and ΓV2 = 88 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It is expected that new precise ex- perimental data at BESIII [59] can be used to further study their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Finally, we would like to stress that thanks to the effects of the isospin combinations, the effective form factors of Σ+, Σ0 and Σ− can be simultaneously reproduced within the same model parameters by using the vector meson dominance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Again, the theoretical results obtained here also indicate that the vecor meson dominance model is a valid tool for studying the baryonic electromagnetic form factors at the time-like region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' More precise data on the e+e− → Y ¯Y reactions can be used to improve our knowledge of hyperon effective form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Acknowledgements We warmly thank Profs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Xiong-Fei Wang and Xiao-Rong Zhou for useful comments and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' This work is partly supported by the National Natural Science Founda- tion of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' 12075288, 11735003, and 11961141012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' It is also supported by the Youth Innovation Promotion Association CAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Gough Eschrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' [SELEX], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content=' Lett.' metadata={'source': 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+page_content='4, 040001 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='1088/1674-1137/44/4/040001 [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} +page_content='05983 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf'} diff --git a/29FKT4oBgHgl3EQfQS0h/content/tmp_files/2301.11766v1.pdf.txt b/29FKT4oBgHgl3EQfQS0h/content/tmp_files/2301.11766v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4efe239643a051cf3ec6eb656849610e543c934a --- /dev/null +++ b/29FKT4oBgHgl3EQfQS0h/content/tmp_files/2301.11766v1.pdf.txt @@ -0,0 +1,1581 @@ +EPJ manuscript No. +(will be inserted by the editor) +Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p +Collisions +R. Abou Yassine6,13, O. Arnold10,9, M. Becker11, P. Bergmann5, A. Blanco1, C. Blum8, M. B¨ohmer10, N. Carolino1, +L. Chlad14,c, P. Chudoba14, I. Ciepał3, J. Dreyer7, W. Esmail5, L. Fabbietti10,9, P. Fonte1,a, J. Friese10, I. Fr¨ohlich8, +T. Galatyuk6,5, J. A. Garz´on15, M. Grunwald17, M. Gumberidze5, S. Harabasz6,b, C. H¨ohne11,5, F. Hojeij13, R. Holzmann5, +H. Huck8, M. Idzik2, B. K¨ampfer7,c, B. Kardan8, V. Kedych6, I. Koenig5, W. Koenig5, M. Kohls8, J. Kolas17, G. Korcyl4, +G. Kornakov17, R. Kotte7, W. Krueger6, A. Kugler14, T. Kunz10, R. Lalik4, F. Linz6,5, L. Lopes1, M. Lorenz8, A. Malige4, +J. Markert5, V. Metag11, J. Michel8, A. Molenda2, C. M¨untz8, M. Nabroth8, L. Naumann7, K. Nowakowski4, J. Orli´nski16, +J.-H. Otto11, Y. Parpottas12, M. Parschau8, V. Pechenov5, O. Pechenova5, K. Piasecki16, J. Pietraszko5, A. Prozorov14,d, +W. Przygoda4, B. Ramstein13, N. Rathod17, J. Ritman5, A. Rost6,5, A. Rustamov5, P. Salabura4, N. Schild6, E. Schwab5, +F. Seck6, U. Singh4, S. Spies8, M. Stefaniak17,5, H. Str¨obele8, J. Stroth8,5, C. Sturm5, K. Sumara4, O. Svoboda14, M. Szala8, +P. Tlusty14, M. Traxler5, H. Tsertos12, V. Wagner14, A.A. Weber11, C. Wendisch5, H.P. Zbroszczyk17, E. Zherebtsova5,e, +M. Zielinski4, and P. Zumbruch5 (HADES collaboration) +1 LIP-Laborat´orio de Instrumentac¸˜ao e F´ısica Experimental de Part´ıculas 3004-516 Coimbra, Portugal +2 AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krak´ow, Poland +3 Institute of Nuclear Physics, Polish Academy of Sciences, 31342 Krak´ow, Poland +4 Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Krak´ow, Poland +5 GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany +6 Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany +7 Institut f¨ur Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany +8 Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany +9 Excellence Cluster ’Origin and Structure of the Universe’, 85748 Garching, Germany +10 Physik Department E62, Technische Universit¨at M¨unchen, 85748 Garching, Germany +11 II.Physikalisches Institut, Justus Liebig Universit¨at Giessen, 35392 Giessen, Germany +12 Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus +13 Laboratoire de Physique des 2 infinis Ir`ene Joliot-Curie, Universit´e Paris-Saclay, CNRS-IN2P3., F-91405 Orsay, France +14 Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech Republic +15 LabCAF. F. F´ısica, Univ. de Santiago de Compostela, 15706 Santiago de Compostela, Spain +16 Uniwersytet Warszawski - Instytut Fizyki Do´swiadczalnej, 02-093 Warszawa, Poland +17 Warsaw University of Technology, 00-662 Warsaw, Poland +a also at Coimbra Polytechnic - ISEC, Coimbra, Portugal +b also at Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Darmstadt, 64390 Darmstadt, Germany +c also at Technische Universit¨at Dresden, 01062 Dresden, Germany +d also at Charles University, Faculty of Mathematics and Physics, 12116 Prague, Czech Republic +e also at University of Wrocław, 50-204 Wrocław, Poland +e-mail: hades-info@gsi.de +Abstract The production of Σ0 hyperons in proton proton collisions at a beam kinetic energy of 3.5 GeV impinging +on a liquid hydrogen target was investigated using data collected with the HADES setup. The total production cross +section is found to be σ(pK+Σ0)[µb] = 17.7 ± 1.7(stat) ± 1.6(syst). Differential cross section distributions of the +exclusive channel pp → pK+Σ0 were analyzed in the center-of-mass, Gottfried-Jackson and helicity reference frames +for the first time at the excess energy of 556 MeV. The data support the interplay between pion and kaon exchange +mechanisms and clearly demonstrate the contribution of interfering nucleon resonances decaying to K+Σ0. The Bonn- +Gatchina partial wave analysis was employed to analyse the data. Due to the limited statistics, it was not possible +to obtain an unambiguous determination of the relative contribution of intermediate nucleon resonances to the final +state. However nucleon resonances with masses around 1.710 GeV/c2 (N∗(1710)) and 1.900 GeV/c2 (N∗(1900) or +∆∗(1900)) are preferred by the fit. +arXiv:2301.11766v1 [nucl-ex] 27 Jan 2023 + +2 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +1 Introduction +Strangeness production at intermediate energies in p+p and +p+A collisions is of particular importance to the field of hadron +physics. The production of baryons with strange quark content, +i.e. hyperons, requires creating a new quark flavor, which can +occur out of the vacuum from the quark sea in the colliding +protons. The s-quark with mass +O(ΛQCD) is distinguished +from the light (u, d) quark flavors but much smaller than heavy +(c, t, b) flavors. The resulting (approximate) SU(3) flavor +symmetry in the u-d-s sector is therefore still a cornerstone of +hadron physics. Since the entrance channel in p+p and p+A +collisions carries no net strangeness, the emergence of an +s-quark can unravel much of the flavor dynamics in hadronic +reactions. The flavor-conserving strong interaction process +requires associate strangeness production, e.g. realized by +simultaneous creation of a single-strange hyperon, such as +Λ or Σ0 and an associated kaon. Therefore, understanding +the production mechanism of strange baryons near threshold +deepens our knowledge of their internal structure and of the +strong interaction in the non-perturbative regime. Strangeness +production is also used as a probe to study hot and dense +nuclear matter in heavy ion collisions both at medium-energy +and in the late stages prior to freeze-out in high-energy +collisions, e.g. at LHC [1]. +The production of the Λ hyperon in p+p and p+A reac- +tions near threshold has been studied extensively by many ex- +periments including HADES [2, 3, 4, 5, 6], yet there are only +few experimental investigations on the Σ0 hyperon [2, 7]. De- +spite there are considerable experimental results and numerous +dedicated theoretical investigations, the strangeness production +mechanism is not yet well understood. In the context of the bo- +son exchange model [8, 9, 10, 11], +it is assumed that the initial protons exchange a virtual me- +son. The interaction between the meson and the initial protons +results in the production of the final state particles, which can +proceed directly or via an intermediate resonance. +The exchange of a virtual meson can be put into one of +two categories. The first category is strange meson exchange, +where strangeness exchange occurs, and no resonances are +involved. In this case, the reaction amplitude KN → KN +is governed by t-channel diagrams. The second category is +non-strange meson exchange, a pion exchange in its simplest +form. At the same time the elementary reaction amplitude +πN → KY is dominated by resonance excitations, which +implies a strong and characteristic energy dependence, where +Y stands for hyperons (Λ, Σ0, ...). +Several experiments have studied the exclusive reaction +pp → pK+Λ and proven that a pure phase space model de- +scription of the data is not sufficient without taking the dynam- +ics of the process into account [2, 6, 12, 13]. It was found that +the Λ hyperon production is dominated by the excitation and +subsequent decay of N∗ resonances to the K+Λ decay chan- +nel. In particular N∗(1650) (JP= 1 +2 +−), N∗(1710) (JP= 1 +2 ++) +and N∗(1720) (JP= 3 +2 ++) were found to contribute. This sup- +ports a picture wherein the exchange of non-strange mesons +is the leading process in the production mechanism. In addi- +tion, a considerable Final State Interaction (FSI) was found +to contribute [14, 15] leading to ΣN → ΛN conversion that +is observed as a ΣN cusp effect in the Λ cross section [16]. +In the pp → pK+Σ0 reaction the proton–hyperon FSI seems +to be negligible, especially at low energies near threshold and +a pure phase space distribution describes the data reasonably +well. The cross section ratio σ(pK+Λ) / σ(pK+Σ0) below ex- +cess energies of ∼ 20 MeV is about 28 in agreement of the +SU(6) prediction and reduces drastically to about 2.5 at excess +energies above 300 MeV [17, 18]. This energy-dependence of +the cross section ratio is strongly affected by FSI effects in the +pp → pK+Λ reaction [19]. +Besides the energy dependence of the cross section, the +differential cross sections at selected energies add much more +stringent tests of the model descriptions. This study fills this +gap and delivers such data which allow some clues about the +involved exchange mesons and resonances, in particular by em- +ploying a partial wave analysis. +Furthermore, +a +theoretical +study +of +the +reaction +pp → pK+Σ0 +based on a chiral dynamical study has +been proposed in [20]. This approach uses the pion and kaon +exchange mechanisms and chiral amplitudes in addition to all +pairs FSI, where the contribution of nucleon resonances appear +naturally using chiral unitary amplitudes. +This paper is organized as follows. In Section 2, the experi- +mental setup is briefly explained. Section 3 is devoted to the Σ0 +selection method, where the particle identification, the Λ hy- +peron reconstruction and the kinematic refit methods were pre- +sented. In Section 3.5 the method for efficiency correction and +differential analysis is described. Sections 5 and 6 presents the +calculated total production cross section and the partial wave +analysis of the exclusive reaction pp → pK+Σ0. In Section 7 +a summary and a short outlook are given. +2 The HADES experiment +The data presented here were collected in April 2007 with the +High Acceptance Di-Electron Spectrometer (HADES) located +at the heavy ion synchrotron SIS18 at GSI Helmholtzzentrum +f¨ur Schwerionenforschung in Darmstadt, Germany. HADES is +characterized by six identical sectors covering almost the full +azimuthal range and polar angles from θ = 18◦ to θ = 85◦. Each +sector of the spectrometer contains a Ring-Imaging Cherenkov +Detector (RICH) operating in a magnetic field-free region that +allows lepton identification over a wide range of momenta. Two +Multi-Wire Drift Chambers (MDCs) are placed in front of a +toroidal magnetic field, and two outer MDCs are placed behind +the magnetic field. The MDCs enable the momentum informa- +tion and the specific energy loss dE/dx to be reconstructed for +each particle track. Two scintillator hodoscopes, the Time Of +Flight (TOF) and TOFino are also placed behind the magnet +and provide a stop time (ts) signal. The TOF and TOFino sys- +tem are used as input to the trigger systems to start the data +readout. A detailed description of the HADES setup can be +found in [21]. +In the present analysis, a proton beam with an intensity of +107 particles/s and kinetic energy T = 3.5 GeV was incident on +a liquid hydrogen target with an areal density of 0.35 g/cm2. +The dimensions of the target were 15 mm in diameter and 50 +mm length located between -65 to -15 mm in the longitudi- +nal direction. The data readout was started by a first level trig- + +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +3 +ger requiring a charged particle multiplicity ≥ 3 (M3). In total, +1.14 × 109 events were recorded under these conditions [3]. +During this experiment HADES included an additional For- +ward Wall (FW) scintillator hodoscope that was placed 7 me- +ters downstream the target in a magnetic field-free region and +covered polar angles from θ = 0.33◦ to θ = 7.17◦ with full az- +imuthal acceptance. The FW measured the hit position and ar- +rival time of the particle track with a time resolution of about +700 ps [22]. +3 Event selection method +In this section, the exclusive reconstruction of the reaction +pp → pK+Σ0 is presented. The Σ0 hyperon is reconstructed +via its electromagnetic decay Σ0 → Λγ (BR ≈ 100%) and the +daughter Λ hyperon is reconstructed with the decay mode +Λ → pπ− (BR = 63.9%). +The Σ0 reconstruction strategy includes the following steps: +a) time of flight (tof) reconstruction, b) charged particle iden- +tification (PID), c) the Λ hyperon reconstruction, and d) the Σ0 +hyperon reconstruction. +3.1 Time of flight reconstruction +The interaction of the high intensity proton beam with the +START detector induced a background and prevented a stable +operation of the RICH detector. Therefore, it was not possible +to use the START detector information during this experiment. +Consequently, the tof of particle tracks were not directly mea- +sured since there was no common start time (t0) reference for +tracks in the same event. The start time has to be reconstructed +in order to obtain a proper time of flight measurement. +The reconstruction algorithm is based on the assumption +that at least one particle has been correctly identified. Since pi- +ons are abundantly produced, it is assumed that any negatively +charged particle track that is geometrically uncorrelated to a +ring in the RICH detector is a π−. The common start time for +each event is calculated by +t0 = ts − d +c × +� +p2 + m2π +p +, +where ts is the stop time of the π−, d is the distance to the TOF +or TOFino hit, mπ is the pion mass, p is the momentum of the +π− and c is the velocity of light. The tof of the other particles +in the same event is the difference between the measured stop +time ts and the common start time t0. +3.2 Particle identification (PID) +The reconstruction of the exclusive reaction pp → pK+pπ−γ +only requires the identification of three particle species, pions +(π−), kaons (K+) and protons (p), since the event is kinemati- +cally complete even without measuring the photon (γ). +As mentioned in the previous section, the π− is identified +as any negatively charged track that is geometrically uncorre- +lated to a ring in the RICH detector. Therefore, the problem +reduces to identifying the positively charged tracks. +In order to minimize systematic bias in the model output, +an auto-encoder [23] implemented in PyTorch framework [24] +is trained simultaneously with both simulated and real events +[25]. The input features used to train the auto-encoder are the +momentum components, the energy loss dE/dx in the MDC and +TOF sub-systems, the reconstructed tof and the distance to the +TOF/TOFino hit. +A classification layer has been stacked on top of the +bottleneck layer of the auto-encoder, which has three output +nodes corresponding to the three classes (π+, K+ and p). Each +node outputs a number between 0 and 1, where all output +numbers sum to 1, so that each number can be interpreted as +a probability of being a specific particle species. The network +is trained by minimizing a cost function that is defined as the +binary cross-entropy loss [26]. Because the network outputs +three probabilities for each particle track, the node with the +largest probability is chosen. +The classification accuracy evaluated on a holdout data-set +is 92% for pions, 76% for kaons and 98% for protons. It is +much lower in the case of kaons since their production rate is +suppressed with respect to the protons and pions. +3.3 Λ hyperon reconstruction +The next step after the PID is to reconstruct the intermediate Λ +hyperon. In this analysis the Λ reconstruction method is two- +fold. In the first case, which is referred to as the Spectrometer +data-set, events with exactly 2 protons, 1 pion and 1 kaon are +required to be within the main HADES detector acceptance. +The other case, referred as the WALL data-set, events were ac- +cepted if exactly 1 proton, 1 pion and 1 kaon are registered in +HADES and in addition one hit in the FW. In the latter case, it is +assumed that the hit registered in the FW is due to the daughter +proton from the Λ decay (marked as pdecay). +A common primary vertex in each event is then defined as +the intersection point or the Point of Closest Approach (PCA) +of the proton and kaon tracks. Since there is more than one pro- +ton in each event in the Spectrometer data-set, the proton-kaon +pair with the smaller Distance of Closest Approach (DCA) is +used to define the primary vertex. To reduce the contribution +from off-target events, a two dimensional selection is applied +on the primary vertex position (x, y, z): +a) -65 < z [mm] < -15 and +b) +� +x2 + y2 < 5 [mm]. +The Spectrometer data-set +Since the daughter Λ decays weakly (cτ = 7.89 cm), it can be +identified by its displaced vertex. First, all possible combina- +tions of the two p and π− candidates were made, leaving the +decision about which is the decay proton (Λ → p π−) for later. +For each combination the decay vertex (the displaced vertex) is +defined as the PCA between the two tracks. The DCA between +the p and π− tracks (marked as dpπ−) is expected to be small if +the tracks stem from the same vertex. Therefore, an upper limit +of dpπ− < 10 mm is imposed in order to reduce Combinatorial + +4 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +Figure 1: (a) The DCA distribution between the p and π− tracks. (b) The DCA distribution between the Λ track and the primary +vertex. In both panels, data are shown by the black points, the blue histogram represents the true Λ and the red histogram +represents the CB, where both the true Λ and the CB were estimated from the simulation. (c) Distribution of the DCA between +the π− track and the primary vertex as a function of the DCA between the p track and the primary vertex. The arrows indicate +the accepted regions. +Background (CB), which originates from combining the wrong +p and π− pairs. Considering momentum and energy conserva- +tion, the p should be emitted in nearly the same direction as the +Λ in the laboratory reference frame, while the π− will have a +different direction. Thus, the DCA between the p track and the +primary vertex (dp,pvtx) is required to be smaller than the DCA +between the π− track and the primary vertex (dπ−,pvtx). Fi- +nally, the DCA between the calculated Λ track and the primary +vertex (dΛ,pvtx) is required to be < 6 mm. The distributions +of the topological variables are shown in Figure 1, where the +selection criteria are indicated by the vertical dashed lines. The +proton used in the Λ reconstruction is tagged as the decay pro- +ton (marked in the following as pdecay), while the other proton +in the event is tagged as the scattered (primary) proton. +To further purify the selected Λ sample, the event kinemat- +ics were constrained to the Σ0 production range. The squared +pΛ missing mass (MM2(ppdecayπ−)) is required to be > 0.2 +GeV/c2 in order to reject the multi-pion production channel +as shown in Figure 2. In this figure, the experimental data +are shown by the black points and the simulations (discussed +in Section 3.4) by different colored histograms. Two peaks +are visible, the first peak at 0.02 GeV/c2 corresponds to the +multi-pion channel via the reaction pp → ppπ+π− (violet +histogram), where a pπ− pair is incorrectly identified as a Λ +candidate and the π+ is incorrectly identified as a K+. The +other broader peak is the sum of pp → pK+Λ, pp → pK+Σ0 +and pp → pK+Λπ0 reactions shown by the red, blue and +green histograms, respectively. The relative normalizations +of the simulated channels have been chosen to best fit the +experimental data as explained in Section 3.4. +The pdecay π− invariant mass distribution is shown in +Figure 3. A peak around the nominal Λ mass is visible on +top of background. The signal has been parameterized by +a Voigt distribution and the background is modeled by a +fourth-order polynomial. Events are further processed if they +are in the range of µ ± 3σ, where the calculated signal to +background ratio in this range is S/B = 2.57 and the number +of Λ candidates is NΛ = 6766. +The WALL data-set +In the WALL data-set the hit in the FW is assumed to be due +to the decay proton. Since the FW is installed in a magnetic +field-free region, the pdecay is reconstructed as a straight line +trajectory from the primary vertex position to the hit in the FW. +The track momentum is calculated from the tof and the dis- +tance from the primary vertex and the FW detector hit, assum- +ing the proton mass. In this case, the topological cuts are not +as effective to suppress the background as in the Spectrometer +data-set. Therefore, events fulfilling the following kinematical +constraints were selected: +(i) MM2(ppdecayπ−) > 0.2 GeV/c2 (Figure 4 a) and +(ii) The squared missing mass of all charged particles is +required to be in the following range: +−0.02 < MM2(pK+pdecayπ−)[GeV2/c4] < 0.01 +be- +cause +only a photon is missing to completely measure the +exclusive final state (Figure 4 b). + +(a) +(b) +(c) +108 +108 +15 +×103 +L Data +4.5 +HADES +TrueA +4 +p(3.5 GeV)+p → pK+z0 +106 +106 +CB +3.5 +ww +0.2 +3 +2.5 +104 +Events +2 +1.5 +102 +102 +1 +0.5 +0 +0 +50 +100 +0 +20 +40 +0 +5 +10 +15 +[mm] +d(p, pvtx) [mm]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +5 +Figure 2: The squared ppdecay π− missing mass distribu- +tion after applying the topological selections. Black points are +the Spectrometer data-set data. The violet histogram is the +pp → ppπ+π− simulation. The pp → pK+Λ, pp → pK+Σ0 +and pp → pK+Λπ0 simulations are shown by the red, blue and +green histograms, respectively. The vertical line and the arrow +indicate the accepted region for the further analysis. +The pdecayπ− invariant mass distribution for the WALL +data-set is shown Figure 5 after applying the selections +mentioned above. Once again, the peak has been fitted by +a Voigt distribution and the background by a fourth-order +polynomial. However, the mass resolution of the Λ peak of +the Spectrometer data-set(Figure 3) is better than the signal +of the WALL data-set, since in the latter case the proton was +detected in the FW, which has a worse momentum resolution. +Events are further processed if they are in the range of µ ± 3σ, +where the calculated signal to background ratio in this range is +S/B = 1.56 and the number of Λ candidates is NΛ = 2340. +3.4 Σ0 hyperon reconstruction +To further suppress the remaining background and to obtain a +better mass resolution, a kinematic fit based on the Lagrange +multiplier method is employed [27]. The fit χ2, expressed as +χ2(η, λ) = (y − η)T V (y)(y − η) + 2λT f(η) , +is minimized by differentiating χ2 with respect to all measured +variables. Here y is a vector containing the initial guesses for +the measured quantities, which are the track parameters pro- +vided by the tracking algorithm, η is an improved set of the +track parameters and V is the covariance matrix comprising +the estimated errors on the measured quantities. The constraint +Figure 3: The pdecay π− invariant mass distribution. The verti- +cal dashed lines indicate the selected mass range. The blue, red +and green curves are for the signal, background and the total +fit. +equations are expressed as a function of η in f(η), where λi are +a set of Lagrange multipliers. +The spherical coordinates used in this analysis for the track +parameterization are defined as follows +y = +� +� +1/p +θ +φ +� +� , +where 1/p is the inverse of the absolute momentum, θ and φ +are the polar and azimthual angles of the track. +Two constraints were applied to both data-sets. The first is +the proton and pion from the Λ decay are constrained to the Λ +mass (MΛ = 1.1157 GeV/c2). The second constraint is that +the missing mass of all the charged final state particles is con- +strained to the photon mass (Mγ = 0 GeV/c2). +The probability that a χ2 of the theoretical distribution is +greater than or equal to the χ2 value found from the fit is known +as the p-value (P(χ2)). The p-value distributions of the Spec- +trometer and the WALL data-sets are shown in Figure 6. Be- +cause both Λ and Σ0 have MM(pK+Λ) = 0, they have simi- +lar distributions, which makes these two reactions difficult to +distinguish. On the other hand, the reaction pp → pK+Λπ0 +should ideally have zero p-value. However, due to the limited +detector resolution it has p-values greater than zero, which is +more pronounced in the WALL data-set. The signal events show +an almost flat distribution between 0 and 1, while events that do +not satisfy the constraint equations have a prominent yield of +p-values close to 0. Therefore, events with p-values > 0.01 are +selected, where the cut was optimized based on a significance +analysis. + +X103 +1.2 + Data +HADES +pK+0 +p(3.5 GeV)+p -→ pK+z0 +Pp元+元 +pK+^ +pK+^元° +0.8 +0.01 +0.6 +Events / +0.4 +0.2 +-0.5 +0 +0.5 +MM? +(pp +元)[GeV2/c4] +decayX103 +1. +Mean +1.114 +Sigma +0.002 +HADES +1.2 +S/B +2.57 +p(3.5 GeV)+p → pK+≥0 +Na +6766 +2 +/ 0.001 GeV/c +0.8 +0.6 +Events / +0.4 +0.2 +1.09 +1.1 +1.11 +1.12 +1.13 +1.14 +1.15 +M. +[GeV/c"] +decay6 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +Figure 4: (a) The squared ppdecay π− missing mass distribution of WALL data-set. (b) The squared ppdecay π− K+ missing mass +distributions. The pp → ppπ+π−, pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 simulations are shown by the violet, red, +blue and green histograms, respectively. The arrows indicate the accepted regions. +Simulation scaling to the experimental data +By inspecting the pK+ missing mass distribution of the com- +bined data-set shown in Figure 7, two peaks corresponding to +the Λ and the Σ0, as well as other minor contributions in the +high mass region, are plainly evident. In order to quantify the +different contributions an incoherent cocktail has been simu- +lated using the Pluto event generator [28]. All the simulated re- +actions have been processed using the same full scale analysis +employed for the experimental data, thus taking into account +the efficiency of the trigger condition, the tracking algorithm +and the analysis procedure. The particle decays, the acceptance +and the particle interactions with the materials of HADES and +the FW have been considered by using GEANT3 [29]. +To determine the contributions of the different channels, a +fit of the simulations to the measured missing mass spectrum +(MM(pK+)) has been carried out by minimizing the quantity +χ2 = +nbins +� +i +(ndata − � +ch(f ch × nch +simulation))2 +σ2 +data + σ2 +simulation +, +where the summation runs over the number of bins of the miss- +ing mass spectrum, ndata is the number of data events in each +bin, nch +simulation is the number of simulated events in each bin +for each channel and fch is a scaling factor for each channel. +The uncertainty for the data and the simulations in each bin is +σdata and σsimulation, respectively. +As can be seen from Figure 7, the experimental data is +primarily described by contributions of pp → pK+Λ, +pp → pK+Σ0 and pp → pK+Λπ0 indicated by the red, blue +and the green histogram, respectively. The other simulated +channels have minor contributions. In total 2613 Σ0 candidates +were collected within the pK+ missing mass range of 1.170- +1.220 GeV/c2, 58% of them are within the main HADES +acceptance and 42% within the FW acceptance. The signal +purity in the mass window calculated from the simulation is +found to be 81%, where the main background contributions are +the reactions pp → pK+Λ (14%) and pp → pK+Λπ0 (5%). +3.5 Efficiency and acceptance correction +The reconstructed experimental distributions are corrected for +the limited detector acceptance and efficiency by using a sim- +ulated phase space distribution that were assigned a weight de- +termined by the best partial wave solution (discussed in Sec- +tion 6), then the events were filtered through the full scale simu- +lation and analysis. The efficiency correction is done in one di- +mension whereas the other three degrees of freedom on which +the efficiency depends are integrated. +The 1D correction matrix (M) is calculated given the ini- +tial 4π distribution (T) for each observable and after filtering + +(a) +(b) +X103 +X103 + Data +45 +HADES +pK+0 +5 +40 +p(3.5 GeV)+p → pK+≥0 +pp元+元 +35 +pK+A +Events / 0.002 GeV2/c4 + pK+A元0 +4 +30 +25 +3 +20 +2 +15 +10F +5 +0 +-1 +-0.5 +0 +0.5 +-0.1 +-0.05 +0 +0.05 +0.1 +MM2 (pp +元) [GeV?/c4] +MM? (pK*p.,) [GeV2/c*]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +7 +Figure 5: The pwallπ− invariant mass distribution. The vertical +dashed lines indicate the selected mass range. The blue, red and +green curves are for the signal, background and the total fit. +through the full scale simulation and analysis (R). To put it an- +other way, a distinct correction matrix M = R/T is constructed +for each angular distribution shown in Figure 8. The inverse +of the correction matrix is then calculated using the Singular +Value Decomposition (SVD) technique [30] implemented in +RooUnfold framework [31]. +3.6 Absolute normalization and systematic +uncertainties +The production cross section of Σ0 can be calculated by nor- +malizing the corrected Σ0 yield to the p+p elastic scattering +yield measured in the same experimental run [32]. This nor- +malization results in a systematic uncertainty of 7%. In addi- +tion, there might be other sources of systematic uncertainty. +The systematic error associated to the exclusive event selection +has been estimated by varying the selection ranges and recal- +culating the cross section. +To test the influence of different selection cuts on the calcu- +lated cross section (see section 5), the whole analysis chain has +been repeated many times under different cut combinations. +Each cut is varied in two steps in either direction. the cross +section for each combination is then calculated by integrating +the yield of the cosθ∗ +Σ0 angular distribution. Following this pro- +cedure the obtained systematic error, defined as the 1σ interval +of the cross sections distribution, is found to be ≈ 2%. +Another source of the systematic errors is the PID, which +is evaluated by activating the dropout layers of the neural net- +work during the inference time as this is equivalent to doing +a Bayesian approximation [33]. The estimated size of the PID +systematic is ≈ 5%. +Table 1: Coefficients of Legendre polynomials determined by +fitting the angular distributions presented in Figure 8. +Angle +A0 [µb] +A1 [µb] +A2 [µb] +cosθ∗ +Σ0 +8.55 ± 0.31 +0.00 +2.75 ± 0.73 +cosθ∗ +p +10.01 ± 0.50 +0.00 +4.33 ± 1.27 +cosθ∗ +K+ +9.83 ± 0.43 +0.00 +-0.13 ± 1.02 +cosθFRpΣ0 +pb,t,p +10.40 ± 0.80 +-0.64 ± 1.73 +2.79 ± 1.85 +cosθFRK+Σ0 +pb,t,K+ +8.55 ± 0.71 +-1.61 ± 1.54 +0.66 ± 1.63 +cosθFRK+p +pb,t,K+ +10.30 ± 1.00 +1.91 ± 1.18 +0.50 ± 2.69 +cosθFRK+Σ0 +p,Σ0 +8.70 ± 0.30 +3.17 ± 0.59 +-0.73 ± 0.75 +cosθFRpΣ0 +p,K+ +8.75 ± 0.29 +-3.52 ± 0.50 +0.37 ± 0.67 +cosθFRK+p +K+,Σ0 +8.81 ± 0.31 +4.84 ± 0.56 +-0.98 ± 0.75 +4 Angular Distributions +This section presents the differential cross section of the reac- +tion pp → pK+Σ0, namely the angular distributions of final +state particles in the center-of-mass (CMS) frame, as well as in +both the Gottfried-Jackson and helicity frames of all two-body +subsystems. All distributions are acceptance and efficiency cor- +rected and then fit with Legendre polynomials dσ/dcosθ = +� +l Al · Pl, with l = 0, 1, 2. The coefficients A1 and A2 are +used to judge the asymmetries and anisotropies of the observed +distributions. The best description of the distribution (indicated +by the blue histogram in Figure 8) was found when the sim- +ulations have been weighted simultaneously with the angular +distribution of the Σ0 hyperon in the CMS frame and the pro- +ton Gottfried-Jackson angular distribution measured in the pΣ0 +rest frame obtained from the data. +Center of mass frame +The angular distributions of the three final state particles in the +CMS are shown in the top row of Figure 8. The Legendre poly- +nomial coefficients obtained from the fits of the angular distri- +butions are listed in Table 1. Since the initial p+p is a symmetric +system, the A1 Legendre parameters of all CMS distributions +were set to zero. The angular distribution of the Σ0 hyperon +(Figure 8 (a)) and proton (Figure 8 (b)) shows an anisotropy, +where it is more pronounced for the proton as quantified by the +A2 parameter listed in Table 1. From the observed anisotropies +and the fit parameters one deduces that a non-zero orbital an- +gular momentum (L) is observed in both the p − K+Σ0 and +Σ0 − pK+ sub-systems. This is in contrast to the kaons, where +the angular distribution is compatible with isotropy. For pure +pion exchange, the final state proton is the leading particle, +since the exchange pion has a small mass, implying a small +4-momentum transfer so that the proton is preferably emitted +in the direction of the initial protons, which could explain the +anisotropy in the proton angular distribution. In this picture, the +Σ0 CMS angular distribution reflects the proton one, while the +kaon has a broader distribution. +The angular distributions in the overall CMS are not suited +to directly draw conclusions on resonant production, which +proceeds as a two step process pp → pR, R → K+ Σ0, where R + +Mean +1.114 +600 +Sigma +0.005 +HADES +S/B +1.56 +p(3.5 GeV)+p -→ pK+Z0 +NA +2340 +500 +2 +ieV/ +5400 +G +.001 +0300 +Events +2200 +100 +1.09 +1.1 +1.11 +1.12 +1.13 +1.14 +1.15 +1.08 +M. +[GeV/c'] +元 +decay8 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +Figure 6: (a) The p-value distributions for the HADES data-set and for (b) the WALL data-set. The insets display the region +of small p-values, where the dashed line and the arrow indicates the accepted region. The pp → pK+Λ, pp → pK+Σ0 and +pp → pK+Λπ0 simulations are shown by the red, blue and green histograms, respectively. +stands for every kind of nucleon resonance, that can be either +an isospin 1/2 N∗ state or an isospin 3/2 ∆∗ state. Therefore +in the following the Gottfreid-Jackson and helicity frames are +presented as a more natural choice for the Lorentzian reference +frames in order to study the reaction properties due to resonant +production. +Gottfried-Jackson frames +The Gottfried-Jackson (G-J) frame first introduced in [34] is +the rest frame of two out of the three produced particles. In the +G-J frame, the G-J angle is defined as the angle between one +of the rest frame particles (e.g. the Σ0) and the initial proton +θRF K+Σ0 +pb,t,Σ0 +, where the label RF stands for reference frame, the +superscript indicates which rest frame is used and the subscript +stands for the two particles, between which the angle is mea- +sured. It should be noted that the two initial protons are indis- +tinguishable. Therefore, the angular distribution is calculated +by using the angle to both protons (pb,t). +In the case of kaon (pion) exchange, the K+p (K+Σ0) rest +frame is equivalent to the rest frame of the exchanged meson +and the initial proton. In this way, the initial 2 → 3 reaction is +reduced to a pure 2 → 2 reaction. If there is a resonant produc- +tion, the internal angular momentum of the resonance is then +reflected in this observable. It has to be noted that the distri- +butions in the G-J frames do not have to be symmetric. The +reason is the asymmetric reaction system, where either a kaon +or a pion collides with a proton. The angular distributions in +the G-J frames are shown in the middle row of Figure 8. +An anisotropy is observed in the pΣ0 G-J frame (Figure 8 +(d)), which could be due to a relative angular momentum in +the pΣ0 system. This effect is related to the above mentioned +anisotropies of the p and Σ0 CMS angular distributions since +they are kinematically related. The angular distribution in the in +K+Σ0 G-J frame (Figure 8 (e)) tends to be asymmetric, which +could be caused by the excitation of nucleon resonances decay- +ing into the K+Σ0 channel [2]. Many of N∗ or ∆∗ resonances +could contribute to the reaction. All these resonances have large +widths and may also contribute through their broad tails to the +reaction. The angular distribution of a true two-body resonance +reaction is asymmetric only if resonances with both parities are +simultaneously excited through interfering amplitudes. Hence, +this distribution in the K+Σ0 G-J frame indicates that more +than one nucleon resonance with opposite parity participates in +the production process [2]. As explained earlier, the K+p rest +frame is equivalent to the rest frame of the exchanged kaon. +Therefore, the deviation from isotropy in the cosθFRK+p +pb,t,K+ an- +gular distribution could be explained by kaon exchange com- +ponent [25]. For a pure pion exchange, the Treiman-Yang (T- +Y) angle measured in the K+Σ0 rest frame is expected to be +an isotropic distribution [35]. Therefore, if a kaon exchange +contributes to the production mechanism it should reflect it- + +(a) +(b) +108 +103 + Data +107 +Spectrometer +107 +Wall +HADES +TTTT +data-set +pp → pK+z0 +data-set +103 +pp →pK+△ +106 +106 +102 +Pp →pK+A元° +105 +Events +10 +104 +10 +10 +E103 +103 +0 +0.005 +0.01 +0.015 +0.02 +0 +0.005 +0.01 +0.015 +0.02 +102 +102 +10 +10 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +P(x2) +P(×2)R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +9 +Figure 7: The pK+ missing mass distribution. The colored +histograms represent the simulated channels, where Y∗ refers +to an excited hyperon (Σ(1385), Λ(1405) or Λ(1520)). The +two peaks are due to the exclusive reactions pp → pK+Λ and +pp → pK+Σ0 as shown by the red and the blue histograms, +respectively. The vertical dashed lines mark the mass window +used to select candidate events of the pp → pK+Σ0 final state. +self in this distribution. The Σ0 hyperon T-Y angle measured in +the K+Σ0 rest frame, shown in Figure 9, shows a clear devia- +tion from isotropy, which could be an indication of a significant +kaon exchange contribution to the reaction mechanism. +Helicity frames +The helicity angle is defined in a similar way as the G-J angle, +but instead of calculating the angle of the respective particle +to the initial proton, the helicity angle is calculated between +one of the rest frame particles and the third produced parti- +cle. The helicity angular distribution thus interrelates the kine- +matics of the three final state particles and it is thus a linear +transformation projection of the Dalitz plot. A uniformly popu- +lated Dalitz plot results in isotropic helicity angle distributions. +On the other hand, if dynamical effects distort the Dalitz plot, +then the helicity angular distribution will be anisotropic. The +helicity angular distributions are shown in the bottom row of +Figure 8. All the distributions are significantly non-isotropic, +which indicates that the reaction is dominated by intermediate +resonances. Therefore, an inclusion of intermediate resonances +is necessary in order to quantitatively describe experimental an- +gular distributions. +Comparison to lower energy +A comparison of the normalized Legendre coefficients between +this measurement and data collected at a lower value of excess +energy ϵ = 162 MeV [2] is listed Table 2. The two sets of coeffi- +cients show striking differences for few coefficients indicating +that the Σ0 production mechanism changes between these val- +ues of excess energy. The CMS distributions are more forward- +backward peaked for the proton and the Σ0 hyperon and less +peaked for the kaon, pointing to a larger relative contribution +of pion with respect to kaon exchange at larger energies. In ad- +dition, the helicity angle distributions have a significant asym- +metry at the highest energy, in contrast with the lower energy +results. +5 Total Cross Section +The total production cross section as function of the excess +energy ϵ is used as a tool to compare the experimental +data to the different theoretical approaches. The result on +the pp → pK+Σ0 production cross section, obtained by +integrating the cosθ∗ +Σ0 angular distribution, is +σ(pK+Σ0)[µb] = 17.7 ± 1.7(stat) ± 1.6(syst) . +The cross section value is included in Figure 10, which +shows a compilation of the pp → pK+Σ0 cross sections as +a function of the excess energy. The present data point corre- +sponds to ϵ = 556 MeV, which is depicted by the green square +and existed in a region where no other measurements have been +performed. This behaviour can not be described by phase space +within experimental uncertainty as clearly seen by the solid +curve σpK+Σ0 = Kϵ2, where the quadratic excess-energy de- +pendence is attributed to a pure (i.e. trivial) three-body phase +space and K is the fit free parameter. +An alternative parametrization proposed by F¨aldt and +Wilkin in [43] that takes the proton-hyperon FSI interaction +into account +σ = C · +ϵ2 +(1 + +� +1 + ϵ/α)2 , +where the parameters C = 7.82 × 102µb GeV −2 and α = +4.57 × 10−2GeV are related to the FSI strength. Interestingly, +the deviations to the pure phase space behavior start showing +up at ϵ > 200 MeV. The displayed data in that region could also +be approximated by σ ≈ 10 µb. +A more appropriate paramerization proposed by Tsushima +in [44] shown by the dotted line is based on a resonance model, +where the hyperon is produced via an intermediate nucleon res- +onance N∗ or ∆∗. This paramerization describes all data points +near threshold up to 1.4 GeV fairly well. +Using +the +pp → pK+Λ +cross +section +measured +by +the HADES collaboration [22], the cross section ratio +σ(pK+Λ)/σ(pK+Σ0) is determined to be 1.90 ± 0.41. +Based on the coupled channel calculation, where the interfer- +ence of the pion and kaon exchange is taken in account, the +cross section ratio can be reproduced by selecting the relative + +X103 +Data +1.2 +pK+0 +HADES +pK+^ + p(3.5 GeV)+p → pK+0 +pK+E元0 +pK++元 +pK+(Y* → A) +0.8 +pK+(Y* → A元°) +pK+(Y* → +元) +50.6 +Simulation Sum +Events / +0.4 +0.2 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +MM (pK*) [GeV/c?]10 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +Figure 8: The corrected angular distributions in the CMS (top row), Gottfried-Jackson (middle row) and helicity frames (bottom +row). The experimental data are shown by the black points, where the error bars are the square root of the quadratic sum of +the statistical and systematic uncertainties. The blue histogram represent the weighted pp → pK+Σ0 phase space simulation +described in the text and the dotted pink histogram indicates the best partial wave analysis solution (discussed in Section 6). +sign for these two mechanism [17]. Figure 11 shows the cross +section ratio as a function of the excess energy together with +a compilation of other measurements [42]. The solid curve +is the ratio of the paramerization of both channels, where +the paramerization proposed by F¨aldt and Wilkin [43] based +on phase space and FSI is used for pp → pK+Λ and the +Tsushima paramerization [44] based on a resonance model is +used for the pp → pK+Σ0 channel. +The observed cross section ratio in the present p+p data is +similar to the corresponding value measured in p+Nb data [7], +despite the large difference in the individual cross sections, thus +corroborating the importance of FSI for these reactions. +6 Partial Wave Analysis +From the results presented above, it was concluded that +the experimental data on angular distributions can not be +described by pure phase space production, but there must be a +resonant component as anticipated in [2]. Therefore, a Partial +Wave Analysis (PWA) using the Bonn-Gatchina Partial Wave +Analysis (Bo-Ga PWA) framework [45] has been applied +with the goal to quantify the relative contributions of different +partial waves. +The Bo-Ga PWA framework takes a list of possible transi- +tion waves as an input that may contribute to the final state. The +non-resonant production proceeds as follows: the proton (JP= + +20 +C +a +HADES +15 +p(3.5 GeV)+p -→ pK+≥0 +10F +5 +-0.5 +0.5 +1 -1 +-0.5 +0.5 +1 -1 +-0.5 +0 +0 +0.5 +0 +cos Q,. +cos Ok+ +20 +(d) +(f) +(e) +15 +[ub] +0 +5 +-0.5 +0.5 +0.5 +1 -1 +0.5 +0 +-1 +-0.5 +0.5 +RFpLC +COS ORF K*20 +cos 0' +COS +Pb.t. K* +20 +(g) +(h) +(i) +15 +10 +5 +-0.5 +-0.5 +0.5 +1-1 +-0.5 +0.5 +1-1 +0.5 +1 +0 +RF Ktp +COS ORF K*20 +p, kt +p, oR. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +11 +Table 2: Comparison of the normalized Legendre coefficients between the present measurement and the data collected by COSY- +TOF experiment at ϵ = 162 MeV [2]. +ϵ = 162 MeV +ϵ = 556 MeV +A1/A0 +A2/A0 +A1/A0 +A2/A0 +cosθCMS +Σ0 +0.0 ± 0.0 +0.03 ± 0.24 +0.0 ± 0.0 +0.32 ± 0.09 +cosθCMS +p +0.0 ± 0.0 +0.25 ± 0.29 +0.0 ± 0.0 +0.43 ± 0.13 +cosθCMS +K+ +0.0 ± 0.0 +0.48 ± 0.22 +0.0 ± 0.0 +-0.01 ± 0.1 +cosθFRpΣ0 +pb,t,p +0.0 ± 0.0 +0.11 ± 0.15 +-0.06 ± 0.17 +0.27 ± 0.18 +cosθFRK+Σ0 +pb,t,K+ +-0.04 ± 0.04 +0.14 ± 0.18 +-0.19 ± 0.18 +0.08 ± 0.19 +cosθFRK+p +pb,t,K+ +-0.07 ± 0.07 +0.57 ± 0.18 +0.19 ± 0.12 +0.05 ± 0.26 +cosθFRK+Σ0 +p,Σ0 +0.27 ± 0.27 +-0.15 ± 0.15 +0.36 ± 0.07 +-0.08 ± 0.09 +cosθFRpΣ0 +p,K+ +-0.22 ± 0.22 +0.0 ± 0.15 +-0.4 ± 0.06 +0.04 ± 0.08 +cosθFRK+p +K+,Σ0 +-0.11 ± 0.11 +0.11 ± 0.18 +0.55 ± 0.07 +-0.11 ± 0.09 +Figure 9: The Σ0 Treiman-Yang angular distribution measured +in the K+Σ0 reference frame. The blue histogram represents +the weighted pp → pK+Σ0 phase space simulation and the +dotted histogram indicates the best partial wave analysis so- +lution (discussed in Section 6). +1 +2 ++) and the hyperon (in this case Σ0 with JP= 1 +2 ++) are com- +bined into a two particle sub-system and then the kaon (JP= +0−) is combined with this sub-system to produce the three- +body final state. In case of the resonant production, the proton +is combined with one of the resonances listed in Table 3 N∗-p, +or ∆∗-p to produce the final state pp → pK+Σ0. Resonance +masses and widths were fixed to the PDG values [46] in order +to reduce the number of the free fit parameters. +The strength (α1) and the phase (α2) of each transition +wave are determined by fitting the partial wave amplitudes +to the experimental data on an event-by-event basis in an +Figure 10: Compilation of cross sections of the reaction +pp → pK+Σ0 from different experiments: COSY-11 [36, 37, +38, 39, 40, 41], COSY-TOF [2] and data points from Landolt- +B¨ornstein (LB) [42]. The production cross section of Σ0 deter- +mined here is shown by the green square. The solid curve rep- +resents a pure phase space fit, the dotted curve is a parametriza- +tion based on the resonance model and the dashed curve is +phase space and FSI as described in the text. +unbinned fit. The fit is based on a log-likelihood minimization +and the fitting procedure is repeated for many iterations until +there is no further improvement of the log-likelihood value. +By comparing the log-likelihood value of many fits the best fit +can be determined through the largest negative value. As an +output, the BG-PWA returns the fitted values of the parameters +α1 and α2 and a list of simulated events that have been used +as an input but with each event being assigned a weight factor, +which gives the contribution of this event to the total yield. +Since the signal region contains background events (mainly +pp → pK+Λ and pp → pK+Λπ0), and because the Bo-Ga + +103 +HaDEs +p(3.5 GeV)+p → pK+z0 +102 +10 +[ub] +COSY-11 +COSY-TOF + LB +HADES +10- + Phase Space +- Phase Space + FSi + Resonance Model +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +E[GeV].25 +do/Φ [degree] +0.2 +0.15 +0. +0.05 +50 +100 +150 +RF K+0 +[degree]HAPESp(3.5 GeV)+p -→ pK+Z0do/db [ub/degree12 +R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +Figure 11: Experimental cross section ratio of the present data +point together with a compilation of the world data: COSY- +11 [36, 37, 38, 39, 40, 41], COSY-TOF [2] and data points +from Landolt-B¨ornstein (LB) [42]. The present data square is +shown by the green square. The solid curve is the ratio of the +paramerization of both channels [43, 44]. +Table 3: A list of N∗ and ∆∗ resonances that might contribute +to the pp → pK+Σ0 reaction. The mass, width and spin-parity +quantum numbers were taken from [46]. +Resonance +Mass +[GeV/c2 ] +Width +[GeV/c2 ] +JP +N∗(1710) +1.710 +0.140 +1 +2 ++ +N∗(1875) +1.875 +0.200 +3 +2 +− +N∗(1880) +1.880 +0.300 +1 +2 ++ +N∗(1895) +1.895 +0.120 +1 +2 +− +N∗(1900) +1.920 +0.200 +3 +2 ++ +∆∗(1900) +1.860 +0.250 +1 +2 +− +∆∗(1910) +1.900 +0.300 +1 +2 ++ +∆∗(1920) +1.920 +0.300 +3 +2 ++ +PWA method works on an event-by-event basis, it is important +to identify whether a particular event belongs to the signal or +the background. The pp → pK+Λ contribution is three times +larger than pp → pK+Λπ0 inside the signal region. Therefore, +the pp → pK+Λ channel is considered the main contributing +background and its kinematics is modeled by performing a +PWA on the pp → pK+Λ-like events. The solutions published +in [22] have been tested and solution No. 8/1 was found +to provide the best description of the experimental data by +including the p+p initial waves 2S+1LJ = 1S0, 3P0, 3P1 and +1D2. +The solution No. 8/1 is then applied to the Λ 4π-phase +space simulations and these events are filtered through the +full simulation and analysis chain. After reconstructing the Λ +events that have been assigned a PWA weight, the missing mass +MM(pK+) spectrum was investigated and the Λ contribution +in the signal region 1.170 < MM(pK+)[GeV/c2] < 1.220 +was determined to be 292 events. Those events are then added +to the signal list with a negative weight. +After subtracting the Λ contribution, the PWA technique +is applied to the pp → pK+Σ0 events. A systematic variation +of the input partial waves was performed and, in addition, the +number of non-resonant and resonant final partial waves was +varied and the quality of the PWA solution was determined by +the negative log-likelihood value of the fit. +The best PWA solution shown by the dashed histograms in +Figures 8 and 9 was obtained by including p+p initial waves +2S+1LJ = 2S0, 3P0, 3P1, 3P2, 1D2 and 3F2. In addition, +nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) were +found to contribute as well as non-resonant partial waves. +However, due to the limited statistics and the large number +of free fit parameters, an unambiguous determination of +the contributions of each resonance is not possible since +these contributions vary significantly for different solutions. +Nevertheless, resonances with masses around 1.710 GeV/c2 +(N∗(1710)) and 1.900 GeV/c2 (N∗(1900) or ∆∗(1900)) are +certainly preferred by the fit. +7 Conclusion and Outlook +The exclusive reconstruction of the reaction pp → pK+Σ0 at +a beam kinetic energy of 3.5 GeV has been presented and the +pp → pK+Σ0 total production cross section was determined +with an accuracy better than 10 % in a region where no data +existed. The dynamics of the reaction was investigated by +studying the angular distributions in the CMS, G-J and helicity +frame. The corrected CMS distributions of the hyperon and +the proton show anisotropies, which it is more pronounced +in the case of the proton. This is the expected behavior if +the pion exchange mechanism dominates the particle pro- +duction process in a simple one-boson exchange formalism. +In addition, an investigation of the Σ0 T-Y angle measured +in the K+Σ0 reference frame, deviates from isotropy, which +hints to a non-negligible contribution of the of kaon exchange +mechanism. +The helicity angular distributions are not isotropic, +which indicates that a pure phase space description with- +out momentum-dependent matrix element(s) is by far not +appropriate. The influence of different nucleon resonances +has been tested by means of a PWA using the Bo-Ga PWA +framework. The best solution was obtained by including the +initial p+p configuration 1S0, 3P0, 3P1, 3P2, 1D2 and 3F2. +Due to the limited statistics, it was not possible to obtain the +exact strength of the individual nucleon resonances. However, +nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) are +preferred by the fit. +Recently, the HADES setup has been upgraded by an elec- +tromagnetic calorimeter (ECAL) and a Forward Detector (FD) +based on PANDA experiment straw tubes [47]. The new data +that was collected in February 2022 offers the opportunity to +perform the same measurement with an upgraded setup at a +higher proton beam energy of 4.5 GeV. This upgrade will allow +the identification of the daughter photon in Σ0 → Λγ via the +ECAL. In addition, it will improve the mass resolution of the + +102 +HADES +(pK+A)/ (pK+0 +10 +0 +COSY-11 +COSY-TOF ++ LB +HADES +10-1 +E[GeV]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions +13 +Λ hyperon in the FD acceptance and consequently improve the +quality of the kinematic refit. Furthermore, the collected data +will provide sufficient statistics to extract quantitative contri- +butions of the different nucleon resonances and a measurement +of their K+Σ0 branching ratios, which will certainly improve +the current measurement. +8 Acknowledgment +The HADES collaboration gratefully acknowledges the support by +SIP JUC Cracow, Cracow (Poland), 2017/26/M/ST2/00600; WUT +Warsaw (Poland) No: 2020/38/E/ST2/00019 (NCN), IDUB-POB- +FWEiTE-3; TU Darmstadt, Darmstadt (Germany), VH-NG-823, +DFG GRK 2128, DFG CRC-TR 211, BMBF:05P18RDFC1, HFHF, +ELEMENTS 500/10.006, GSI F&E, EMMI at GSI Darmstadt; +Goethe-University, +Frankfurt +(Germany), +BMBF:05P12RFGHJ, +GSI F&E, HIC for FAIR (LOEWE), EMMI at GSI Darmstadt; +JLU Giessen, Giessen (Germany),BMBF:05P12RGGHM; IJCLab +Orsay, Orsay (France), CNRS/IN2P3; NPI CAS, Rez, Rez (Czech +Republic), MSMT LTT17003, MSMT LM2018112, MSMT OP VVV +CZ.02.1.01/0.0/0.0/18 046/0016066; +European +Union’s +Horizon +2020, no. 824093 (STRONG2020). +This project has received funding from the programme ”Netzwerke +2021”, an initiative of the Ministry of Culture and Science of the State +of Northrhine Westphalia. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Zherebtsova5,e, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Zielinski4, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Zumbruch5 (HADES collaboration) 1 LIP-Laborat´orio de Instrumentac¸˜ao e F´ısica Experimental de Part´ıculas 3004-516 Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Portugal 2 AGH University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Faculty of Physics and Applied Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 30-059 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland 3 Institute of Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 31342 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland 4 Smoluchowski Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Jagiellonian University of Cracow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 30-059 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland 5 GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 64291 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 6 Technische Universit¨at Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 64289 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 7 Institut f¨ur Strahlenphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Helmholtz-Zentrum Dresden-Rossendorf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 01314 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 8 Institut f¨ur Kernphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Goethe-Universit¨at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 60438 Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 9 Excellence Cluster ’Origin and Structure of the Universe’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 10 Physik Department E62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany 11 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='Physikalisches Institut, Justus Liebig Universit¨at Giessen, 35392 Giessen, Germany 12 Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus 13 Laboratoire de Physique des 2 infinis Ir`ene Joliot-Curie, Universit´e Paris-Saclay, CNRS-IN2P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=', F-91405 Orsay, France 14 Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech Republic 15 LabCAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' F´ısica, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' de Santiago de Compostela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 15706 Santiago de Compostela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Spain 16 Uniwersytet Warszawski - Instytut Fizyki Do´swiadczalnej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 02-093 Warszawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland 17 Warsaw University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 00-662 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland a also at Coimbra Polytechnic - ISEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Portugal b also at Helmholtz Research Academy Hesse for FAIR (HFHF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Campus Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 64390 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany c also at Technische Universit¨at Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 01062 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Germany d also at Charles University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 12116 Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Czech Republic e also at University of Wrocław,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 50-204 Wrocław,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Poland e-mail: hades-info@gsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='de Abstract The production of Σ0 hyperons in proton proton collisions at a beam kinetic energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV impinging on a liquid hydrogen target was investigated using data collected with the HADES setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The total production cross section is found to be σ(pK+Σ0)[µb] = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='7(stat) ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6(syst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Differential cross section distributions of the exclusive channel pp → pK+Σ0 were analyzed in the center-of-mass, Gottfried-Jackson and helicity reference frames for the first time at the excess energy of 556 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The data support the interplay between pion and kaon exchange mechanisms and clearly demonstrate the contribution of interfering nucleon resonances decaying to K+Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Bonn- Gatchina partial wave analysis was employed to analyse the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Due to the limited statistics, it was not possible to obtain an unambiguous determination of the relative contribution of intermediate nucleon resonances to the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' However nucleon resonances with masses around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='710 GeV/c2 (N∗(1710)) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='900 GeV/c2 (N∗(1900) or ∆∗(1900)) are preferred by the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11766v1 [nucl-ex] 27 Jan 2023 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 1 Introduction Strangeness production at intermediate energies in p+p and p+A collisions is of particular importance to the field of hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The production of baryons with strange quark content, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' hyperons, requires creating a new quark flavor, which can occur out of the vacuum from the quark sea in the colliding protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The s-quark with mass O(ΛQCD) is distinguished from the light (u, d) quark flavors but much smaller than heavy (c, t, b) flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The resulting (approximate) SU(3) flavor symmetry in the u-d-s sector is therefore still a cornerstone of hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since the entrance channel in p+p and p+A collisions carries no net strangeness, the emergence of an s-quark can unravel much of the flavor dynamics in hadronic reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The flavor-conserving strong interaction process requires associate strangeness production, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' realized by simultaneous creation of a single-strange hyperon, such as Λ or Σ0 and an associated kaon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, understanding the production mechanism of strange baryons near threshold deepens our knowledge of their internal structure and of the strong interaction in the non-perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Strangeness production is also used as a probe to study hot and dense nuclear matter in heavy ion collisions both at medium-energy and in the late stages prior to freeze-out in high-energy collisions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' at LHC [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The production of the Λ hyperon in p+p and p+A reac- tions near threshold has been studied extensively by many ex- periments including HADES [2, 3, 4, 5, 6], yet there are only few experimental investigations on the Σ0 hyperon [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' De- spite there are considerable experimental results and numerous dedicated theoretical investigations, the strangeness production mechanism is not yet well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the context of the bo- son exchange model [8, 9, 10, 11], it is assumed that the initial protons exchange a virtual me- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The interaction between the meson and the initial protons results in the production of the final state particles, which can proceed directly or via an intermediate resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The exchange of a virtual meson can be put into one of two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The first category is strange meson exchange, where strangeness exchange occurs, and no resonances are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this case, the reaction amplitude KN → KN is governed by t-channel diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The second category is non-strange meson exchange, a pion exchange in its simplest form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' At the same time the elementary reaction amplitude πN → KY is dominated by resonance excitations, which implies a strong and characteristic energy dependence, where Y stands for hyperons (Λ, Σ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Several experiments have studied the exclusive reaction pp → pK+Λ and proven that a pure phase space model de- scription of the data is not sufficient without taking the dynam- ics of the process into account [2, 6, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' It was found that the Λ hyperon production is dominated by the excitation and subsequent decay of N∗ resonances to the K+Λ decay chan- nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In particular N∗(1650) (JP= 1 2 −), N∗(1710) (JP= 1 2 +) and N∗(1720) (JP= 3 2 +) were found to contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This sup- ports a picture wherein the exchange of non-strange mesons is the leading process in the production mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In addi- tion, a considerable Final State Interaction (FSI) was found to contribute [14, 15] leading to ΣN → ΛN conversion that is observed as a ΣN cusp effect in the Λ cross section [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the pp → pK+Σ0 reaction the proton–hyperon FSI seems to be negligible, especially at low energies near threshold and a pure phase space distribution describes the data reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The cross section ratio σ(pK+Λ) / σ(pK+Σ0) below ex- cess energies of ∼ 20 MeV is about 28 in agreement of the SU(6) prediction and reduces drastically to about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 at excess energies above 300 MeV [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This energy-dependence of the cross section ratio is strongly affected by FSI effects in the pp → pK+Λ reaction [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Besides the energy dependence of the cross section, the differential cross sections at selected energies add much more stringent tests of the model descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This study fills this gap and delivers such data which allow some clues about the involved exchange mesons and resonances, in particular by em- ploying a partial wave analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Furthermore, a theoretical study of the reaction pp → pK+Σ0 based on a chiral dynamical study has been proposed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This approach uses the pion and kaon exchange mechanisms and chiral amplitudes in addition to all pairs FSI, where the contribution of nucleon resonances appear naturally using chiral unitary amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In Section 2, the experi- mental setup is briefly explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Section 3 is devoted to the Σ0 selection method, where the particle identification, the Λ hy- peron reconstruction and the kinematic refit methods were pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 the method for efficiency correction and differential analysis is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Sections 5 and 6 presents the calculated total production cross section and the partial wave analysis of the exclusive reaction pp → pK+Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In Section 7 a summary and a short outlook are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 2 The HADES experiment The data presented here were collected in April 2007 with the High Acceptance Di-Electron Spectrometer (HADES) located at the heavy ion synchrotron SIS18 at GSI Helmholtzzentrum f¨ur Schwerionenforschung in Darmstadt, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' HADES is characterized by six identical sectors covering almost the full azimuthal range and polar angles from θ = 18◦ to θ = 85◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Each sector of the spectrometer contains a Ring-Imaging Cherenkov Detector (RICH) operating in a magnetic field-free region that allows lepton identification over a wide range of momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Two Multi-Wire Drift Chambers (MDCs) are placed in front of a toroidal magnetic field, and two outer MDCs are placed behind the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The MDCs enable the momentum informa- tion and the specific energy loss dE/dx to be reconstructed for each particle track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Two scintillator hodoscopes, the Time Of Flight (TOF) and TOFino are also placed behind the magnet and provide a stop time (ts) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The TOF and TOFino sys- tem are used as input to the trigger systems to start the data readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A detailed description of the HADES setup can be found in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the present analysis, a proton beam with an intensity of 107 particles/s and kinetic energy T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV was incident on a liquid hydrogen target with an areal density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='35 g/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The dimensions of the target were 15 mm in diameter and 50 mm length located between -65 to -15 mm in the longitudi- nal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The data readout was started by a first level trig- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 3 ger requiring a charged particle multiplicity ≥ 3 (M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In total, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='14 × 109 events were recorded under these conditions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' During this experiment HADES included an additional For- ward Wall (FW) scintillator hodoscope that was placed 7 me- ters downstream the target in a magnetic field-free region and covered polar angles from θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='33◦ to θ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='17◦ with full az- imuthal acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The FW measured the hit position and ar- rival time of the particle track with a time resolution of about 700 ps [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3 Event selection method In this section, the exclusive reconstruction of the reaction pp → pK+Σ0 is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Σ0 hyperon is reconstructed via its electromagnetic decay Σ0 → Λγ (BR ≈ 100%) and the daughter Λ hyperon is reconstructed with the decay mode Λ → pπ− (BR = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Σ0 reconstruction strategy includes the following steps: a) time of flight (tof) reconstruction, b) charged particle iden- tification (PID), c) the Λ hyperon reconstruction, and d) the Σ0 hyperon reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 Time of flight reconstruction The interaction of the high intensity proton beam with the START detector induced a background and prevented a stable operation of the RICH detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, it was not possible to use the START detector information during this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Consequently, the tof of particle tracks were not directly mea- sured since there was no common start time (t0) reference for tracks in the same event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The start time has to be reconstructed in order to obtain a proper time of flight measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The reconstruction algorithm is based on the assumption that at least one particle has been correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since pi- ons are abundantly produced, it is assumed that any negatively charged particle track that is geometrically uncorrelated to a ring in the RICH detector is a π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The common start time for each event is calculated by t0 = ts − d c × � p2 + m2π p , where ts is the stop time of the π−, d is the distance to the TOF or TOFino hit, mπ is the pion mass, p is the momentum of the π− and c is the velocity of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The tof of the other particles in the same event is the difference between the measured stop time ts and the common start time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 Particle identification (PID) The reconstruction of the exclusive reaction pp → pK+pπ−γ only requires the identification of three particle species, pions (π−), kaons (K+) and protons (p), since the event is kinemati- cally complete even without measuring the photon (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' As mentioned in the previous section, the π− is identified as any negatively charged track that is geometrically uncorre- lated to a ring in the RICH detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, the problem reduces to identifying the positively charged tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In order to minimize systematic bias in the model output, an auto-encoder [23] implemented in PyTorch framework [24] is trained simultaneously with both simulated and real events [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The input features used to train the auto-encoder are the momentum components, the energy loss dE/dx in the MDC and TOF sub-systems, the reconstructed tof and the distance to the TOF/TOFino hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A classification layer has been stacked on top of the bottleneck layer of the auto-encoder, which has three output nodes corresponding to the three classes (π+, K+ and p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Each node outputs a number between 0 and 1, where all output numbers sum to 1, so that each number can be interpreted as a probability of being a specific particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The network is trained by minimizing a cost function that is defined as the binary cross-entropy loss [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Because the network outputs three probabilities for each particle track, the node with the largest probability is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The classification accuracy evaluated on a holdout data-set is 92% for pions, 76% for kaons and 98% for protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' It is much lower in the case of kaons since their production rate is suppressed with respect to the protons and pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='3 Λ hyperon reconstruction The next step after the PID is to reconstruct the intermediate Λ hyperon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this analysis the Λ reconstruction method is two- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the first case, which is referred to as the Spectrometer data-set, events with exactly 2 protons, 1 pion and 1 kaon are required to be within the main HADES detector acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The other case, referred as the WALL data-set, events were ac- cepted if exactly 1 proton, 1 pion and 1 kaon are registered in HADES and in addition one hit in the FW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the latter case, it is assumed that the hit registered in the FW is due to the daughter proton from the Λ decay (marked as pdecay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A common primary vertex in each event is then defined as the intersection point or the Point of Closest Approach (PCA) of the proton and kaon tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since there is more than one pro- ton in each event in the Spectrometer data-set, the proton-kaon pair with the smaller Distance of Closest Approach (DCA) is used to define the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' To reduce the contribution from off-target events, a two dimensional selection is applied on the primary vertex position (x, y, z): a) -65 < z [mm] < -15 and b) � x2 + y2 < 5 [mm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Spectrometer data-set Since the daughter Λ decays weakly (cτ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='89 cm), it can be identified by its displaced vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' First, all possible combina- tions of the two p and π− candidates were made, leaving the decision about which is the decay proton (Λ → p π−) for later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' For each combination the decay vertex (the displaced vertex) is defined as the PCA between the two tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The DCA between the p and π− tracks (marked as dpπ−) is expected to be small if the tracks stem from the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, an upper limit of dpπ− < 10 mm is imposed in order to reduce Combinatorial 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions Figure 1: (a) The DCA distribution between the p and π− tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (b) The DCA distribution between the Λ track and the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In both panels, data are shown by the black points, the blue histogram represents the true Λ and the red histogram represents the CB, where both the true Λ and the CB were estimated from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (c) Distribution of the DCA between the π− track and the primary vertex as a function of the DCA between the p track and the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The arrows indicate the accepted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Background (CB), which originates from combining the wrong p and π− pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Considering momentum and energy conserva- tion, the p should be emitted in nearly the same direction as the Λ in the laboratory reference frame, while the π− will have a different direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Thus, the DCA between the p track and the primary vertex (dp,pvtx) is required to be smaller than the DCA between the π− track and the primary vertex (dπ−,pvtx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Fi- nally, the DCA between the calculated Λ track and the primary vertex (dΛ,pvtx) is required to be < 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The distributions of the topological variables are shown in Figure 1, where the selection criteria are indicated by the vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The proton used in the Λ reconstruction is tagged as the decay pro- ton (marked in the following as pdecay), while the other proton in the event is tagged as the scattered (primary) proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' To further purify the selected Λ sample, the event kinemat- ics were constrained to the Σ0 production range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The squared pΛ missing mass (MM2(ppdecayπ−)) is required to be > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 GeV/c2 in order to reject the multi-pion production channel as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this figure, the experimental data are shown by the black points and the simulations (discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4) by different colored histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Two peaks are visible, the first peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02 GeV/c2 corresponds to the multi-pion channel via the reaction pp → ppπ+π− (violet histogram), where a pπ− pair is incorrectly identified as a Λ candidate and the π+ is incorrectly identified as a K+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The other broader peak is the sum of pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 reactions shown by the red, blue and green histograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The relative normalizations of the simulated channels have been chosen to best fit the experimental data as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pdecay π− invariant mass distribution is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A peak around the nominal Λ mass is visible on top of background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The signal has been parameterized by a Voigt distribution and the background is modeled by a fourth-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Events are further processed if they are in the range of µ ± 3σ, where the calculated signal to background ratio in this range is S/B = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='57 and the number of Λ candidates is NΛ = 6766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The WALL data-set In the WALL data-set the hit in the FW is assumed to be due to the decay proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since the FW is installed in a magnetic field-free region, the pdecay is reconstructed as a straight line trajectory from the primary vertex position to the hit in the FW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The track momentum is calculated from the tof and the dis- tance from the primary vertex and the FW detector hit, assum- ing the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this case, the topological cuts are not as effective to suppress the background as in the Spectrometer data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, events fulfilling the following kinematical constraints were selected: (i) MM2(ppdecayπ−) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 GeV/c2 (Figure 4 a) and (ii) The squared missing mass of all charged particles is required to be in the following range: −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02 < MM2(pK+pdecayπ−)[GeV2/c4] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 be- cause only a photon is missing to completely measure the exclusive final state (Figure 4 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (a) (b) (c) 108 108 15 ×103 L Data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 HADES TrueA 4 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p → pK+z0 106 106 CB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 ww 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 104 Events 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 102 102 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 0 50 100 0 20 40 0 5 10 15 [mm] d(p, pvtx) [mm]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 5 Figure 2: The squared ppdecay π− missing mass distribu- tion after applying the topological selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Black points are the Spectrometer data-set data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The violet histogram is the pp → ppπ+π− simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 simulations are shown by the red, blue and green histograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The vertical line and the arrow indicate the accepted region for the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pdecayπ− invariant mass distribution for the WALL data-set is shown Figure 5 after applying the selections mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Once again, the peak has been fitted by a Voigt distribution and the background by a fourth-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' However, the mass resolution of the Λ peak of the Spectrometer data-set(Figure 3) is better than the signal of the WALL data-set, since in the latter case the proton was detected in the FW, which has a worse momentum resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Events are further processed if they are in the range of µ ± 3σ, where the calculated signal to background ratio in this range is S/B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='56 and the number of Λ candidates is NΛ = 2340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 Σ0 hyperon reconstruction To further suppress the remaining background and to obtain a better mass resolution, a kinematic fit based on the Lagrange multiplier method is employed [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The fit χ2, expressed as χ2(η, λ) = (y − η)T V (y)(y − η) + 2λT f(η) , is minimized by differentiating χ2 with respect to all measured variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Here y is a vector containing the initial guesses for the measured quantities, which are the track parameters pro- vided by the tracking algorithm, η is an improved set of the track parameters and V is the covariance matrix comprising the estimated errors on the measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The constraint Figure 3: The pdecay π− invariant mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The verti- cal dashed lines indicate the selected mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The blue, red and green curves are for the signal, background and the total fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' equations are expressed as a function of η in f(η), where λi are a set of Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The spherical coordinates used in this analysis for the track parameterization are defined as follows y = � � 1/p θ φ � � , where 1/p is the inverse of the absolute momentum, θ and φ are the polar and azimthual angles of the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Two constraints were applied to both data-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The first is the proton and pion from the Λ decay are constrained to the Λ mass (MΛ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1157 GeV/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The second constraint is that the missing mass of all the charged final state particles is con- strained to the photon mass (Mγ = 0 GeV/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The probability that a χ2 of the theoretical distribution is greater than or equal to the χ2 value found from the fit is known as the p-value (P(χ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The p-value distributions of the Spec- trometer and the WALL data-sets are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Be- cause both Λ and Σ0 have MM(pK+Λ) = 0, they have simi- lar distributions, which makes these two reactions difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' On the other hand, the reaction pp → pK+Λπ0 should ideally have zero p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' However, due to the limited detector resolution it has p-values greater than zero, which is more pronounced in the WALL data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The signal events show an almost flat distribution between 0 and 1, while events that do not satisfy the constraint equations have a prominent yield of p-values close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, events with p-values > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 are selected, where the cut was optimized based on a significance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' X103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 Data HADES pK+0 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p -→ pK+z0 Pp元+元 pK+^ pK+^元° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 MM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (pp 元)[GeV2/c4] decayX103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='114 Sigma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='002 HADES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 S/B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='57 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p → pK+≥0 Na 6766 2 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='001 GeV/c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' [GeV/c"] decay6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions Figure 4: (a) The squared ppdecay π− missing mass distribution of WALL data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (b) The squared ppdecay π− K+ missing mass distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pp → ppπ+π−, pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 simulations are shown by the violet, red, blue and green histograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The arrows indicate the accepted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Simulation scaling to the experimental data By inspecting the pK+ missing mass distribution of the com- bined data-set shown in Figure 7, two peaks corresponding to the Λ and the Σ0, as well as other minor contributions in the high mass region, are plainly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In order to quantify the different contributions an incoherent cocktail has been simu- lated using the Pluto event generator [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' All the simulated re- actions have been processed using the same full scale analysis employed for the experimental data, thus taking into account the efficiency of the trigger condition, the tracking algorithm and the analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The particle decays, the acceptance and the particle interactions with the materials of HADES and the FW have been considered by using GEANT3 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' To determine the contributions of the different channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' a fit of the simulations to the measured missing mass spectrum (MM(pK+)) has been carried out by minimizing the quantity χ2 = nbins � i (ndata − � ch(f ch × nch simulation))2 σ2 data + σ2 simulation ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' where the summation runs over the number of bins of the miss- ing mass spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' ndata is the number of data events in each bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' nch simulation is the number of simulated events in each bin for each channel and fch is a scaling factor for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The uncertainty for the data and the simulations in each bin is σdata and σsimulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' As can be seen from Figure 7, the experimental data is primarily described by contributions of pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 indicated by the red, blue and the green histogram, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The other simulated channels have minor contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In total 2613 Σ0 candidates were collected within the pK+ missing mass range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='170- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='220 GeV/c2, 58% of them are within the main HADES acceptance and 42% within the FW acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The signal purity in the mass window calculated from the simulation is found to be 81%, where the main background contributions are the reactions pp → pK+Λ (14%) and pp → pK+Λπ0 (5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 Efficiency and acceptance correction The reconstructed experimental distributions are corrected for the limited detector acceptance and efficiency by using a sim- ulated phase space distribution that were assigned a weight de- termined by the best partial wave solution (discussed in Sec- tion 6), then the events were filtered through the full scale simu- lation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The efficiency correction is done in one di- mension whereas the other three degrees of freedom on which the efficiency depends are integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The 1D correction matrix (M) is calculated given the ini- tial 4π distribution (T) for each observable and after filtering (a) (b) X103 X103 Data 45 HADES pK+0 5 40 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p → pK+≥0 pp元+元 35 pK+A Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='002 GeV2/c4 pK+A元0 4 30 25 3 20 2 15 10F 5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 MM2 (pp 元) [GeV?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='/c4] MM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' (pK*p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=',) [GeV2/c*]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 7 Figure 5: The pwallπ− invariant mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The vertical dashed lines indicate the selected mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The blue, red and green curves are for the signal, background and the total fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' through the full scale simulation and analysis (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' To put it an- other way, a distinct correction matrix M = R/T is constructed for each angular distribution shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The inverse of the correction matrix is then calculated using the Singular Value Decomposition (SVD) technique [30] implemented in RooUnfold framework [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 Absolute normalization and systematic uncertainties The production cross section of Σ0 can be calculated by nor- malizing the corrected Σ0 yield to the p+p elastic scattering yield measured in the same experimental run [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This nor- malization results in a systematic uncertainty of 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In addi- tion, there might be other sources of systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The systematic error associated to the exclusive event selection has been estimated by varying the selection ranges and recal- culating the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' To test the influence of different selection cuts on the calcu- lated cross section (see section 5), the whole analysis chain has been repeated many times under different cut combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Each cut is varied in two steps in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' the cross section for each combination is then calculated by integrating the yield of the cosθ∗ Σ0 angular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Following this pro- cedure the obtained systematic error, defined as the 1σ interval of the cross sections distribution, is found to be ≈ 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Another source of the systematic errors is the PID, which is evaluated by activating the dropout layers of the neural net- work during the inference time as this is equivalent to doing a Bayesian approximation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The estimated size of the PID systematic is ≈ 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Table 1: Coefficients of Legendre polynomials determined by fitting the angular distributions presented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Angle A0 [µb] A1 [µb] A2 [µb] cosθ∗ Σ0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='73 cosθ∗ p 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='27 cosθ∗ K+ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='13 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02 cosθFRpΣ0 pb,t,p 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='64 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='85 cosθFRK+Σ0 pb,t,K+ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='61 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='63 cosθFRK+p pb,t,K+ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='50 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='69 cosθFRK+Σ0 p,Σ0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='75 cosθFRpΣ0 p,K+ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='67 cosθFRK+p K+,Σ0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='75 4 Angular Distributions This section presents the differential cross section of the reac- tion pp → pK+Σ0, namely the angular distributions of final state particles in the center-of-mass (CMS) frame, as well as in both the Gottfried-Jackson and helicity frames of all two-body subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' All distributions are acceptance and efficiency cor- rected and then fit with Legendre polynomials dσ/dcosθ = � l Al · Pl, with l = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The coefficients A1 and A2 are used to judge the asymmetries and anisotropies of the observed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The best description of the distribution (indicated by the blue histogram in Figure 8) was found when the sim- ulations have been weighted simultaneously with the angular distribution of the Σ0 hyperon in the CMS frame and the pro- ton Gottfried-Jackson angular distribution measured in the pΣ0 rest frame obtained from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Center of mass frame The angular distributions of the three final state particles in the CMS are shown in the top row of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Legendre poly- nomial coefficients obtained from the fits of the angular distri- butions are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since the initial p+p is a symmetric system, the A1 Legendre parameters of all CMS distributions were set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The angular distribution of the Σ0 hyperon (Figure 8 (a)) and proton (Figure 8 (b)) shows an anisotropy, where it is more pronounced for the proton as quantified by the A2 parameter listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' From the observed anisotropies and the fit parameters one deduces that a non-zero orbital an- gular momentum (L) is observed in both the p − K+Σ0 and Σ0 − pK+ sub-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This is in contrast to the kaons, where the angular distribution is compatible with isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' For pure pion exchange, the final state proton is the leading particle, since the exchange pion has a small mass, implying a small 4-momentum transfer so that the proton is preferably emitted in the direction of the initial protons, which could explain the anisotropy in the proton angular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this picture, the Σ0 CMS angular distribution reflects the proton one, while the kaon has a broader distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The angular distributions in the overall CMS are not suited to directly draw conclusions on resonant production, which proceeds as a two step process pp → pR, R → K+ Σ0, where R Mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='114 600 Sigma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='005 HADES S/B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='56 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p -→ pK+Z0 NA 2340 500 2 ieV/ 5400 G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='001 0300 Events 2200 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='08 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=" [GeV/c'] 元 decay8 R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions Figure 6: (a) The p-value distributions for the HADES data-set and for (b) the WALL data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The insets display the region of small p-values, where the dashed line and the arrow indicates the accepted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 simulations are shown by the red, blue and green histograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' stands for every kind of nucleon resonance, that can be either an isospin 1/2 N∗ state or an isospin 3/2 ∆∗ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore in the following the Gottfreid-Jackson and helicity frames are presented as a more natural choice for the Lorentzian reference frames in order to study the reaction properties due to resonant production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Gottfried-Jackson frames The Gottfried-Jackson (G-J) frame first introduced in [34] is the rest frame of two out of the three produced particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the G-J frame, the G-J angle is defined as the angle between one of the rest frame particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' the Σ0) and the initial proton θRF K+Σ0 pb,t,Σ0 , where the label RF stands for reference frame, the superscript indicates which rest frame is used and the subscript stands for the two particles, between which the angle is mea- sured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' It should be noted that the two initial protons are indis- tinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, the angular distribution is calculated by using the angle to both protons (pb,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In the case of kaon (pion) exchange, the K+p (K+Σ0) rest frame is equivalent to the rest frame of the exchanged meson and the initial proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In this way, the initial 2 → 3 reaction is reduced to a pure 2 → 2 reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' If there is a resonant produc- tion, the internal angular momentum of the resonance is then reflected in this observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' It has to be noted that the distri- butions in the G-J frames do not have to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The reason is the asymmetric reaction system, where either a kaon or a pion collides with a proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The angular distributions in the G-J frames are shown in the middle row of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' An anisotropy is observed in the pΣ0 G-J frame (Figure 8 (d)), which could be due to a relative angular momentum in the pΣ0 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This effect is related to the above mentioned anisotropies of the p and Σ0 CMS angular distributions since they are kinematically related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The angular distribution in the in K+Σ0 G-J frame (Figure 8 (e)) tends to be asymmetric, which could be caused by the excitation of nucleon resonances decay- ing into the K+Σ0 channel [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Many of N∗ or ∆∗ resonances could contribute to the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' All these resonances have large widths and may also contribute through their broad tails to the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The angular distribution of a true two-body resonance reaction is asymmetric only if resonances with both parities are simultaneously excited through interfering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Hence, this distribution in the K+Σ0 G-J frame indicates that more than one nucleon resonance with opposite parity participates in the production process [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' As explained earlier, the K+p rest frame is equivalent to the rest frame of the exchanged kaon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, the deviation from isotropy in the cosθFRK+p pb,t,K+ an- gular distribution could be explained by kaon exchange com- ponent [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' For a pure pion exchange, the Treiman-Yang (T- Y) angle measured in the K+Σ0 rest frame is expected to be an isotropic distribution [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, if a kaon exchange contributes to the production mechanism it should reflect it- (a) (b) 108 103 Data 107 Spectrometer 107 Wall HADES TTTT data-set pp → pK+z0 data-set 103 pp →pK+△ 106 106 102 Pp →pK+A元° 105 Events 10 104 10 10 E103 103 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02 102 102 10 10 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 P(x2) P(×2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 9 Figure 7: The pK+ missing mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The colored histograms represent the simulated channels, where Y∗ refers to an excited hyperon (Σ(1385), Λ(1405) or Λ(1520)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The two peaks are due to the exclusive reactions pp → pK+Λ and pp → pK+Σ0 as shown by the red and the blue histograms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The vertical dashed lines mark the mass window used to select candidate events of the pp → pK+Σ0 final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' self in this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Σ0 hyperon T-Y angle measured in the K+Σ0 rest frame, shown in Figure 9, shows a clear devia- tion from isotropy, which could be an indication of a significant kaon exchange contribution to the reaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Helicity frames The helicity angle is defined in a similar way as the G-J angle, but instead of calculating the angle of the respective particle to the initial proton, the helicity angle is calculated between one of the rest frame particles and the third produced parti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The helicity angular distribution thus interrelates the kine- matics of the three final state particles and it is thus a linear transformation projection of the Dalitz plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A uniformly popu- lated Dalitz plot results in isotropic helicity angle distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' On the other hand, if dynamical effects distort the Dalitz plot, then the helicity angular distribution will be anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The helicity angular distributions are shown in the bottom row of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' All the distributions are significantly non-isotropic, which indicates that the reaction is dominated by intermediate resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, an inclusion of intermediate resonances is necessary in order to quantitatively describe experimental an- gular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Comparison to lower energy A comparison of the normalized Legendre coefficients between this measurement and data collected at a lower value of excess energy ϵ = 162 MeV [2] is listed Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The two sets of coeffi- cients show striking differences for few coefficients indicating that the Σ0 production mechanism changes between these val- ues of excess energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The CMS distributions are more forward- backward peaked for the proton and the Σ0 hyperon and less peaked for the kaon, pointing to a larger relative contribution of pion with respect to kaon exchange at larger energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In ad- dition, the helicity angle distributions have a significant asym- metry at the highest energy, in contrast with the lower energy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 5 Total Cross Section The total production cross section as function of the excess energy ϵ is used as a tool to compare the experimental data to the different theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The result on the pp → pK+Σ0 production cross section, obtained by integrating the cosθ∗ Σ0 angular distribution, is σ(pK+Σ0)[µb] = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='7(stat) ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6(syst) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The cross section value is included in Figure 10, which shows a compilation of the pp → pK+Σ0 cross sections as a function of the excess energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The present data point corre- sponds to ϵ = 556 MeV, which is depicted by the green square and existed in a region where no other measurements have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This behaviour can not be described by phase space within experimental uncertainty as clearly seen by the solid curve σpK+Σ0 = Kϵ2, where the quadratic excess-energy de- pendence is attributed to a pure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' trivial) three-body phase space and K is the fit free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' An alternative parametrization proposed by F¨aldt and Wilkin in [43] that takes the proton-hyperon FSI interaction into account σ = C · ϵ2 (1 + � 1 + ϵ/α)2 , where the parameters C = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='82 × 102µb GeV −2 and α = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='57 × 10−2GeV are related to the FSI strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Interestingly, the deviations to the pure phase space behavior start showing up at ϵ > 200 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The displayed data in that region could also be approximated by σ ≈ 10 µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A more appropriate paramerization proposed by Tsushima in [44] shown by the dotted line is based on a resonance model, where the hyperon is produced via an intermediate nucleon res- onance N∗ or ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This paramerization describes all data points near threshold up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 GeV fairly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Using the pp → pK+Λ cross section measured by the HADES collaboration [22], the cross section ratio σ(pK+Λ)/σ(pK+Σ0) is determined to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Based on the coupled channel calculation, where the interfer- ence of the pion and kaon exchange is taken in account, the cross section ratio can be reproduced by selecting the relative X103 Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 pK+0 HADES pK+^ p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p → pK+0 pK+E元0 pK++元 pK+(Y* → A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 pK+(Y* → A元°) pK+(Y* → +元) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 Simulation Sum Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 MM (pK*) [GeV/c?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' ]10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions Figure 8: The corrected angular distributions in the CMS (top row), Gottfried-Jackson (middle row) and helicity frames (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The experimental data are shown by the black points, where the error bars are the square root of the quadratic sum of the statistical and systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The blue histogram represent the weighted pp → pK+Σ0 phase space simulation described in the text and the dotted pink histogram indicates the best partial wave analysis solution (discussed in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' sign for these two mechanism [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Figure 11 shows the cross section ratio as a function of the excess energy together with a compilation of other measurements [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The solid curve is the ratio of the paramerization of both channels, where the paramerization proposed by F¨aldt and Wilkin [43] based on phase space and FSI is used for pp → pK+Λ and the Tsushima paramerization [44] based on a resonance model is used for the pp → pK+Σ0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The observed cross section ratio in the present p+p data is similar to the corresponding value measured in p+Nb data [7], despite the large difference in the individual cross sections, thus corroborating the importance of FSI for these reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 6 Partial Wave Analysis From the results presented above, it was concluded that the experimental data on angular distributions can not be described by pure phase space production, but there must be a resonant component as anticipated in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, a Partial Wave Analysis (PWA) using the Bonn-Gatchina Partial Wave Analysis (Bo-Ga PWA) framework [45] has been applied with the goal to quantify the relative contributions of different partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The Bo-Ga PWA framework takes a list of possible transi- tion waves as an input that may contribute to the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The non-resonant production proceeds as follows: the proton (JP= 20 C a HADES 15 p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p -→ pK+≥0 10F 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1 -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1 -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 cos Q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' cos Ok+ 20 (d) (f) (e) 15 [ub] 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1 -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content="5 RFpLC COS ORF K*20 cos 0' COS Pb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' K* 20 (g) (h) (i) 15 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 1 0 RF Ktp COS ORF K*20 p, kt p, oR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 11 Table 2: Comparison of the normalized Legendre coefficients between the present measurement and the data collected by COSY- TOF experiment at ϵ = 162 MeV [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' ϵ = 162 MeV ϵ = 556 MeV A1/A0 A2/A0 A1/A0 A2/A0 cosθCMS Σ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='09 cosθCMS p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='13 cosθCMS K+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1 cosθFRpΣ0 pb,t,p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 cosθFRK+Σ0 pb,t,K+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='19 cosθFRK+p pb,t,K+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='26 cosθFRK+Σ0 p,Σ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='09 cosθFRpΣ0 p,K+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='08 cosθFRK+p K+,Σ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='09 Figure 9: The Σ0 Treiman-Yang angular distribution measured in the K+Σ0 reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The blue histogram represents the weighted pp → pK+Σ0 phase space simulation and the dotted histogram indicates the best partial wave analysis so- lution (discussed in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 1 2 +) and the hyperon (in this case Σ0 with JP= 1 2 +) are com- bined into a two particle sub-system and then the kaon (JP= 0−) is combined with this sub-system to produce the three- body final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In case of the resonant production, the proton is combined with one of the resonances listed in Table 3 N∗-p, or ∆∗-p to produce the final state pp → pK+Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Resonance masses and widths were fixed to the PDG values [46] in order to reduce the number of the free fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The strength (α1) and the phase (α2) of each transition wave are determined by fitting the partial wave amplitudes to the experimental data on an event-by-event basis in an Figure 10: Compilation of cross sections of the reaction pp → pK+Σ0 from different experiments: COSY-11 [36, 37, 38, 39, 40, 41], COSY-TOF [2] and data points from Landolt- B¨ornstein (LB) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The production cross section of Σ0 deter- mined here is shown by the green square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The solid curve rep- resents a pure phase space fit, the dotted curve is a parametriza- tion based on the resonance model and the dashed curve is phase space and FSI as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' unbinned fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The fit is based on a log-likelihood minimization and the fitting procedure is repeated for many iterations until there is no further improvement of the log-likelihood value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' By comparing the log-likelihood value of many fits the best fit can be determined through the largest negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' As an output, the BG-PWA returns the fitted values of the parameters α1 and α2 and a list of simulated events that have been used as an input but with each event being assigned a weight factor, which gives the contribution of this event to the total yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Since the signal region contains background events (mainly pp → pK+Λ and pp → pK+Λπ0), and because the Bo-Ga 103 HaDEs p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p → pK+z0 102 10 [ub] COSY-11 COSY-TOF LB HADES 10- Phase Space Phase Space + FSi Resonance Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='4 E[GeV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='25 do/Φ [degree] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='05 50 100 150 RF K+0 [degree]HAPESp(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p -→ pK+Z0do/db [ub/degree12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions Figure 11: Experimental cross section ratio of the present data point together with a compilation of the world data: COSY- 11 [36, 37, 38, 39, 40, 41], COSY-TOF [2] and data points from Landolt-B¨ornstein (LB) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The present data square is shown by the green square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The solid curve is the ratio of the paramerization of both channels [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Table 3: A list of N∗ and ∆∗ resonances that might contribute to the pp → pK+Σ0 reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The mass, width and spin-parity quantum numbers were taken from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Resonance Mass [GeV/c2 ] Width [GeV/c2 ] JP N∗(1710) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='140 1 2 + N∗(1875) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='200 3 2 − N∗(1880) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='300 1 2 + N∗(1895) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='120 1 2 − N∗(1900) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='200 3 2 + ∆∗(1900) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='250 1 2 − ∆∗(1910) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='300 1 2 + ∆∗(1920) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='300 3 2 + PWA method works on an event-by-event basis, it is important to identify whether a particular event belongs to the signal or the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The pp → pK+Λ contribution is three times larger than pp → pK+Λπ0 inside the signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Therefore, the pp → pK+Λ channel is considered the main contributing background and its kinematics is modeled by performing a PWA on the pp → pK+Λ-like events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The solutions published in [22] have been tested and solution No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 8/1 was found to provide the best description of the experimental data by including the p+p initial waves 2S+1LJ = 1S0, 3P0, 3P1 and 1D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The solution No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 8/1 is then applied to the Λ 4π-phase space simulations and these events are filtered through the full simulation and analysis chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' After reconstructing the Λ events that have been assigned a PWA weight, the missing mass MM(pK+) spectrum was investigated and the Λ contribution in the signal region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='170 < MM(pK+)[GeV/c2] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='220 was determined to be 292 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Those events are then added to the signal list with a negative weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' After subtracting the Λ contribution, the PWA technique is applied to the pp → pK+Σ0 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' A systematic variation of the input partial waves was performed and, in addition, the number of non-resonant and resonant final partial waves was varied and the quality of the PWA solution was determined by the negative log-likelihood value of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The best PWA solution shown by the dashed histograms in Figures 8 and 9 was obtained by including p+p initial waves 2S+1LJ = 2S0, 3P0, 3P1, 3P2, 1D2 and 3F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In addition, nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) were found to contribute as well as non-resonant partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' However, due to the limited statistics and the large number of free fit parameters, an unambiguous determination of the contributions of each resonance is not possible since these contributions vary significantly for different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Nevertheless, resonances with masses around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='710 GeV/c2 (N∗(1710)) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='900 GeV/c2 (N∗(1900) or ∆∗(1900)) are certainly preferred by the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 7 Conclusion and Outlook The exclusive reconstruction of the reaction pp → pK+Σ0 at a beam kinetic energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV has been presented and the pp → pK+Σ0 total production cross section was determined with an accuracy better than 10 % in a region where no data existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The dynamics of the reaction was investigated by studying the angular distributions in the CMS, G-J and helicity frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The corrected CMS distributions of the hyperon and the proton show anisotropies, which it is more pronounced in the case of the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This is the expected behavior if the pion exchange mechanism dominates the particle pro- duction process in a simple one-boson exchange formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In addition, an investigation of the Σ0 T-Y angle measured in the K+Σ0 reference frame, deviates from isotropy, which hints to a non-negligible contribution of the of kaon exchange mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The helicity angular distributions are not isotropic, which indicates that a pure phase space description with- out momentum-dependent matrix element(s) is by far not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The influence of different nucleon resonances has been tested by means of a PWA using the Bo-Ga PWA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The best solution was obtained by including the initial p+p configuration 1S0, 3P0, 3P1, 3P2, 1D2 and 3F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Due to the limited statistics, it was not possible to obtain the exact strength of the individual nucleon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' However, nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) are preferred by the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Recently, the HADES setup has been upgraded by an elec- tromagnetic calorimeter (ECAL) and a Forward Detector (FD) based on PANDA experiment straw tubes [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The new data that was collected in February 2022 offers the opportunity to perform the same measurement with an upgraded setup at a higher proton beam energy of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This upgrade will allow the identification of the daughter photon in Σ0 → Λγ via the ECAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' In addition, it will improve the mass resolution of the 102 HADES (pK+A)/ (pK+0 10 0 COSY-11 COSY-TOF + LB HADES 10-1 E[GeV]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Abou Yassine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' : Investigation of the Σ0 Production Mechanism in p(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='5 GeV)+p Collisions 13 Λ hyperon in the FD acceptance and consequently improve the quality of the kinematic refit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Furthermore, the collected data will provide sufficient statistics to extract quantitative contri- butions of the different nucleon resonances and a measurement of their K+Σ0 branching ratios, which will certainly improve the current measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 8 Acknowledgment The HADES collaboration gratefully acknowledges the support by SIP JUC Cracow, Cracow (Poland), 2017/26/M/ST2/00600;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' WUT Warsaw (Poland) No: 2020/38/E/ST2/00019 (NCN), IDUB-POB- FWEiTE-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' TU Darmstadt, Darmstadt (Germany), VH-NG-823, DFG GRK 2128, DFG CRC-TR 211, BMBF:05P18RDFC1, HFHF, ELEMENTS 500/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='006, GSI F&E, EMMI at GSI Darmstadt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Goethe-University, Frankfurt (Germany), BMBF:05P12RFGHJ, GSI F&E, HIC for FAIR (LOEWE), EMMI at GSI Darmstadt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' JLU Giessen, Giessen (Germany),BMBF:05P12RGGHM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' IJCLab Orsay, Orsay (France), CNRS/IN2P3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' NPI CAS, Rez, Rez (Czech Republic), MSMT LTT17003, MSMT LM2018112, MSMT OP VVV CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='0/18 046/0016066;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' European Union’s Horizon 2020, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' 824093 (STRONG2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' This project has received funding from the programme ”Netzwerke 2021”, an initiative of the Ministry of Culture and Science of the State of Northrhine Westphalia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The sole responsibility for the content of this publication lies with the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' The following colleagues from Russian institutes did contribute to the results presented in this publication but are not listed as authors following the decision of the HADES Collaboration Board on March 23, 2022: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Agakishiev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Belyaev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Fateev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Ierusalimov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Ladygin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Vasiliev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Golubeva, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Guber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content=' Ivashkin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} +page_content='1140/epja/s10050-021-00388- w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FKT4oBgHgl3EQfQS0h/content/2301.11766v1.pdf'} diff --git a/29FST4oBgHgl3EQfYTgs/content/tmp_files/2301.13787v1.pdf.txt b/29FST4oBgHgl3EQfYTgs/content/tmp_files/2301.13787v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1047578e434d4baf3f80bdf214c7d6ff6f62972a --- /dev/null +++ b/29FST4oBgHgl3EQfYTgs/content/tmp_files/2301.13787v1.pdf.txt @@ -0,0 +1,1290 @@ +Designing Covalent Organic Framework-based Light-driven Microswimmers +towards Intraocular Theranostic Applications + +Varun Sridhar1,+, Erdost Yildiz1,+, Andrés Rodríguez-Camargo,2,3, Xianglong Lyu1, Liang Yao2, Paul +Wrede1, Amirreza Aghakhani1, Mukrime Birgul Akolpoglu1, Filip Podjaski2,4,5,*, Bettina V. +Lotsch2,3,5,6,*, Metin Sitti1,7,8,* + +1 Physical Intelligence Department, Max Planck Institute for Intelligent Systems, 70569 Stuttgart, +Germany +2 Nanochemistry Department, Max Planck Institute for Solid State Research, 70569 Stuttgart, +Germany +3 Department of Chemistry, University of Stuttgart, 70569 Stuttgart, Germany +4 Department of Chemistry, Imperial College London, W12 0BZ London, United Kingdom +5 Cluster of Excellence E-conversion, Lichtenbergstrasse 4, 85748 Garching, Germany +6 Department of Chemistry, University of Munich (LMU), Munich, Germany +7 Institute for Biomedical Engineering, ETH Zurich, 8092 Zurich, Switzerland +8 School of Medicine and College of Engineering, Koç University, 34450 Istanbul, Turkey + ++ These authors contributed equally to this article. +* Correspondence to: sitti@is.mpg.de, f.podjaski@imperial.ac.uk, b.lotsch@fkf.mpg.de + + + +Abstract +Even micromachines with tailored functionalities enable targeted therapeutic applications in +biological environments, their controlled motion in biological media and drug delivery functions +usually require sophisticated designs and complex propulsion apparatuses for practical +applications. Covalent organic frameworks (COFs), new chemically versatile and nanoporous +materials, offer microscale multi-purpose solutions, which are not explored in light-driven +micromachines. We describe and compare two different types of COFs, uniformly spherical TABP- +PDA-COF sub-micron particles and texturally highly nanoporous, irregular, micron-sized TpAzo- +COF particles as light-driven microrobots. They can be used as highly efficient visible-light-driven +drug carriers in aqueous ionic and cellular media, even in intraocular fluids. Their absorption +ranging down to red light enables phototaxis even in deeper biological media and the organic +nature of COFs enables their biocompatibility. The inherently porous structure with ~2.5 nm +structural pores, and large surface areas allow for targeted and efficient drug loading even for +insoluble drugs and peptides, which can be released on demand. Also, indocyanine green (ICG) +dye loading in the pores enables photoacoustic imaging or optical coherence tomography and +hyperthermia in operando conditions. The real-time visualization of the drug-loaded COF +microswimmers enables new insights into the function of porous organic micromachines, which +will be useful to solve various drug delivery problems. + +Keywords: Covalent organic framework, light-driven, microswimmer, targeted drug delivery, +optical coherence tomography + + +Introduction +Microrobots are tiny machines that are tailored to be controlled externally to perform individual +tasks. In order to achieve external control as well as multi-purpose functionality, micro/nanobots +typically require a sophisticated and specifically adapted design enabling targeted control and +applications. Their primary area of use is the large field of biomedicine.1, 2, 3, 4, 5 The microrobots +should not elicit any immune response and should be compatible with the cells to enable +biomedical applications.6, 7 Also, the propulsion method should be as noninvasive as possible, +and non-toxic, excluding the use of dedicated toxic fuels.8, 9 For an efficient microrobot function, +motion control is the first requirement, which can become more and more challenging if the +liquid they are propelled in contains species that hinder propulsion or external control. Wireless +motion control requires external energy input and is typically realized by magnetic or acoustic +actuation,10 but can also be realized by ultraviolet (UV) or blue light, even in biological conditions, +as evidenced very recently.11 However, UV light, which is typically used for light-driven +microswimmers12, 13 is incompatible with biological tissues. Also, visible light control, usually +reported with high-intensity blue light,12 limits applications to transparent conditions since tissue +penetration requires more red light or near-infrared light, which is a significant challenge to the +field. +The critical tasks of mobile microrobots are cargo uptake and delivery, often linked to +biopharmaceutical classes and properties of drugs, after actively navigating to a target diseased +tissue region.4, 7, 14, 15 Drug uptake and its controlled release were typically realized efficiently by +encapsulation structures being a separate part of the microrobots; these were then opened in +the desired conditions or where a release could be triggered otherwise. More recently, inherently +porous structures, such as metal-organic frameworks16, 17, 18 and porous carbon nitrides were +used for such applications since their sizeable inner pore volume, reminiscent of a sponge, +enables high and, even environmentally stable drug loading.11 However, porous particle +structures with many textural pores of different sizes as part of their inner surface area leave +challenges for controlled loading and release from their volume. +As a last critical step to clinical applications, cell viability, the absence of foreign body reactions, +and tissue biocompatibility are necessary conditions for microrobots to be used in biological + +contexts, which is not always easy to ensure, with all the desired functions being fulfilled at a +time.4, 7, 14 For this purpose, typically biocompatible metal coatings, such as gold, titanium, or +polymers are employed, but also organic-based materials are up-and-coming and were used +recently without coatings, such as carbon nitrides.11, 17, 19 Compared to inorganic structures, +organic materials not only offer potential biodegradability but also the high flexibility of chemical +design of organic materials, especially in terms of surface functionalities and porosity, might +enable more efficient and targeted biomedical applications, such as drug delivery or +hyperthermia.20 Even the most sophisticated designs with biocompatible metallic structures fail +during actuation inside heterogenic biological fluids and specific-targeted drug delivery and +imaging in live tissues.21, 22, 23 Because of that, new materials and actuation methods should be +investigated for the basic tasks for the clinical obstacles, such as intraocular motion and drug +delivery. +In this work, we introduce covalent organic frameworks (COFs) as a tailorable active component +to the field of micro- and nanomachines, or more precisely, light-driven microswimmers. These +highly porous and crystalline materials can fulfill all the requirements listed above since their +molecular structure, and also their morphology, can be designed and tuned bottom-up while +enabling targeted properties.20, 24, 25 Simultaneously, they can use visible light for photocatalytic +reactions with their environment, which can also be used for active particle propulsion.19, 26, 27 +Depending on the propulsion mechanism and particle structure, the propulsion can be self- +diffusiophoretic or self-electrophoretic, while allowing light-induced directional motion control +via phototaxis.28 The tailorable properties of the COF building blocks enable tuning of not only +their absorption wavelength but also pore size and volume, surface polarity, and chemical +affinity, which enable the loading of large and small molecule cargo based on the specific +application requirements. Since such organic structures are non-magnetic per se, unless so-called +Janus or hybrid (encapsulation) structures were to be employed, the use of light is a highly +promising and convenient method not only to propel them but also to trigger functions within +them and to image their behavior.29 Thanks to their promising and ion-tolerant visible light- +driven propulsion properties, we especially focused on their use in ophthalmological +applications. In addition to their light-driven propulsion, their configurable particle sizes enable + +them to pass through the fibrillar mesh of vitreous humor (~500 nm pore size).30 For this purpose, +we selected therapeutic agents and imaging modalities accordingly. +Here, these possibilities are investigated and exemplified. We study and compare two very +different modified COFs, namely TAPB-PDA-COF made from the condensation of 1,3,5-tris(4- +aminophenyl)benzene +and +terephthaldehyde,31 +and +TpAzo-COF +made +from +1,3,5- +triformylphloroglucinol and 4,4′-azodianiline,32 as light-driven microswimmer examples, in order +to explore the microrobotic possibilities for this class of materials and to establish versatile +applications. We describe their light-controlled propulsion in biological media, their +biocompatibility, as well as their uptake and release of drugs that can be physisorbed to the pores +of the material.25, 33 Since these two COFs have distinct structures and morphologies, we derive +design guidelines for their propulsion and cargo-related functions. For theranostic delivery +functions of microrobots, we used doxorubicin, insulin, and indocyanine green (ICG), which +covers the breath of small molecules and peptides used as therapeutic and imaging agents. We +further image the motion of the COFs in real-time and potential in-vivo conditions using optical +coherence tomography as well as photoacoustic imaging of COF particle swarms loaded with a +near-infrared active dye (ICG). The water-soluble or insoluble drugs and the contrast agents can +be loaded to track their release, allowing for first insights into the action of porous drug carriers +in real-time clinical imaging modalities. In this way, we designed and investigated in detail the +first photoactive intraocular drug carriers for various theranostic applications. + +Results and Discussion +COF synthesis and characterization +The COF structures to be used as microswimmers were selected based on their structural and +optical properties and synthesized akin to a procedure reported earlier.31, 32 In brief, the TAPB- +PDA-COF nanospheres were obtained by using TAPB (1,3,5-tris(4-aminophenyl)benzene) and +PDA (terephthaldehyde) as building blocks, with acetonitrile and Sc(Otf)3 as solvent and catalyst, +respectively (see Materials & Methods section for more details) to yield a two-dimensional (2D), +imide linked organic network (Fig. 1a). These 2D sheets are stacked on each together forming an + +ordered 3D structure with hexagonal pore channels of a diameter of approx. 3.4 nm, as reported +earlier and confirmed for the here modified synthesis by nitrogen sorption, powder XRD, and FT- +IR analysis (Fig. 1b and Fig. S1).34 The obtained COF submicron particles (henceforth called +nanoparticles for better discrimination) have an almost perfectly spherical shape (Fig. 1c), which +is beneficial for propulsion in fluids.35 The synthesis yields a very homogeneous product with a +Brunauer, Emmett, and Teller (BET) surface area as high as 685 m²/g (Fig. S1c) and a narrow size +distribution of approx. 452 ± 74 nm (Fig. S2a,b). TEM analysis reveals that these nanoparticles +consist of agglomerated individual crystallites with sizes between 50 and 100 nm (Fig. S1c). +The TpAzo-COF presented here for comparison is synthesized by solvothermal condensation +between 1,3,5-triformylphloroglucinol (Tp) and 4,4´-azodianiline (Azo), forming a tautomeric +ketoenamine COF (Fig. 1d). A highly crystalline product with a 2D molecular structure is obtained +(see Fig. S3 for structural analysis), with slightly smaller structural pores of 2.6 nm and a similar +BET surface area of 635 m²/g (Fig. 1e). In contrast to the first COF, the particle morphology is +much less defined, leaving open large textural voids reminiscent of a sponge (Fig. 1f and Fig. S4). +The overall particle sizes are broadly distributed (7 ± 18 µm), hence being much larger. The +primary crystallites forming the COF particle are only 20 nm (Fig. S4d), and the TpAzo-COF +particles appear to be agglomerates of those. +Light-induced swimming in aqueous media +To enable light-triggered propulsion, we first investigate the light absorption properties. UV-Vis +spectroscopy and Kubelka-Munck analysis show that TAPB-PDA-COF and TpAzo-COF have an +optical band gap of 464 nm (Fig. 2a) and 616 nm (Fig. 2f), respectively with a small absorption +tail commonly arising from defect states. Hence, visible light propulsion can be extended up to a +wavelength of ~470 nm (blue light) for the small TABP-PDA particles, and to green or even red +parts of the spectrum for the large TpAzo-COF. Their light-induced propulsion was studied under +a microscope in a microfluidic chamber under ambient conditions to test the phototaxis +capabilities while diluting them to 100 µg/ml. First, we focus on propulsion in distilled water. +While in the dark, COF particles show only local Brownian motion with a mean displacement +speed of 4.5 µm/s for the TAPB-PDA and 3.7 µm/s for the TpAzo-COF, respectively (Fig. 2b,g, +dashed line). When light from the photodiode is focused on the microswimmers through the + +microscope, their propulsion speed is significantly enhanced and becomes ballistic, as seen in +Video S1. The particles move towards the center of the light, then upwards. This way, the light- +driven collective assembly or trapping of the microswimmers is made possible. UV excitation at +385 nm propels the TABP-PDA-COF with 13.2 ± 2.4 µm/s, while 470 nm blue light gives an even +increased speed of 16.4 ± 3.1 µm/s (~36 bodylengths/s (BLPS)). At 510 nm illumination, no light- +enhanced swimming was observed, consistent with the absorption spectrum (Fig. 2b). The Tp- +Azo-COF is propelled with 4.9 ± 1.2, 12.1 ± 2.1 (~2 BLPS), 8.2 ± 1.7, and 4.2 ± 1.2 µm/s at 385, +470, 560 nm, and 630 nm, respectively (Fig. 2g). UV and red light hence do not increase +propulsion significantly below Brownian motion and the absolute propulsion speed is lower, but +it can be triggered even by yellow light (560 nm). +Ion tolerance for light-driven microswimmers and phototaxis behavior +Ionic conditions represent a major challenge for light-driven microswimmers.36, 37, 38 The +presence of pores, both structural and textural (=morphological), was suggested previously to +enable the propulsion of microswimmers in ionic environments, which includes most of the +biological fluids and cell culture media.11 To confirm this and to widen the insights from different +and better controlled structural features present in our model COFs, we first tested them in +increasing concentrations of salt (NaCl), see Fig. 2c,h. +When propelled at 470 nm, the TABP-PDA-COF does not decrease the speed compared to +distilled water up to concentrations as large as 1000 mM. Therefore, the ionic concentration in +the media at which the microswimmers’ speed is halved (EI50) cannot be attributed.39 Also, we +observe a slightly increasing propulsion speed between 0.5 and 10 mM, with a maximum value +of 25.2 ± 3.7 µm/s (54% increase vs. distilled water, 0 mM) at 1 mM NaCl (Fig. 2c). An explanation +for this non-linear behavior remains to be found. Ionic interactions can be influencing the +Helmholtz and Debye layers, as well as the materials' inner space charge layer. As such, light- +induced charge carrier stability or recombination will also be affected. On the other hand, the +chlorine evolution reaction due to dissolved NaCl may another photocatalytic pathway possibly +increasing the reaction rate, and thereby the propulsion speed.11, 40, 41, 42 These factors currently +cannot be studied or disentangled on such size and complex reaction interface. However, their + +propulsion speed even surpasses our previously reported PHI microswimmers, the only reported +system with comparable ionic tolerance.11 +Similarly, for the TpAzo-COFs, an increased propulsion speed compared to pure water is observed +in all ionic conditions (1-1000 mM) at 560 nm illumination, peaking at 1 mM (14.7 ± 2.7 µm/s, +79% increase vs. distilled water) and followed by a 28% relative decay to 10.5 µm/s at 1000 mM. +When increasing the wavelength, no active propulsion is observed for TABP-PDA-COF (Fig. 2b), +but the Tp-AZO-COF exhibits slightly enhanced propulsion even at 630 nm (4.2 µm/s) (Fig. 2g). +Next, three standard biological media are studied, namely, Dulbecco’s phosphate-buffered saline +(dPBS), minimum essential medium (MEM), and MEM plus fetal bovine serum (FBS) (Fig. 2d,i), +which slightly differ in their components: dPBS contains NaCl, KCl, Na2HPO4, and KH2PO4 at ca. 10 +g/L (~150 mM) in total; MEM contains the same components as dPBS and additional two amino +acids, vitamins, and glucose, some of which can be redox-active agents that help extract not only +electrons but especially holes from the microswimmers under illumination to power them.11, 43 +FBS, slightly more viscous, adds nutrients for cell growth and imitates the conditions found within +the body.11 At 470 nm illumination, the mean speeds of the TAPB-PDA-COF microswimmers in +dPBS, MEM, and MEM + FBS are 20.4 ± 4.5 µm/s, 20.9 ± 2.8µm/s and 17.6 ± 3.3 µm/s, +respectively. These speeds are again higher than in distilled water (16.4 ± 3.1 µs/s). The slight +decrease upon FBS addition can be attributed to the increasing viscosity or other surface +interactions with the proteins present in the FBS. +Very similar behavior is observed with the TpAzo-COF at 560 nm, where the swimming speeds +are equivalent to the maximum value in 1mM NaCl, or even slightly higher (13.8 ± 3.4 µm/s, 16.2± +3.4 µm/s, 14.7 ± 3.6 µm/s, and 11.8 ± 2.2 µm/s in dPBS, MEM (with and without glucose), and +MEM + FBS). A difference however is observed when glucose, a well-oxidizable fuel,11, 43 is absent +– the speed is reduced. Its vital role as fuel for propulsion is clearly visible when illuminating +TpAzo at 630 nm in MEM that contains glucose, where efficient propulsion, independent of FBS, +is observed (10 ± 2.7 µm/s and 7.4 ± 1.8 µm/s respectively). This purely red light-induced +photocatalytic motion in the presence of high ion concentrations and without using potent and +toxic fuels is unprecedented.29, 44 However, the still efficient propulsion at 560 nm without + +glucose in MEM confirms that the other ingredients (including dissolved oxygen11, 19) may also +assist motion induced by photocatalysis, or at least do not hamper it. These experiments not only +show the superiority in performance over current inorganic microswimmers in high-salinity +media but also highlight how crucial facile redox species are that can act as fuel for propulsion, +akin to photocatalysis in general, and especially if sub-band gap trap states might be partially +involved (630 nm illumination).43, 45, 46 Such a substantial shift toward the red part of the spectrum +that can penetrate deeper tissues makes organic and small band gap microswimmers (especially +with trap states in the gap) attractive for micromachines not just in-vitro, but even for in-vivo +conditions. +Light-driven directional propulsion control +Phototaxis is the property by which microswimmers swim towards or away from the direction of +incident light (i.e., positive or negative phototaxis), which often depends on their surface +charge.47, 48 It enables direction control, opposite to random ballistic displacement usually +observed with Janus particles.11, 49 When the COF microswimmers were illuminated by a directed +light source from the side with a 45° angle, both TABP-PDA-COF and TpAzo-COF microswimmers +exhibit positive phototaxis, and swim toward the light that can propel them (Fig. 2 e, j, and video +S2). TABP-PDA-COF and TpAzo-COF particles move with mean speeds of 13.3 ± 1.8 µm/s and 7.6 +± 0.8 µm/s, at 470 nm and 630 nm illumination in water and MEM, respectively. This apparent +increase in the particle speed compared to vertical illumination could be attributed to the larger +parallel component of the light direction to the propulsion direction when the samples were +illuminated from the side. When the samples are illuminated from the bottom, only the side- +wise motion component is measured as a common standard, artificially decreasing the actual +velocity.50, 51 Similar findings have been found on carbon nitride microswimmers, which were +discussed in more detail in our previous study.11 The required symmetry breaking is created by +the side-wise illumination and, thereby, an artificially created Janus structure results from the +self-shadowing of the microswimmers.13, 47 +Biocompatibility of COFs +In order to be used in potential biomedical applications and to ascertain biocompatibility, +microswimmers should have no significant cytotoxicity. Hence, we tested the cytotoxicity of the + +microswimmers with human umbilical vein endothelial cells (HUVEC) in dMEM with FBS. +Different concentrations of TAPB-PDA-COF and TpAzo-COF microswimmers (3.1-25 µg/ml) were +incubated with HUVECs in the dark, and their viability was investigated with calcein-based +live/dead fluorescence staining of the cells after 24 hours. The cells with TABP-PDA COF were +completely viable, and they did not show any significant decrease in viability even at high +concentrations, both with illumination and without illumination at 470 nm with maximum light +intensity, 10 mW/cm2, for 30 minutes), as seen in Fig. 3 a, which is visible also in live cell +fluorescent images in Fig. 3 b. TpAzo-COF (Fig. 3 c,d) shows lower cell viability in comparison with +TAPB-PDA-COF, with 93% and 75% HUVEC cell viability in 25 µg/ml concentration (in dark and +with 630 nm illumination (10 mW/cm2), respectively). Also, at concentrations of 3.1 µg/ml, the +viability is decreased to 88% in comparison to the TABP-PDA COF. However, this fairly good +viability indicates that also the TpAZo COF can be used at lower concentrations for drug delivery +applications. Generally, illumination seems not to affect the viability at low concentrations (3.1 +and 6.25 µg/ml), and only slightly at 12.5 and 25 µg/ml for both COFs. These results also suggest +that light-induced propulsion induces only minimal cytotoxicity in the range of light-driven +propulsion periods. Compared to carbon nitride microswimmers, which have a larger band gap +(2.5 eV, 450 nm) and a very low-lying valance band, and therefore enable more redox reactions +with organic matter, including cells in principle, the use of 470 nm or 630 nm light with our TABP- +PDA COFs and TpAzo COFs shows potential for reduced cell death [with 97% and 88% cell viability +after 30 minutes of light in 3.1 µg/ml concentrations of TAPB-PDA-COFs and TpAzo-COFs, +respectively] and makes especially the TABP-PDA COFs more applicable to practical applications +such as drug delivery.11 A previous study with primary cells from mouse splenocytes further +confirmed no detectable level of IL-12 (a pro-inflammatory cytokine) in the untreated samples in +concentrations used above in the dark.52 +Drug loading, drug delivery, and hyperthermia +To explore the COF microswimmer’s applicability to biological environments, we also studied +their potential as drug carriers with different pharmacological agents. The differently +pronounced textural and structural porosity of the TABP-PDA and TpAzo-COFs (see Fig. 1 and Fig. +S1-S4), which enables ionic tolerance (Fig. 2c,h), is not only beneficial for motion but also as space + +to take up, transport and deliver therapeutic drugs. We studied and compared how the structural +features enable interactions with such cargo in the following experiments. For this reason, we +chose an imaging agent, indocyanine green (ICG), and two different pharmacological agents with +different Biopharmaceutics Classification System (BCS) classes: doxorubicin (DOX) (Class III) and +insulin (Class I).53 Also both pharmacological agents are currently used to treat common ocular +disorders.54 +First, we tested the loading of DOX, a chemotherapeutic agent against various cancer types, +including retinoblastoma.55 200 µg of DOX was added to a suspension of 100 µg of COF +microswimmers dispersed in 1 mL MEM, resulting in 138 µg DOX encapsulated (loading efficiency +of 138%) on the TABP-PDA-COF microswimmers after 24 hours, and 75% for TpAzo-COF. Due to +the small molecular size of DOX (~1.1 nm approximate molecular diameter), the molecule should +fit into the structural pores of both COF structures (3.4 nm and 2.5 nm), while adsorbing also on +the inner textural surface. The overall negative surface charge on both COF microswimmers +attracts the positively charged DOX molecules in physiological pH values and gives rise to stable +loading. Since the overall surface areas are similar within 10%, it appears that differences in +polarity or hydrogen bonding, possibly mediated by the carbonyl groups of TpAzo-COF, enable +electrostatic repulsions with the DOX molecules and interfere with DOX uptake in TpAzo-COF +structures, which is also correlated with the zeta potential measurements. While the positive +zeta potential of the TABP-PDA-COF (ζTABP-PDA-COF = 12.13 ± 1.28 mV) reduces agglomeration and +enables sufficient drug loading values, the negative zeta potential of the TpAzo-COF (ζTpAzo-COF = - +19.67 ± 0.68 mV) leads to agglomerations and reduces drug loading due to electrostatic +repulsions.56 In addition, a lower crystallinity and thereby, possibly decreased accessible pore +volume of TpAzo-COF are expected to lead to reduced DOX uptake. Overall, the DOX uptake of +both COF materials is among the highest reported, relative to other artificial structures using +physical encapsulation.11, 57 +The DOX release can be achieved by changing the pH to slightly more acidic conditions, i.e., from +pH = 7.2 to 5.5 (Fig. 3 e, g), which is achieved by adding HCl to PBS. The TABP-PDA microswimmers +release 95 µg of DOX within 60 minutes, which is significantly boosted compared to the weak, +passive release also observed (12 µg). The passive release is commonly observed when drugs + +such as DOX are not entirely trapped or encapsulated within porous structures but physisorbed +to the surface. Encapsulation within the TpAzo-COF, with a more open texture, appears more +stable, as evidenced by the lower passive release at pH 7.2 (5 µg in 60 minutes). In line, a +reduction of pH to 5 only releases 7% in 60 min, whereas a pH 3.5 yields 25% and is more +reasonable as a release trigger. The acid-triggered DOX release in the TABP-PDA-COF and TpAzo- +COF microswimmers can be seen in fluorescence imaging in Figure 3f,h, respectively. The +enhanced drug delivery of microswimmers at lower pH has the potential to enable the targeted +therapy in tumor or infection environments, which typically have acidic pHs.58, 59 +We also studied the loading and release of peptide (insulin), a frequently used drug in diabetic +retinopathy and convenient for light-controlled drug release applications.60, 61 Its larger molecular +size of ~3 nm makes larger pore sizes on the COFs desirable to allow for an efficiently +encapsulated loading. Indeed, insulin loading was observed on both COFS, 60% for TABP-PDA- +COF (3.5 nm pore size] and 40% for TpAzo-COF (2.5 nm pore size) (Fig. 3i,k), which suggests that +physisorption of the drugs occurs on the outer surface of the textural pores and that the +structural pores can assist stable uptake. +Similar to DOX release from the COF structures, changing pH enables insulin release from both +COFs. While the TABP-PDA-COF shows a continuously increasing cumulative release of +approximately 35 µg/ml within 60 min at pH 5 already, which may be desirable for slower dosing, +the TpAzo-COF releases its cargo rather instantly (within 10 min), and at lower amounts (~10 +µg/ml in more acidic pH 3.3 again). With both drugs, no visible light-triggered release was +observed, opposite to the carbon nitride systems reported earlier with DOX. However, as seen +herein, the absence of such a property can be very beneficial since it enables the decoupling of +motion control and drug release, which would otherwise have to co-occur.11 +As a third theranostic agent to load onto COFs, we used ICG dye, commonly used in diagnosing +retinal diseases.62 Firstly, we investigated ICG loading and near-infrared laser-induced +hyperthermia capabilities; then, we focused on medical imaging of ICG-loaded COF +microswimmers with photoacoustic imaging and optical coherence tomography. As is the case +with drug delivery, TABP-PDA-COF has a pore size larger than the size of the ICG (~2.9 nm +molecular diameter on its longest axis); hence, the drug is presumably loaded better into the + +structural pores of the TABP-PDA COF (3.5 nm), while in the case of TpAzo-COF, it appears to +dominantly bond to the bigger, textural pores (Fig. 4a). After ICG was loaded onto the both, +TpAzo-COF and TAPB-PDA-COF microswimmers at two different loading levels (50% and 100%, +w/w), they were irradiated with a near-infrared (NIR) laser at 808 nm.63 ICG-loaded TABP-PDA- +COFs achieved quick heating to 66 oC and 69 oC after only 3 minutes of 808 nm NIR irradiation for +50% and 100% loading, respectively. Compared to TABP-PDA-COFs, ICG-loaded TpAzo-COFs +heated up to 42 oC and 45 oC for 50% and 100% loading under the same NIR illumination +conditions (Fig 4 b, c). Heat generation and accumulation are always affected by heat transport +to the environment. Assuming similar absorption and hence heat generation at the same +loadings, these findings indicate that indeed, ICG transfers the heat slightly better by binding to +TABP-PDA COF, and that the TpAzo COF dissipates accumulated heat faster to the environment +due to its more open shape, and thereby reaches lower temperatures over extended times. In +both cases, this NIR-controlled hyperthermia behavior of both COFs could be helpful for novel +intraocular photodynamic therapy application, which is already in the clinical trial phase for ICG +dye.64 Compared to other novel intraocular photothermal therapy agents in the recent literature, +especially TABP-PDA-COFs with pores enabling ICG uptake into the material’s structural pores +and intense heating from 25 oC to 69 oC in 3 minutes, shows significant potential for the +photodynamic combined therapy applications that are used to degrade cells by heat +generation.65, 66 +Photoacoustic imaging and optical coherence tomography +Imaging microswimmers as they move in different fluids is one of the most critical enablers for +their potential in vivo applications.67 For this purpose, we selected to study two clinical imaging +methods: optical coherence tomography (OCT) and photoacoustic (PA) imaging. OCT is the gold +standard high-resolution clinical imaging method to observe intraocular structures and is +accessible in most ophthalmology clinics worldwide.68 PA is an emerging imaging technique that +combines the resolution of optical imaging with the depth of penetration of ultrasound imaging. +In recent years, PA has been used in ophthalmology as it shows significant advantages in imaging +deep ocular structures, such as lymphatic drainage and choroidal vasculature.69, 70 While in the +PA imaging method ICG was used as a contrast agent to enhance the visualization of COF + +microswimmers in the complex environment of intraocular fluids, COFs were imaged in +intraocular structures and ocular fluids without any contrast agent during OCT imaging. Different +concentrations of ICG are loaded onto the COF microswimmers and imaged under photoacoustic +imaging (Fig. 4d,e). While TAPB-PDA-COFs achieve up to 500 mean pixel intensity (MPI) at 815 +nm, which is the highest peak in the emission spectrum of ICG, TpAzo-COFs achieve 250 MPI +under the same imaging conditions. These signal intensity increases correlate with the +concentration of the ICG in the COF loading suspension and also the drug uptake ability of both +COFs, which correlate with other drug loading experiments. Imaging uptake and delivery of +therapeutic agents on microswimmers will be helpful in the targeted in vivo drug delivery +experiments.71 +As a next step, the light-driven propulsion of the COF microswimmers in intraocular fluids was +observed using PA imaging. In both vitreous and aqueous fluids, COFs were illuminated in the +same fashion as in the light-induced swimming experiments in various media and then observed +with photoacoustic imaging for 30 minutes (Fig. 5a-c). Except for TpAzo-COF in vitreous humor +under 630 nm light illumination, an increased ICG emission signal was observed in the focus areas +for all experimental groups. These results indicate that the light-driven collective motion of both +COF microswimmer types could be trackable under PA imaging. +For clinical applicability, we observed and measured the light-driven swimming of COF +microswimmers in intraocular fluids under real-time OCT. While the mean speeds of the smaller +and spherical TAPB-PDA-COFs were 12.1 ± 1.7 µm/s in aqueous humor and 7.6 ± 0.8 µm/s (~16.8 +BLPS) in the vitreous humor, mean speeds of TpAzo-COFs were slightly increased to 14.2 ± 1.5 +µm/s in aqueous humor and 8.8 ± 1.0 µm/s (~1.25 BLPS) in the vitreous humor under 470 nm +light illumination (Fig. 5d and Video S3). Compared to the previous intraocular microrobotic +studies employing magnetic actuation of helical microswimmers, the speed of the +microswimmers in terms of BLPS was significantly higher, ~16.8 BLPS in the current study vs. ~5.3 +BLPS for the fastest magnetic intraocular microswimmers previously.23 Light-driven accumulation +behavior of both COF microswimmer types in the focus of the light was trackable under real-time +OCT imaging without any contrast agent loading (Fig. 5e and Video S4). Additionally, their light- +driven propulsion in 470 nm wavelength light was also trackable even inside an ex vivo porcine + +eye with anterior segment OCT imaging (Video S5). The COF-based microswimmers are the first +intraocular microswimmers that can swim and be trackable inside the eye without any contrast +agent or surface modification. TpAzo-COFs were actuated faster, opposite to the previous +experiments, which highlights that a perfectly spherical shape of TAPB-PDA-COFs alone is not of +dominating benefit for mesh-like heterogeneous structures. Although the reasons for this +inverted swimming speed remain to be clarified and likely depend on photocatalytic reaction +rates in the respective environment, it is possibly also linked to the increased viscosity and +fibrillary mesh structures in the aqueous and vitreous humor that overall decrease the propulsion +speed of both COF microswimmer types compared with previous aqueous conditions.30 These +results show that both COF microswimmer types are suitable microrobotic drug delivery agents +under both PA and OCT imaging, while enabling actual biomedical applications inside body fluids, +especially for intraocular structures. With the help of their promising drug delivery and NIR- +based hyperthermia abilities, they could solve the active retinal drug delivery problems in various +ocular disorders.54 They could be easily loaded with DOX for chemotherapy without adverse +effects on retinoblastoma patients or with insulin to treat increased ocular pressure.72, 73 COF- +based microswimmers can easily be controllable with visible light, instead of other passive +nanomedicine agents in ophthalmology clinics and they do not require complex and unalterable +magnetic coil setups with narrow working spaces.21, 23 + +Conclusion and Outlook +In this manuscript, we have studied two structurally and texturally distinct COF microswimmer +types with tunable nanopore sizes towards their potential intraocular medical applications as +multifunctional microswimmers. This comparison of COFs from two different families with +distinct morphologies and drug loading capabilities yielded promising results in terms of +biocompatibility, imaging, drug delivery, and visible light-induced propulsion in ionic and +biological media, surpassing the applicability of current magnetically actuated microswimmer- +based systems – without a need of further structural modification or sophisticated structural +engineering. Simultaneously, the COF microswimmers can be propelled by visible and even red + +light in ionic and biological conditions (Fig. 2). Although some medium-dependent propulsion +trends at low salt concentrations remain to be clarified, their porous structure, coupled with +photocatalytic activity, seems key to efficient photocatalytic motion without dedicated toxic fuels +or harm to the tissue. A compact spherical shape, as achieved by the size-modified synthesis of +the TABP-PDA COFs, appears beneficial for fast propulsion, enabling bubble-free motion at 36 +BLPS while opening up possibilities for mobility in the intraocular region. On the other hand, large +and texturally more porous structures, as observed for the TpAZo-COF, enable similar absolute +propulsion speeds in ionic conditions, albeit at a much-reduced speed relative to their size (~2 +BLPS). The explanation for this behavior remains to be found and rationalized by numerical +models, especially since simple fluid dynamics and the applicability of Reynolds numbers, which +do not include inner flow, are not suited for these systems.74 Both microswimmers allow for +precise motion control as single particles by their phototactic properties, enabling complex +curvilinear navigation around obstacles in principle and collective motion for particle (re- +)assembly (Fig. 2).11, 19, 75 We show that large structural and textural pores enable the loading of +different drugs and dyes (e.g., insulin, ICG, and DOX) but that the pore size itself only plays a +partial role in (stable) uptake, since textural surface area also contributes to drug binding, as +clearly visible by the different uptake properties of the small drug DOX (Fig. 3), whereas larger +molecules, such as insulin or ICG, can stay more stably bound, even in lower loading amounts +(Fig. 4). Since the drug binding and release is also affected by chemical interactions between the +COF backbone and the drug, independent of pore size and surface area, future material design +should focus on optimizing these interaction factors to broaden our insights. +The versatility of COFs, not only on the morphological but especially on a molecular level, is +anticipated to enable tailored approaches to tune the adsorption and desorption properties of +drugs, akin to their use on gas sorption.76 Modifications of these interactions, especially by +external stimuli, such as pH changes, light, viscosity changes, and oxygen content in the vicinity, +can enable the desired interaction strength with the cargo and its release kinetics.11 This +possibility is anticipated to enable tailored, targeted, and especially semi-autonomous therapy +not only for in vitro but also for in vivo applications.77, 78 + +We further demonstrated medical imaging of the ICG-loaded COFs, enabled by photoacoustic +imaging and optical coherence tomography. In principle, both of them enable the visualization of +swarms and motion of large individual particles, providing more detailed insights into local +propulsion and release properties inside the eye or soft tissues where visible light cannot +penetrate easily. Since the ICG loading can be kept very low in the porous COFs while maintaining +a high signal intensity (Fig. 4). Optical coherence tomography inside eye tissue also enables real- +time imaging studies of drug-loaded microswimmers and evaluation in intraocular fluids and +structures, laying the grounds for a more detailed understanding of release properties and burst +kinetics for various theranostic agents. By decoupling COF microswimmers’ motion control and +release mechanism, a broad range of independent functionalities is made possible on these +porous organic structures in parallel. We anticipate that especially simultaneous imaging, drug +release, and NIR light-assisted photothermal therapy capabilities will offer additional theranostic +abilities beyond what current state-of-art noninvasive photodynamic therapy techniques could +achieve.79 In the near future, they could be functionalized in ophthalmology clinics for +multimodal therapy and imaging of retinal diseases, such as retinoblastoma, diabetic +retinopathy, or glaucoma. + +Materials & Methods +Synthesis and preparation of covalent organic frameworks +Synthesis of TAPB-PDA-COF was carried out according to a previous report with minor changes.34 +In a typical colloidal reaction, 1,3,5- tris(4-aminophenyl)benzene (TAPB) (0.030 mmol, 10.4 mg) +and terephthaldehyde (PDA) (0.044 mmol, 5.96 mg) were dissolved in 14 mL acetonitrile. After +10 minutes of sonication, a solution of Sc(OTf)3 (0.014 mmol, 7.00 mg) in 7 mL acetonitrile was +added dropwise at room temperature under slight stirring. After 24 hours of reaction, the solvent +was exchanged for distilled water by centrifugation for five times (795 g for 10 minutes each). +For solids characterization, the particles were precipitated by adding 0.5 mL of 1 M NaCl solution, +washed with methanol, and dried by supercritical CO2 on a Leica EM CPD300 instrument. TpAzo- +COF was synthesized according to a previous report.80 + +Brunauer–Emmett–Teller (BET) measurements and analysis +Nitrogen sorption measurements were performed on a Quantachrome Instruments Autosorb iQ +MP at 77 K. Before the gas adsorption studies, the samples were degassed for 12 h at 120 °C +under a vacuum. Multipoint BET surface area calculations and pressure ranges were chosen +according to the linear region on the BET plot in the range between 0.05 and 0.35 P/P0. Pore size +distribution was determined from Nitrogen adsorption isotherms using the NLDFT cylindrical +pores in the carbon model for nitrogen at 77 K. +PXRD measurements and analysis +Powder X-ray diffraction experiments were performed on a Stoe Stadi P diffractometer (Cu-Kα1, +Ge(111) in Debye-Scherrer geometry. The samples were measured in sealed glass capillaries (OD += 1.0 mm) and spun for improved particle statistics. +Transmission electron microscopy (TEM) and scanning electron microscopy (SEM) +Transmission electron microscopy was performed with a Philips CM30 ST (300kV, LaB6 cathode). +The samples were prepared dry onto a copper lacey carbon grid (Plano). Images were recorded +with a TVIPS TemCam-F216 CMOS camera. The program EM-Menu 4.0 Extended was used for +analysis. +SEM images were obtained on a Zeiss Merlin or a VEGA TS 5130MM (TESCAN) with an InLens +detector using electron energy of 1.5 kV. The samples were cast on indium-doped tin oxide (ITO) +substrates, and a 3 nm-thick iridium film was sputtered on them to reduce charging. +UV-VIS measurements and analysis +For diffuse reflectance UV–visible absorption, spectra were collected on a Cary 5000 +spectrometer (referenced to barium sulfate). Absorption spectra were calculated from the +reflectance data using the Kubelka-Munk and assuming a direct band gap.81 +Zeta potential measurements +The Z potential was determined using a Malvern nano Zs zetasizer. Dispersions of 0.5 mg/mL COF +in 10 mM aqueous NaCl were sonicated 15 min before zeta potential experiments. Surface charge +values represent the mean of 3 experiments and their standard deviation is indicated. + +Light-driven propulsion experiments +The spectral irradiance of the illumination in the microscope was measured at the place of the +sample chamber with a calibrated Ocean Optics OCEAN-FX-XR1-ES spectrophotometer after +attenuation by a neutral density filter. The results have been normalized to the filter attenuation +and the spot size of the light beam in the microscope. It was measured to be 2.0 ± 0.5 mm in +diameter, resulting in a relative experimental error of 50% after the error propagation +calculation. In the case of visible light propulsion, a broad-spectrum low-intensity white LED is +illuminated from the top, and lights with various wavelengths (385 nm, 470 nm, 510 nm, 560 nm, +and 630 nm) are illuminated through the microscope objective. The intensity of the microscope +light (1 mW/cm2 for the control experiments in the dark and 2 mW/cm2 for imaging during UV +light-based propulsion) was increased to 10 mW/cm2 for visible light propulsion. For +photocatalytic and PEC experiments, a calibrated Thorlabs S425C/PM100D optical power meter +directly measured the light intensity.19 All light intensities are used in the light propulsion +experiments under the ocular safety limit (54 mW/cm2) for ophthalmic devices.82 +Biocompatibility experiments +Human umbilical vein endothelial cells (CRL-1730 [HUVEC], ATCC, Manassas, VA) were grown in +dMEM supplemented with 10% (v/v) FBS and 1% (v/v) penicillin/streptomycin (Gibco, Grand +Island, NY, USA) at 37°C in a 5% CO2, 95% air-humidified atmosphere. Cells were reseeded after +growing to confluence into μ-Slide eight-well plates (Ibidi GmbH, Gräfelfing, Germany) at a cell +density of 25 x 103 cells/well and incubated for two days. HUVEC cells were incubated with TAPB- +PDA or TpAzo COF microswimmers at varying concentrations (3.1 to 25 μg/ml) for cytotoxicity +testing. Then, the cell viability was measured using a LIVE/DEAD assay (Thermo Fisher Scientific, +Waltham, MA) incorporating calcein-AM (green) and ethidium homodimer-1 (red) dyes. After 24 +hours of incubation with the COF microswimmers, live-dead cell numbers were calculated from +fluorescence microscopy images. Furthermore, cytotoxicity of microswimmers during light +actuation (470 nm for TAPB-PDA and 630 nm for TpAzo, 10 mW/cm2 and 4 mW/cm2, respectively) +was tested by live/dead staining of HUVEC cells right after and 24 hours after actuation of COF +microswimmers for 30 minutes.11 + +Drug & ICG loading and release tests +The loading efficiency was measured by centrifuging the DOX (44583, Sigma-Aldrich, St. Louis, +USA) or insulin (I3661, Sigma-Aldrich, St. Louis, USA) loaded microswimmers and comparing the +optical density (OD) of the supernatant with the precalibrated OD of DOX or insulin (200 μg/ml) +at 480 nm. Both COF microswimmers (100 μg/ml) were dispersed with DOX or insulin (200 +μg/ml), and this solution was stirred in the dark for 24 hours to allow the drugs to be adsorbed. +After 24 hours, the suspension was centrifuged, and the supernatant was used for measuring the +drug loading. The drug-loaded COF solution was washed three times with water and stored in +dPBS at +4°C for further delivery experiments. For the pH release, the pH of the resulting HCl- +diluted PBS solution was checked using a pH meter to confirm the stability of the pH during the +release experiments.11 +NIR-based remote heating of ICG-loaded COF particles +TpAzo-COF and TAPB-PDA-COF loaded with 50% and 100% ICG were loaded in microtubes and +irradiated with a NIR laser (808 nm, 0.6 W/cm2). Thermal images were obtained, and temperature +information was recorded with a thermal infrared camera (ETS320, FLIR Systems). +Photoacoustic imaging measurements and analysis +The photoacoustic (PA) signal characterizations were performed inside a Multispectral +Optoacoustic Tomography device (MSOT 512-element transducer, iThera Medical) system with +three scanning steps of 0.2 mm at different wavelengths. The samples with different +concentrations were prepared inside a transparent stripe and embedded in an agar phantom (1.5 +g/100 mL agar-DI water). The same preparation was done for the control sample. The agar +phantom was placed at the center of the transducer arrays. The measurements were then taken +for a range of wavelengths (660 – 980 nm), and each image was repeated three times for each +laser pulse and then averaged. A circular region of interest (ROI) was chosen for calculating the +PA signal at each wavelength. Finally, the diagrams were plotted against the control sample for +all concentrations. +For PA imaging of light-induced motion of nanoparticles, a handheld 3D photoacoustic probe +(256-element transducer, iThera Medical) was used for real-time tracking. The laser wavelength +was set at 800 nm, and the image sequences were taken at 10 frames per second. Then, a + +volumetric image of 20 × 20 × 20 mm³ was constructed from three orthogonal imaging planes. +The real-time change in the signal intensity at the light actuation spot indicated the movement +of the nanoparticles. +Optical coherence tomography (OCT) +The fresh porcine eyes were purchased from Ulmer Fleisch food factory, Ulm, Germany. Within +six hours after the euthanasia of the animals, a set of enucleated eyes stabilized to the holder, +and COFs were injected with a 30G syringe in the anterior chambers of the porcine eyes before +OCT imaging. Besides that, aqueous humor was removed from another set of fresh porcine eyes +with the help of 30G trocar and cannula. For vitreous collection, a classical vitrectomy procedure +is followed.83 The intraocular fluids with COFs were injected into a cylindrical tubing and +observed via OCT (TEL320C1 – Spectral Domain OCT System, Thorlabs). The motion inside the leg +was recorded with an image speed at a medium sensitivity (76 kHz). The refractive index was set +to 1.00, and the Hann filter was used for the apodization window. The A-scan averaging was set +to 1, and the B-scan averaging to 1 with a pixel size of 6.5 μm. + +Author contributions: F.P., V.S., B.V.L., and M.S. conceived and designed the project. F.P., V.S., +and E.Y. wrote the manuscript, with input and corrections from all authors. A.R. and L.Y +synthesized and characterized the materials. V.S. and X.L. performed the light propulsion +experiments and analyzed the data. E.Y. performed and analyzed in vitro biocompatibility tests. +X.L. and M.B.A. performed and analyzed drug loading experiments. B. A. performed and analyzed +NIR hyperthermia experiments. A.A., E.Y., and P.W. performed and analyzed the photoacoustic +imaging. E.Y. isolated porcine intraocular fluids and performed optical coherence tomography. +M.S., F.P., and B.V.L. supervised the research. All authors contributed to the discussion of the +data and overall results. +Data availability: All data are available from the corresponding author upon reasonable request. +Acknowledgments: The authors acknowledge Viola Duppel for SEM and TEM image acquisition. +We thank Julia Kröger for the fruitful discussions. Support by the Max Planck Society, the Bavarian +Research Network SolTech (B.V.L.), and the Deutsche Forschungsgemeinschaft (DFG) via the + +cluster of excellence “e-conversion” (project number EXC2089/1–390776260) is gratefully +acknowledged. 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Journal of ophthalmology 2017, 2017. + + + + + +Graphical Abstract: + + + + +Conceptual illustration of light-driven and light-steered COF microswimmers towards targeted +intraocular drug delivery and photothermal therapy applications under optical coherence +tomography-based real-time imaging. + + +Drug Loaded COF +Microswimmers +Light-driven propulsion +Visible Light Laser +Optical Trapping & +Source +Real-time Imaging +Targeted Drug Release & +Photothermal Therapy +Central Retina +Eye +Optical Coherence +Tomography +Figure 1: Structural properties of the two types of COF particles used as light-powered +microswimmers. a-c: Imine-linked TABP-PDA-COF nanoparticles. a: Precursors for synthesis and +molecular structure of the 2D covalent organic framework that stacks in the third dimension. b: +Calculated pore size distribution from nitrogen sorption isotherms at 77 K (see Fig. S1, S2 for +details), highlighting a fairly uniform pore diameter of 3.4 nm. c: SEM image of TABP-PDA COF +nanoparticles with a narrow diameter distribution around 450 nm. d-f: Azo-linked TpAzo-COF +microparticles. d: Precursors for synthesis and molecular structure of the 2D network that stacks +in the 3rd dimension. e: Calculated pore size distribution from nitrogen sorption isotherms at 77 +K (see Fig. S3, S4 for details), highlighting a relatively uniform pore diameter of 2.6 nm. f: SEM +images of the TpAzo-COF microparticles with a sponge-like structure and high levels of textural +porosity, including macropores and heterogeneous size distribution (6.97 ± 17.62 µm, see Fig. +S3, S4). + + + +685m²/g +0.3 +3.41nm +200nm +0.25- +NH2 +Y +0.2 +(p)^p +TAPB-PDA +0.1 +3.4nm +0.05- +PDA +H2N +NH2 +10 +20 +30 +40 +TAPB +Pore width (nm) +d +0.2 +TpAzo-COF +2.54 mm +500.mm +OHI +HO +OH +TpAzo +00 +H2N +OH +Azo +Tp +80 +Pana width (nm) + +Figure 2: Optical properties and propulsion of TABP-PDA-COF and TpAzo-COF microswimmers +in water and ionic media and their phototaxis behavior. a, f: Absorbance properties and optical +band gap extracted from UV-Vis diffuse reflectance spectra of TABP-PDA-COF (a) and TpAzo-COF +(f) particles, respectively, measured in the solid state. b, g: Mean speeds of the COF +microswimmers illuminated in distilled water at different wavelengths under the microscope. +The dashed line denotes the local Brownian motion speed. Density: 100 µg/ml, N = 50 particles. +Error bar = S.D. c, h: Propulsion in NaCl with increasing concentration and wavelength highlighting +strong ionic tolerance for light-driven propulsion. d, i: Comparison of propulsion speed in +different commonly used biological media (dPBS, MEM) and MEM modified by removing glucose +or adding FBS. Density: 100 µg/ml, N = 50 particles (a-d). Mean ± S.D. e, j: Phototactic control of +diluted COF microswimmer particles following illumination from the side (S=start, E=end of +trajectory). + + + +Mumination directbion +Figure 3: COF microswimmer biocompatibility, drug loading, and triggered release properties. +a-d: In vitro cell viability results for COF microswimmers a, c: cell viability percentages of HUVEC +cells in the presence of increasing TAPB-PDA-COF and TpAzo-COF microswimmer concentrations +with/without 470 nm and 630 nm illumination, respectively, for 30 minutes, mean ± S.D. b, d: +Corresponding fluorescence images of live cells (green) and dead cells (red) with 25 μg/ml, 30 +minutes, 470 nm and 630 nm, respectively. e-h: DOX uptake & release results for COF +microswimmers. e: TAPB-PDA-COF loading and release capacity with Doxorubicin (DOX) in MEM +at different pH over time, reaching 138% for TABP-PDA-COF loaded in MEM. f: Corresponding +fluorescence image of DOX (red) loaded TAPB-PDA-COFs at 25 µg/ml concentration. g: TpAzo- +COF with 75% loading and their subsequent stepwise release at different pH conditions; in + +LiV +:Dead +Live.Dead +DoX Loaded Particles +DOX Loaded Particles +nsulin Loaded Particle +Insulin Loaded Particlesneutral pH (7.2), slightly acidic conditions (pH=5), and acidic (3.3) as encountered around cancer +cells. h: Corresponding fluorescence image of DOX (red) loaded TpAzo-COFs at 25 µg/ml +concentration. i-l: Insulin uptake & release results for COF microswimmers. i: Insulin loading of +TAPB-PDA-COF with 60 % loading in MEM and release in different pH values over time. j: +Corresponding fluorescence images of FITC (green) labeled insulin-loaded TAPB-PDA COFs. k: +Insulin loading of TpAzo-COF with 40% loading in MEM and its release at different pH values over +time. l: Corresponding fluorescence images of FITC (green) labeled insulin-loaded TpAzo-COFs. +All scale bars are 100 μm. + + + + +Figure 4: Indocyanine green (ICG) loading, imaging, and hyperthermia functions of both COF +microswimmer types. a: ICG uptake into suitable structural pores (TAPB-PDA-COF) or texturally +porous structures (Tp-Azo-COF). b,c: NIR-based heating of 50% and 100% ICG-loaded COF +particles. d: Intensity of photoacoustic signal vs. ICG loading, highlighting high sensitivity regimes +at low loading concentrations for TAPB-PDA-COF microswimmers. e: The photoacoustic signal +intensity vs. ICG loading highlights high sensitivity regimes at low loading concentrations for +TpAzo COF microswimmers. + + +2.9nm + structure +texture +Figure 5: Real-time imaging of COF motion by photoacoustic and optical coherence tomography +imaging modalities. a-c: Photoacoustic imaging of focused light-driven actuation of ICG-loaded +COFs in both intraocular fluids. After 30 min, the accumulation of COF microswimmers in the +focus of the light with different wavelengths is visible. d: Mean speeds of COF microswimmer +particles illuminated with 470 nm light in intraocular fluids (Video S3). e: Optical coherence +images of COFs in aqueous humor (Video S4). The COF swimmers’ light-driven movement on the +tubing’s light-applied side is visible. The scale bar is 500 μm on each axis. + + +No Light +Light +No LighitSupporting Information + +Designing Covalent Organic Framework-based Light-driven Microswimmers +towards Intraocular Theranostic Applications + + + + +Figure S1. TABP-PDA COF structural analysis. a: Powder XRD after washing. b: FT-IR of the +precursors and the COF. c: BET surface area measurement for overall surface area analysis. + + + + +Figure S2. TABP-PDA COF particle morphology and structure. a: SEM image illustrating uniform +size distribution of the washed COF microparticles. b: Particle size distribution showing high +uniformity. c: TEM image showing a single COF nanoparticle consisting of crystalline domains +with a lateral size of approx. 50 nm. + + +300 +Experimental +Simulated +N-H +Intensity (a.u.) +Transmittance +mmn +C-H +=0 +C=N +50) +-Adsorption +TAPB +PDA +IDesaption +TAPB-PDA COF +5 +10 +15 +20 +25 +30 +35 +40 +4000 +3500 +3000 +2500 +2000 +1500 +1000 +500 +0 +0.4. +20 (Degrees) +Wavenumber(cm-1) +PAP3μm +20- +15- +5. +250300350400450500550 +100nm +Particle size (nm) +Figure S3: TpAzo-COF structural analysis. a: Powder XRD after washing. b: FTIR of the COF. c: BET +surface area measurement for overall surface area analysis. + + + +Figure S4. TpAzo-COF particle morphology and structure. a: SEM image illustrating the +agglomerated structure of TpAzo-COF microparticles. b: SEM image (zoomed in) showing sponge- +like inner structure with macropores. c: Particle size distribution showing non-uniformity of the +particle agglomerates. The particle size is centered around 7 µm. d: TEM image showing the +interconnection of crystalline COF nanosheets with a domain size of approximately 50 nm or less. + +b +a +TpAzo exp. +100 +BET surface +TpAzosim. +600 +90 +Intensity (a.u.) +area: 635 m2 g-1 +80 +? +400 +300 +70 +200 +Volume +09 +100- +a-Adsorption +Desorption +50- +0 +0.2 +0.4 +0.6 +4000350030002500200015001000500 +0.8 +5 +10 +15 +20 +25 +30 +35 +40 +P/Po +2e(Degree) +Wavenumber(cm-1)500nm +20μm +100nmSupporting Videos + +Video S1. Light-driven propulsion of 100 µg/ml TABP-PDA and TpAzo COF microswimmers +inside distilled water with a 470-nm wavelength light source + +Video S2. Phototaxis behavior of TABP-PDA and TpAzo COF microswimmers inside MEM using a +directional 470-nm wavelength light source + +Video S3. TABP-PDA COF and TpAzo COF microswimmer propulsion inside the porcine aqueous +and porcine vitreous humor fluid + +Video S4. Optical coherence tomography (OCT) imaging and guided trapping of TABP-PDA and +TpAzo COF microswimmers inside the aqueous humor fluid + +Video S5. Optical coherence tomography (OCT) imaging and guided propulsion of TABP-PDA +and TpAzo COF microswimmers inside the anterior chambers of the porcine eye + + diff --git a/29FST4oBgHgl3EQfYTgs/content/tmp_files/load_file.txt b/29FST4oBgHgl3EQfYTgs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f80460777469aec182a9ff23c95ea57b027379c --- /dev/null +++ b/29FST4oBgHgl3EQfYTgs/content/tmp_files/load_file.txt @@ -0,0 +1,985 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf,len=984 +page_content='Designing Covalent Organic Framework based Light driven Microswimmers towards Intraocular Theranostic Applications Varun Sridhar1,+, Erdost Yildiz1,+, Andrés Rodríguez-Camargo,2,3, Xianglong Lyu1, Liang Yao2, Paul Wrede1, Amirreza Aghakhani1, Mukrime Birgul Akolpoglu1, Filip Podjaski2,4,5,*, Bettina V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Lotsch2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Metin Sitti1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='* 1 Physical Intelligence Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Max Planck Institute for Intelligent Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Germany 2 Nanochemistry Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Max Planck Institute for Solid State Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Germany 3 Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Germany 4 Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' W12 0BZ London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' United Kingdom 5 Cluster of Excellence E-conversion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Lichtenbergstrasse 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Germany 6 Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' University of Munich (LMU),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Germany 7 Institute for Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 8092 Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Switzerland 8 School of Medicine and College of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Koç University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 34450 Istanbul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Turkey + These authors contributed equally to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' * Correspondence to: sitti@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='de, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='podjaski@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='uk, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='lotsch@fkf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='de Abstract Even micromachines with tailored functionalities enable targeted therapeutic applications in biological environments, their controlled motion in biological media and drug delivery functions usually require sophisticated designs and complex propulsion apparatuses for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Covalent organic frameworks (COFs), new chemically versatile and nanoporous materials, offer microscale multi-purpose solutions, which are not explored in light-driven micromachines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We describe and compare two different types of COFs, uniformly spherical TABP- PDA-COF sub-micron particles and texturally highly nanoporous, irregular, micron-sized TpAzo- COF particles as light-driven microrobots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' They can be used as highly efficient visible-light-driven drug carriers in aqueous ionic and cellular media, even in intraocular fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Their absorption ranging down to red light enables phototaxis even in deeper biological media and the organic nature of COFs enables their biocompatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The inherently porous structure with ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 nm structural pores, and large surface areas allow for targeted and efficient drug loading even for insoluble drugs and peptides, which can be released on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Also, indocyanine green (ICG) dye loading in the pores enables photoacoustic imaging or optical coherence tomography and hyperthermia in operando conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The real-time visualization of the drug-loaded COF microswimmers enables new insights into the function of porous organic micromachines, which will be useful to solve various drug delivery problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Keywords: Covalent organic framework, light-driven, microswimmer, targeted drug delivery, optical coherence tomography Introduction Microrobots are tiny machines that are tailored to be controlled externally to perform individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In order to achieve external control as well as multi-purpose functionality, micro/nanobots typically require a sophisticated and specifically adapted design enabling targeted control and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Their primary area of use is the large field of biomedicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1, 2, 3, 4, 5 The microrobots should not elicit any immune response and should be compatible with the cells to enable biomedical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6, 7 Also, the propulsion method should be as noninvasive as possible, and non-toxic, excluding the use of dedicated toxic fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8, 9 For an efficient microrobot function, motion control is the first requirement, which can become more and more challenging if the liquid they are propelled in contains species that hinder propulsion or external control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Wireless motion control requires external energy input and is typically realized by magnetic or acoustic actuation,10 but can also be realized by ultraviolet (UV) or blue light, even in biological conditions, as evidenced very recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 However, UV light, which is typically used for light-driven microswimmers12, 13 is incompatible with biological tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Also, visible light control, usually reported with high-intensity blue light,12 limits applications to transparent conditions since tissue penetration requires more red light or near-infrared light, which is a significant challenge to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The critical tasks of mobile microrobots are cargo uptake and delivery, often linked to biopharmaceutical classes and properties of drugs, after actively navigating to a target diseased tissue region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4, 7, 14, 15 Drug uptake and its controlled release were typically realized efficiently by encapsulation structures being a separate part of the microrobots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' these were then opened in the desired conditions or where a release could be triggered otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' More recently, inherently porous structures, such as metal-organic frameworks16, 17, 18 and porous carbon nitrides were used for such applications since their sizeable inner pore volume, reminiscent of a sponge, enables high and, even environmentally stable drug loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 However, porous particle structures with many textural pores of different sizes as part of their inner surface area leave challenges for controlled loading and release from their volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' As a last critical step to clinical applications, cell viability, the absence of foreign body reactions, and tissue biocompatibility are necessary conditions for microrobots to be used in biological contexts, which is not always easy to ensure, with all the desired functions being fulfilled at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4, 7, 14 For this purpose, typically biocompatible metal coatings, such as gold, titanium, or polymers are employed, but also organic-based materials are up-and-coming and were used recently without coatings, such as carbon nitrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 17, 19 Compared to inorganic structures, organic materials not only offer potential biodegradability but also the high flexibility of chemical design of organic materials, especially in terms of surface functionalities and porosity, might enable more efficient and targeted biomedical applications, such as drug delivery or hyperthermia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='20 Even the most sophisticated designs with biocompatible metallic structures fail during actuation inside heterogenic biological fluids and specific-targeted drug delivery and imaging in live tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='21, 22, 23 Because of that, new materials and actuation methods should be investigated for the basic tasks for the clinical obstacles, such as intraocular motion and drug delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In this work, we introduce covalent organic frameworks (COFs) as a tailorable active component to the field of micro- and nanomachines, or more precisely, light-driven microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These highly porous and crystalline materials can fulfill all the requirements listed above since their molecular structure, and also their morphology, can be designed and tuned bottom-up while enabling targeted properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='20, 24, 25 Simultaneously, they can use visible light for photocatalytic reactions with their environment, which can also be used for active particle propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='19, 26, 27 Depending on the propulsion mechanism and particle structure, the propulsion can be self- diffusiophoretic or self-electrophoretic, while allowing light-induced directional motion control via phototaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='28 The tailorable properties of the COF building blocks enable tuning of not only their absorption wavelength but also pore size and volume, surface polarity, and chemical affinity, which enable the loading of large and small molecule cargo based on the specific application requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Since such organic structures are non-magnetic per se, unless so-called Janus or hybrid (encapsulation) structures were to be employed, the use of light is a highly promising and convenient method not only to propel them but also to trigger functions within them and to image their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='29 Thanks to their promising and ion-tolerant visible light- driven propulsion properties, we especially focused on their use in ophthalmological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In addition to their light-driven propulsion, their configurable particle sizes enable them to pass through the fibrillar mesh of vitreous humor (~500 nm pore size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='30 For this purpose, we selected therapeutic agents and imaging modalities accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Here, these possibilities are investigated and exemplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We study and compare two very different modified COFs, namely TAPB-PDA-COF made from the condensation of 1,3,5-tris(4- aminophenyl)benzene and terephthaldehyde,31 and TpAzo-COF made from 1,3,5- triformylphloroglucinol and 4,4′-azodianiline,32 as light-driven microswimmer examples, in order to explore the microrobotic possibilities for this class of materials and to establish versatile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We describe their light-controlled propulsion in biological media, their biocompatibility, as well as their uptake and release of drugs that can be physisorbed to the pores of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='25, 33 Since these two COFs have distinct structures and morphologies, we derive design guidelines for their propulsion and cargo-related functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For theranostic delivery functions of microrobots, we used doxorubicin, insulin, and indocyanine green (ICG), which covers the breath of small molecules and peptides used as therapeutic and imaging agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We further image the motion of the COFs in real-time and potential in-vivo conditions using optical coherence tomography as well as photoacoustic imaging of COF particle swarms loaded with a near-infrared active dye (ICG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The water-soluble or insoluble drugs and the contrast agents can be loaded to track their release, allowing for first insights into the action of porous drug carriers in real-time clinical imaging modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In this way, we designed and investigated in detail the first photoactive intraocular drug carriers for various theranostic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Results and Discussion COF synthesis and characterization The COF structures to be used as microswimmers were selected based on their structural and optical properties and synthesized akin to a procedure reported earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='31, 32 In brief, the TAPB- PDA-COF nanospheres were obtained by using TAPB (1,3,5-tris(4-aminophenyl)benzene) and PDA (terephthaldehyde) as building blocks, with acetonitrile and Sc(Otf)3 as solvent and catalyst, respectively (see Materials & Methods section for more details) to yield a two-dimensional (2D), imide linked organic network (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These 2D sheets are stacked on each together forming an ordered 3D structure with hexagonal pore channels of a diameter of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 nm, as reported earlier and confirmed for the here modified synthesis by nitrogen sorption, powder XRD, and FT- IR analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='34 The obtained COF submicron particles (henceforth called nanoparticles for better discrimination) have an almost perfectly spherical shape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1c), which is beneficial for propulsion in fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='35 The synthesis yields a very homogeneous product with a Brunauer, Emmett, and Teller (BET) surface area as high as 685 m²/g (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S1c) and a narrow size distribution of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 452 ± 74 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S2a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TEM analysis reveals that these nanoparticles consist of agglomerated individual crystallites with sizes between 50 and 100 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The TpAzo-COF presented here for comparison is synthesized by solvothermal condensation between 1,3,5-triformylphloroglucinol (Tp) and 4,4´-azodianiline (Azo), forming a tautomeric ketoenamine COF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A highly crystalline product with a 2D molecular structure is obtained (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S3 for structural analysis), with slightly smaller structural pores of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 nm and a similar BET surface area of 635 m²/g (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In contrast to the first COF, the particle morphology is much less defined, leaving open large textural voids reminiscent of a sponge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1f and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The overall particle sizes are broadly distributed (7 ± 18 µm), hence being much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The primary crystallites forming the COF particle are only 20 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S4d), and the TpAzo-COF particles appear to be agglomerates of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Light-induced swimming in aqueous media To enable light-triggered propulsion, we first investigate the light absorption properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' UV-Vis spectroscopy and Kubelka-Munck analysis show that TAPB-PDA-COF and TpAzo-COF have an optical band gap of 464 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2a) and 616 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2f), respectively with a small absorption tail commonly arising from defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Hence, visible light propulsion can be extended up to a wavelength of ~470 nm (blue light) for the small TABP-PDA particles, and to green or even red parts of the spectrum for the large TpAzo-COF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Their light-induced propulsion was studied under a microscope in a microfluidic chamber under ambient conditions to test the phototaxis capabilities while diluting them to 100 µg/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' First, we focus on propulsion in distilled water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' While in the dark, COF particles show only local Brownian motion with a mean displacement speed of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 µm/s for the TAPB-PDA and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 µm/s for the TpAzo-COF, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2b,g, dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' When light from the photodiode is focused on the microswimmers through the microscope, their propulsion speed is significantly enhanced and becomes ballistic, as seen in Video S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The particles move towards the center of the light, then upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' This way, the light- driven collective assembly or trapping of the microswimmers is made possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' UV excitation at 385 nm propels the TABP-PDA-COF with 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 µm/s, while 470 nm blue light gives an even increased speed of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 µm/s (~36 bodylengths/s (BLPS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' At 510 nm illumination, no light- enhanced swimming was observed, consistent with the absorption spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The Tp- Azo-COF is propelled with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 (~2 BLPS), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 µm/s at 385, 470, 560 nm, and 630 nm, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' UV and red light hence do not increase propulsion significantly below Brownian motion and the absolute propulsion speed is lower, but it can be triggered even by yellow light (560 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Ion tolerance for light-driven microswimmers and phototaxis behavior Ionic conditions represent a major challenge for light-driven microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='36, 37, 38 The presence of pores, both structural and textural (=morphological), was suggested previously to enable the propulsion of microswimmers in ionic environments, which includes most of the biological fluids and cell culture media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 To confirm this and to widen the insights from different and better controlled structural features present in our model COFs, we first tested them in increasing concentrations of salt (NaCl), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2c,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' When propelled at 470 nm, the TABP-PDA-COF does not decrease the speed compared to distilled water up to concentrations as large as 1000 mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Therefore, the ionic concentration in the media at which the microswimmers’ speed is halved (EI50) cannot be attributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='39 Also, we observe a slightly increasing propulsion speed between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 and 10 mM, with a maximum value of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 µm/s (54% increase vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' distilled water, 0 mM) at 1 mM NaCl (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' An explanation for this non-linear behavior remains to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=" Ionic interactions can be influencing the Helmholtz and Debye layers, as well as the materials' inner space charge layer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' As such, light- induced charge carrier stability or recombination will also be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' On the other hand, the chlorine evolution reaction due to dissolved NaCl may another photocatalytic pathway possibly increasing the reaction rate, and thereby the propulsion speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 40, 41, 42 These factors currently cannot be studied or disentangled on such size and complex reaction interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' However, their propulsion speed even surpasses our previously reported PHI microswimmers, the only reported system with comparable ionic tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 Similarly, for the TpAzo-COFs, an increased propulsion speed compared to pure water is observed in all ionic conditions (1-1000 mM) at 560 nm illumination, peaking at 1 mM (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 µm/s, 79% increase vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' distilled water) and followed by a 28% relative decay to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 µm/s at 1000 mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' When increasing the wavelength, no active propulsion is observed for TABP-PDA-COF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2b), but the Tp-AZO-COF exhibits slightly enhanced propulsion even at 630 nm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 µm/s) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Next, three standard biological media are studied, namely, Dulbecco’s phosphate-buffered saline (dPBS), minimum essential medium (MEM), and MEM plus fetal bovine serum (FBS) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2d,i), which slightly differ in their components: dPBS contains NaCl, KCl, Na2HPO4, and KH2PO4 at ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 10 g/L (~150 mM) in total;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' MEM contains the same components as dPBS and additional two amino acids, vitamins, and glucose, some of which can be redox-active agents that help extract not only electrons but especially holes from the microswimmers under illumination to power them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 43 FBS, slightly more viscous, adds nutrients for cell growth and imitates the conditions found within the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 At 470 nm illumination, the mean speeds of the TAPB-PDA-COF microswimmers in dPBS, MEM, and MEM + FBS are 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 µm/s, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8µm/s and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3 µm/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These speeds are again higher than in distilled water (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 µs/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The slight decrease upon FBS addition can be attributed to the increasing viscosity or other surface interactions with the proteins present in the FBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Very similar behavior is observed with the TpAzo-COF at 560 nm, where the swimming speeds are equivalent to the maximum value in 1mM NaCl, or even slightly higher (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 µm/s, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 µm/s, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 µm/s, and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 µm/s in dPBS, MEM (with and without glucose), and MEM + FBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A difference however is observed when glucose, a well-oxidizable fuel,11, 43 is absent – the speed is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Its vital role as fuel for propulsion is clearly visible when illuminating TpAzo at 630 nm in MEM that contains glucose, where efficient propulsion, independent of FBS, is observed (10 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 µm/s and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 µm/s respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' This purely red light-induced photocatalytic motion in the presence of high ion concentrations and without using potent and toxic fuels is unprecedented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='29, 44 However, the still efficient propulsion at 560 nm without glucose in MEM confirms that the other ingredients (including dissolved oxygen11, 19) may also assist motion induced by photocatalysis, or at least do not hamper it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These experiments not only show the superiority in performance over current inorganic microswimmers in high-salinity media but also highlight how crucial facile redox species are that can act as fuel for propulsion, akin to photocatalysis in general, and especially if sub-band gap trap states might be partially involved (630 nm illumination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='43, 45, 46 Such a substantial shift toward the red part of the spectrum that can penetrate deeper tissues makes organic and small band gap microswimmers (especially with trap states in the gap) attractive for micromachines not just in-vitro, but even for in-vivo conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Light-driven directional propulsion control Phototaxis is the property by which microswimmers swim towards or away from the direction of incident light (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', positive or negative phototaxis), which often depends on their surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='47, 48 It enables direction control, opposite to random ballistic displacement usually observed with Janus particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 49 When the COF microswimmers were illuminated by a directed light source from the side with a 45° angle, both TABP-PDA-COF and TpAzo-COF microswimmers exhibit positive phototaxis, and swim toward the light that can propel them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2 e, j, and video S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TABP-PDA-COF and TpAzo-COF particles move with mean speeds of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 µm/s and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 µm/s, at 470 nm and 630 nm illumination in water and MEM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' This apparent increase in the particle speed compared to vertical illumination could be attributed to the larger parallel component of the light direction to the propulsion direction when the samples were illuminated from the side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' When the samples are illuminated from the bottom, only the side- wise motion component is measured as a common standard, artificially decreasing the actual velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='50, 51 Similar findings have been found on carbon nitride microswimmers, which were discussed in more detail in our previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 The required symmetry breaking is created by the side-wise illumination and, thereby, an artificially created Janus structure results from the self-shadowing of the microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='13, 47 Biocompatibility of COFs In order to be used in potential biomedical applications and to ascertain biocompatibility, microswimmers should have no significant cytotoxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Hence, we tested the cytotoxicity of the microswimmers with human umbilical vein endothelial cells (HUVEC) in dMEM with FBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Different concentrations of TAPB-PDA-COF and TpAzo-COF microswimmers (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1-25 µg/ml) were incubated with HUVECs in the dark, and their viability was investigated with calcein-based live/dead fluorescence staining of the cells after 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The cells with TABP-PDA COF were completely viable, and they did not show any significant decrease in viability even at high concentrations, both with illumination and without illumination at 470 nm with maximum light intensity, 10 mW/cm2, for 30 minutes), as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3 a, which is visible also in live cell fluorescent images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TpAzo-COF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3 c,d) shows lower cell viability in comparison with TAPB-PDA-COF, with 93% and 75% HUVEC cell viability in 25 µg/ml concentration (in dark and with 630 nm illumination (10 mW/cm2), respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Also, at concentrations of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 µg/ml, the viability is decreased to 88% in comparison to the TABP-PDA COF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' However, this fairly good viability indicates that also the TpAZo COF can be used at lower concentrations for drug delivery applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Generally, illumination seems not to affect the viability at low concentrations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='25 µg/ml), and only slightly at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 and 25 µg/ml for both COFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These results also suggest that light-induced propulsion induces only minimal cytotoxicity in the range of light-driven propulsion periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Compared to carbon nitride microswimmers, which have a larger band gap (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 eV, 450 nm) and a very low-lying valance band, and therefore enable more redox reactions with organic matter, including cells in principle, the use of 470 nm or 630 nm light with our TABP- PDA COFs and TpAzo COFs shows potential for reduced cell death [with 97% and 88% cell viability after 30 minutes of light in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 µg/ml concentrations of TAPB-PDA-COFs and TpAzo-COFs, respectively] and makes especially the TABP-PDA COFs more applicable to practical applications such as drug delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 A previous study with primary cells from mouse splenocytes further confirmed no detectable level of IL-12 (a pro-inflammatory cytokine) in the untreated samples in concentrations used above in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='52 Drug loading, drug delivery, and hyperthermia To explore the COF microswimmer’s applicability to biological environments, we also studied their potential as drug carriers with different pharmacological agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The differently pronounced textural and structural porosity of the TABP-PDA and TpAzo-COFs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S1-S4), which enables ionic tolerance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2c,h), is not only beneficial for motion but also as space to take up, transport and deliver therapeutic drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We studied and compared how the structural features enable interactions with such cargo in the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For this reason, we chose an imaging agent, indocyanine green (ICG), and two different pharmacological agents with different Biopharmaceutics Classification System (BCS) classes: doxorubicin (DOX) (Class III) and insulin (Class I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='53 Also both pharmacological agents are currently used to treat common ocular disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='54 First, we tested the loading of DOX, a chemotherapeutic agent against various cancer types, including retinoblastoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='55 200 µg of DOX was added to a suspension of 100 µg of COF microswimmers dispersed in 1 mL MEM, resulting in 138 µg DOX encapsulated (loading efficiency of 138%) on the TABP-PDA-COF microswimmers after 24 hours, and 75% for TpAzo-COF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Due to the small molecular size of DOX (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 nm approximate molecular diameter), the molecule should fit into the structural pores of both COF structures (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 nm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 nm), while adsorbing also on the inner textural surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The overall negative surface charge on both COF microswimmers attracts the positively charged DOX molecules in physiological pH values and gives rise to stable loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Since the overall surface areas are similar within 10%, it appears that differences in polarity or hydrogen bonding, possibly mediated by the carbonyl groups of TpAzo-COF, enable electrostatic repulsions with the DOX molecules and interfere with DOX uptake in TpAzo-COF structures, which is also correlated with the zeta potential measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' While the positive zeta potential of the TABP-PDA-COF (ζTABP-PDA-COF = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='13 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='28 mV) reduces agglomeration and enables sufficient drug loading values, the negative zeta potential of the TpAzo-COF (ζTpAzo-COF = - 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='68 mV) leads to agglomerations and reduces drug loading due to electrostatic repulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='56 In addition, a lower crystallinity and thereby, possibly decreased accessible pore volume of TpAzo-COF are expected to lead to reduced DOX uptake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Overall, the DOX uptake of both COF materials is among the highest reported, relative to other artificial structures using physical encapsulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 57 The DOX release can be achieved by changing the pH to slightly more acidic conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', from pH = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3 e, g), which is achieved by adding HCl to PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The TABP-PDA microswimmers release 95 µg of DOX within 60 minutes, which is significantly boosted compared to the weak, passive release also observed (12 µg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The passive release is commonly observed when drugs such as DOX are not entirely trapped or encapsulated within porous structures but physisorbed to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Encapsulation within the TpAzo-COF, with a more open texture, appears more stable, as evidenced by the lower passive release at pH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 (5 µg in 60 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In line, a reduction of pH to 5 only releases 7% in 60 min, whereas a pH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 yields 25% and is more reasonable as a release trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The acid-triggered DOX release in the TABP-PDA-COF and TpAzo- COF microswimmers can be seen in fluorescence imaging in Figure 3f,h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The enhanced drug delivery of microswimmers at lower pH has the potential to enable the targeted therapy in tumor or infection environments, which typically have acidic pHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='58, 59 We also studied the loading and release of peptide (insulin), a frequently used drug in diabetic retinopathy and convenient for light-controlled drug release applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='60, 61 Its larger molecular size of ~3 nm makes larger pore sizes on the COFs desirable to allow for an efficiently encapsulated loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Indeed, insulin loading was observed on both COFS, 60% for TABP-PDA- COF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 nm pore size] and 40% for TpAzo-COF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 nm pore size) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3i,k), which suggests that physisorption of the drugs occurs on the outer surface of the textural pores and that the structural pores can assist stable uptake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Similar to DOX release from the COF structures, changing pH enables insulin release from both COFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' While the TABP-PDA-COF shows a continuously increasing cumulative release of approximately 35 µg/ml within 60 min at pH 5 already, which may be desirable for slower dosing, the TpAzo-COF releases its cargo rather instantly (within 10 min), and at lower amounts (~10 µg/ml in more acidic pH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3 again).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' With both drugs, no visible light-triggered release was observed, opposite to the carbon nitride systems reported earlier with DOX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' However, as seen herein, the absence of such a property can be very beneficial since it enables the decoupling of motion control and drug release, which would otherwise have to co-occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 As a third theranostic agent to load onto COFs, we used ICG dye, commonly used in diagnosing retinal diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='62 Firstly, we investigated ICG loading and near-infrared laser-induced hyperthermia capabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' then, we focused on medical imaging of ICG-loaded COF microswimmers with photoacoustic imaging and optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' As is the case with drug delivery, TABP-PDA-COF has a pore size larger than the size of the ICG (~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='9 nm molecular diameter on its longest axis);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' hence, the drug is presumably loaded better into the structural pores of the TABP-PDA COF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 nm), while in the case of TpAzo-COF, it appears to dominantly bond to the bigger, textural pores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After ICG was loaded onto the both, TpAzo-COF and TAPB-PDA-COF microswimmers at two different loading levels (50% and 100%, w/w), they were irradiated with a near-infrared (NIR) laser at 808 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='63 ICG-loaded TABP-PDA- COFs achieved quick heating to 66 oC and 69 oC after only 3 minutes of 808 nm NIR irradiation for 50% and 100% loading, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Compared to TABP-PDA-COFs, ICG-loaded TpAzo-COFs heated up to 42 oC and 45 oC for 50% and 100% loading under the same NIR illumination conditions (Fig 4 b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Heat generation and accumulation are always affected by heat transport to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Assuming similar absorption and hence heat generation at the same loadings, these findings indicate that indeed, ICG transfers the heat slightly better by binding to TABP-PDA COF, and that the TpAzo COF dissipates accumulated heat faster to the environment due to its more open shape, and thereby reaches lower temperatures over extended times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In both cases, this NIR-controlled hyperthermia behavior of both COFs could be helpful for novel intraocular photodynamic therapy application, which is already in the clinical trial phase for ICG dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='64 Compared to other novel intraocular photothermal therapy agents in the recent literature, especially TABP-PDA-COFs with pores enabling ICG uptake into the material’s structural pores and intense heating from 25 oC to 69 oC in 3 minutes, shows significant potential for the photodynamic combined therapy applications that are used to degrade cells by heat generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='65, 66 Photoacoustic imaging and optical coherence tomography Imaging microswimmers as they move in different fluids is one of the most critical enablers for their potential in vivo applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='67 For this purpose, we selected to study two clinical imaging methods: optical coherence tomography (OCT) and photoacoustic (PA) imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' OCT is the gold standard high-resolution clinical imaging method to observe intraocular structures and is accessible in most ophthalmology clinics worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='68 PA is an emerging imaging technique that combines the resolution of optical imaging with the depth of penetration of ultrasound imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In recent years, PA has been used in ophthalmology as it shows significant advantages in imaging deep ocular structures, such as lymphatic drainage and choroidal vasculature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='69, 70 While in the PA imaging method ICG was used as a contrast agent to enhance the visualization of COF microswimmers in the complex environment of intraocular fluids, COFs were imaged in intraocular structures and ocular fluids without any contrast agent during OCT imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Different concentrations of ICG are loaded onto the COF microswimmers and imaged under photoacoustic imaging (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 4d,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' While TAPB-PDA-COFs achieve up to 500 mean pixel intensity (MPI) at 815 nm, which is the highest peak in the emission spectrum of ICG, TpAzo-COFs achieve 250 MPI under the same imaging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These signal intensity increases correlate with the concentration of the ICG in the COF loading suspension and also the drug uptake ability of both COFs, which correlate with other drug loading experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Imaging uptake and delivery of therapeutic agents on microswimmers will be helpful in the targeted in vivo drug delivery experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='71 As a next step, the light-driven propulsion of the COF microswimmers in intraocular fluids was observed using PA imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In both vitreous and aqueous fluids, COFs were illuminated in the same fashion as in the light-induced swimming experiments in various media and then observed with photoacoustic imaging for 30 minutes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 5a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Except for TpAzo-COF in vitreous humor under 630 nm light illumination, an increased ICG emission signal was observed in the focus areas for all experimental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' These results indicate that the light-driven collective motion of both COF microswimmer types could be trackable under PA imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For clinical applicability, we observed and measured the light-driven swimming of COF microswimmers in intraocular fluids under real-time OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' While the mean speeds of the smaller and spherical TAPB-PDA-COFs were 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='7 µm/s in aqueous humor and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 µm/s (~16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 BLPS) in the vitreous humor, mean speeds of TpAzo-COFs were slightly increased to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 µm/s in aqueous humor and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='0 µm/s (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='25 BLPS) in the vitreous humor under 470 nm light illumination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 5d and Video S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Compared to the previous intraocular microrobotic studies employing magnetic actuation of helical microswimmers, the speed of the microswimmers in terms of BLPS was significantly higher, ~16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 BLPS in the current study vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3 BLPS for the fastest magnetic intraocular microswimmers previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='23 Light-driven accumulation behavior of both COF microswimmer types in the focus of the light was trackable under real-time OCT imaging without any contrast agent loading (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 5e and Video S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Additionally, their light- driven propulsion in 470 nm wavelength light was also trackable even inside an ex vivo porcine eye with anterior segment OCT imaging (Video S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The COF-based microswimmers are the first intraocular microswimmers that can swim and be trackable inside the eye without any contrast agent or surface modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TpAzo-COFs were actuated faster, opposite to the previous experiments, which highlights that a perfectly spherical shape of TAPB-PDA-COFs alone is not of dominating benefit for mesh-like heterogeneous structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Although the reasons for this inverted swimming speed remain to be clarified and likely depend on photocatalytic reaction rates in the respective environment, it is possibly also linked to the increased viscosity and fibrillary mesh structures in the aqueous and vitreous humor that overall decrease the propulsion speed of both COF microswimmer types compared with previous aqueous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='30 These results show that both COF microswimmer types are suitable microrobotic drug delivery agents under both PA and OCT imaging, while enabling actual biomedical applications inside body fluids, especially for intraocular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' With the help of their promising drug delivery and NIR- based hyperthermia abilities, they could solve the active retinal drug delivery problems in various ocular disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='54 They could be easily loaded with DOX for chemotherapy without adverse effects on retinoblastoma patients or with insulin to treat increased ocular pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='72, 73 COF- based microswimmers can easily be controllable with visible light, instead of other passive nanomedicine agents in ophthalmology clinics and they do not require complex and unalterable magnetic coil setups with narrow working spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='21, 23 Conclusion and Outlook In this manuscript, we have studied two structurally and texturally distinct COF microswimmer types with tunable nanopore sizes towards their potential intraocular medical applications as multifunctional microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' This comparison of COFs from two different families with distinct morphologies and drug loading capabilities yielded promising results in terms of biocompatibility, imaging, drug delivery, and visible light-induced propulsion in ionic and biological media, surpassing the applicability of current magnetically actuated microswimmer- based systems – without a need of further structural modification or sophisticated structural engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Simultaneously, the COF microswimmers can be propelled by visible and even red light in ionic and biological conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Although some medium-dependent propulsion trends at low salt concentrations remain to be clarified, their porous structure, coupled with photocatalytic activity, seems key to efficient photocatalytic motion without dedicated toxic fuels or harm to the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A compact spherical shape, as achieved by the size-modified synthesis of the TABP-PDA COFs, appears beneficial for fast propulsion, enabling bubble-free motion at 36 BLPS while opening up possibilities for mobility in the intraocular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' On the other hand, large and texturally more porous structures, as observed for the TpAZo-COF, enable similar absolute propulsion speeds in ionic conditions, albeit at a much-reduced speed relative to their size (~2 BLPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The explanation for this behavior remains to be found and rationalized by numerical models, especially since simple fluid dynamics and the applicability of Reynolds numbers, which do not include inner flow, are not suited for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='74 Both microswimmers allow for precise motion control as single particles by their phototactic properties, enabling complex curvilinear navigation around obstacles in principle and collective motion for particle (re- )assembly (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11, 19, 75 We show that large structural and textural pores enable the loading of different drugs and dyes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', insulin, ICG, and DOX) but that the pore size itself only plays a partial role in (stable) uptake, since textural surface area also contributes to drug binding, as clearly visible by the different uptake properties of the small drug DOX (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 3), whereas larger molecules, such as insulin or ICG, can stay more stably bound, even in lower loading amounts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Since the drug binding and release is also affected by chemical interactions between the COF backbone and the drug, independent of pore size and surface area, future material design should focus on optimizing these interaction factors to broaden our insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The versatility of COFs, not only on the morphological but especially on a molecular level, is anticipated to enable tailored approaches to tune the adsorption and desorption properties of drugs, akin to their use on gas sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='76 Modifications of these interactions, especially by external stimuli, such as pH changes, light, viscosity changes, and oxygen content in the vicinity, can enable the desired interaction strength with the cargo and its release kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 This possibility is anticipated to enable tailored, targeted, and especially semi-autonomous therapy not only for in vitro but also for in vivo applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='77, 78 We further demonstrated medical imaging of the ICG-loaded COFs, enabled by photoacoustic imaging and optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In principle, both of them enable the visualization of swarms and motion of large individual particles, providing more detailed insights into local propulsion and release properties inside the eye or soft tissues where visible light cannot penetrate easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Since the ICG loading can be kept very low in the porous COFs while maintaining a high signal intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Optical coherence tomography inside eye tissue also enables real- time imaging studies of drug-loaded microswimmers and evaluation in intraocular fluids and structures, laying the grounds for a more detailed understanding of release properties and burst kinetics for various theranostic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' By decoupling COF microswimmers’ motion control and release mechanism, a broad range of independent functionalities is made possible on these porous organic structures in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We anticipate that especially simultaneous imaging, drug release, and NIR light-assisted photothermal therapy capabilities will offer additional theranostic abilities beyond what current state-of-art noninvasive photodynamic therapy techniques could achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='79 In the near future, they could be functionalized in ophthalmology clinics for multimodal therapy and imaging of retinal diseases, such as retinoblastoma, diabetic retinopathy, or glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Materials & Methods Synthesis and preparation of covalent organic frameworks Synthesis of TAPB-PDA-COF was carried out according to a previous report with minor changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='34 In a typical colloidal reaction, 1,3,5- tris(4-aminophenyl)benzene (TAPB) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='030 mmol, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 mg) and terephthaldehyde (PDA) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='044 mmol, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='96 mg) were dissolved in 14 mL acetonitrile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After 10 minutes of sonication, a solution of Sc(OTf)3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='014 mmol, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='00 mg) in 7 mL acetonitrile was added dropwise at room temperature under slight stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After 24 hours of reaction, the solvent was exchanged for distilled water by centrifugation for five times (795 g for 10 minutes each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For solids characterization, the particles were precipitated by adding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 mL of 1 M NaCl solution, washed with methanol, and dried by supercritical CO2 on a Leica EM CPD300 instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TpAzo- COF was synthesized according to a previous report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='80 Brunauer–Emmett–Teller (BET) measurements and analysis Nitrogen sorption measurements were performed on a Quantachrome Instruments Autosorb iQ MP at 77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Before the gas adsorption studies, the samples were degassed for 12 h at 120 °C under a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Multipoint BET surface area calculations and pressure ranges were chosen according to the linear region on the BET plot in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='35 P/P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Pore size distribution was determined from Nitrogen adsorption isotherms using the NLDFT cylindrical pores in the carbon model for nitrogen at 77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' PXRD measurements and analysis Powder X-ray diffraction experiments were performed on a Stoe Stadi P diffractometer (Cu-Kα1, Ge(111) in Debye-Scherrer geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The samples were measured in sealed glass capillaries (OD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='0 mm) and spun for improved particle statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Transmission electron microscopy (TEM) and scanning electron microscopy (SEM) Transmission electron microscopy was performed with a Philips CM30 ST (300kV, LaB6 cathode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The samples were prepared dry onto a copper lacey carbon grid (Plano).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Images were recorded with a TVIPS TemCam-F216 CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The program EM-Menu 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='0 Extended was used for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' SEM images were obtained on a Zeiss Merlin or a VEGA TS 5130MM (TESCAN) with an InLens detector using electron energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The samples were cast on indium-doped tin oxide (ITO) substrates, and a 3 nm-thick iridium film was sputtered on them to reduce charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' UV-VIS measurements and analysis For diffuse reflectance UV–visible absorption, spectra were collected on a Cary 5000 spectrometer (referenced to barium sulfate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Absorption spectra were calculated from the reflectance data using the Kubelka-Munk and assuming a direct band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='81 Zeta potential measurements The Z potential was determined using a Malvern nano Zs zetasizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Dispersions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 mg/mL COF in 10 mM aqueous NaCl were sonicated 15 min before zeta potential experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Surface charge values represent the mean of 3 experiments and their standard deviation is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Light-driven propulsion experiments The spectral irradiance of the illumination in the microscope was measured at the place of the sample chamber with a calibrated Ocean Optics OCEAN-FX-XR1-ES spectrophotometer after attenuation by a neutral density filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The results have been normalized to the filter attenuation and the spot size of the light beam in the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' It was measured to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 mm in diameter, resulting in a relative experimental error of 50% after the error propagation calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In the case of visible light propulsion, a broad-spectrum low-intensity white LED is illuminated from the top, and lights with various wavelengths (385 nm, 470 nm, 510 nm, 560 nm, and 630 nm) are illuminated through the microscope objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The intensity of the microscope light (1 mW/cm2 for the control experiments in the dark and 2 mW/cm2 for imaging during UV light-based propulsion) was increased to 10 mW/cm2 for visible light propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For photocatalytic and PEC experiments, a calibrated Thorlabs S425C/PM100D optical power meter directly measured the light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='19 All light intensities are used in the light propulsion experiments under the ocular safety limit (54 mW/cm2) for ophthalmic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='82 Biocompatibility experiments Human umbilical vein endothelial cells (CRL-1730 [HUVEC], ATCC, Manassas, VA) were grown in dMEM supplemented with 10% (v/v) FBS and 1% (v/v) penicillin/streptomycin (Gibco, Grand Island, NY, USA) at 37°C in a 5% CO2, 95% air-humidified atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Cells were reseeded after growing to confluence into μ-Slide eight-well plates (Ibidi GmbH, Gräfelfing, Germany) at a cell density of 25 x 103 cells/well and incubated for two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' HUVEC cells were incubated with TAPB- PDA or TpAzo COF microswimmers at varying concentrations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 to 25 μg/ml) for cytotoxicity testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Then, the cell viability was measured using a LIVE/DEAD assay (Thermo Fisher Scientific, Waltham, MA) incorporating calcein-AM (green) and ethidium homodimer-1 (red) dyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After 24 hours of incubation with the COF microswimmers, live-dead cell numbers were calculated from fluorescence microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Furthermore, cytotoxicity of microswimmers during light actuation (470 nm for TAPB-PDA and 630 nm for TpAzo, 10 mW/cm2 and 4 mW/cm2, respectively) was tested by live/dead staining of HUVEC cells right after and 24 hours after actuation of COF microswimmers for 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 Drug & ICG loading and release tests The loading efficiency was measured by centrifuging the DOX (44583, Sigma-Aldrich, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Louis, USA) or insulin (I3661, Sigma-Aldrich, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Louis, USA) loaded microswimmers and comparing the optical density (OD) of the supernatant with the precalibrated OD of DOX or insulin (200 μg/ml) at 480 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Both COF microswimmers (100 μg/ml) were dispersed with DOX or insulin (200 μg/ml), and this solution was stirred in the dark for 24 hours to allow the drugs to be adsorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After 24 hours, the suspension was centrifuged, and the supernatant was used for measuring the drug loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The drug-loaded COF solution was washed three times with water and stored in dPBS at +4°C for further delivery experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For the pH release, the pH of the resulting HCl- diluted PBS solution was checked using a pH meter to confirm the stability of the pH during the release experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='11 NIR-based remote heating of ICG-loaded COF particles TpAzo-COF and TAPB-PDA-COF loaded with 50% and 100% ICG were loaded in microtubes and irradiated with a NIR laser (808 nm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 W/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Thermal images were obtained, and temperature information was recorded with a thermal infrared camera (ETS320, FLIR Systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Photoacoustic imaging measurements and analysis The photoacoustic (PA) signal characterizations were performed inside a Multispectral Optoacoustic Tomography device (MSOT 512-element transducer, iThera Medical) system with three scanning steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 mm at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The samples with different concentrations were prepared inside a transparent stripe and embedded in an agar phantom (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 g/100 mL agar-DI water).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The same preparation was done for the control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The agar phantom was placed at the center of the transducer arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The measurements were then taken for a range of wavelengths (660 – 980 nm), and each image was repeated three times for each laser pulse and then averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A circular region of interest (ROI) was chosen for calculating the PA signal at each wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Finally, the diagrams were plotted against the control sample for all concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For PA imaging of light-induced motion of nanoparticles, a handheld 3D photoacoustic probe (256-element transducer, iThera Medical) was used for real-time tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The laser wavelength was set at 800 nm, and the image sequences were taken at 10 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Then, a volumetric image of 20 × 20 × 20 mm³ was constructed from three orthogonal imaging planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The real-time change in the signal intensity at the light actuation spot indicated the movement of the nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Optical coherence tomography (OCT) The fresh porcine eyes were purchased from Ulmer Fleisch food factory, Ulm, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Within six hours after the euthanasia of the animals, a set of enucleated eyes stabilized to the holder, and COFs were injected with a 30G syringe in the anterior chambers of the porcine eyes before OCT imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Besides that, aqueous humor was removed from another set of fresh porcine eyes with the help of 30G trocar and cannula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' For vitreous collection, a classical vitrectomy procedure is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='83 The intraocular fluids with COFs were injected into a cylindrical tubing and observed via OCT (TEL320C1 – Spectral Domain OCT System, Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The motion inside the leg was recorded with an image speed at a medium sensitivity (76 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The refractive index was set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='00, and the Hann filter was used for the apodization window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The A-scan averaging was set to 1, and the B-scan averaging to 1 with a pixel size of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Author contributions: F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' conceived and designed the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' wrote the manuscript, with input and corrections from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Y synthesized and characterized the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' performed the light propulsion experiments and analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' performed and analyzed in vitro biocompatibility tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' performed and analyzed drug loading experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' performed and analyzed NIR hyperthermia experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' performed and analyzed the photoacoustic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' isolated porcine intraocular fluids and performed optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=', and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' supervised the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' All authors contributed to the discussion of the data and overall results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Data availability: All data are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Acknowledgments: The authors acknowledge Viola Duppel for SEM and TEM image acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' We thank Julia Kröger for the fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Support by the Max Planck Society, the Bavarian Research Network SolTech (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' ), and the Deutsche Forschungsgemeinschaft (DFG) via the cluster of excellence “e-conversion” (project number EXC2089/1–390776260) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' has received and acknowledges UKRI funding under the grant reference EP/X027449/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' E.' metadata={'source': 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carriers for in vivo drug delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Nature communications 2018, 9(1): 2785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Benyettou F, Kaddour N, Prakasam T, Das G, Sharma SK, Thomas SA, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' In vivo oral insulin delivery via covalent organic frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Chemical Science 2021, 12(17): 6037- 6047.' metadata={'source': 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Kandambeth S, BabaRao R, Marathe Y, Kunjir SM, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Phosphoric Acid Loaded Azo (−N═N−) Based Covalent Organic Framework for Proton Conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Journal of the American Chemical Society 2014, 136(18): 6570-6573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Chen Z, Dinh HN, Miller E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Photoelectrochemical water splitting, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Yan B, Vakulenko M, Min SH, Hauswirth WW, Nirenberg S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Maintaining ocular safety with light exposure, focusing on devices for optogenetic stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Vision Res 2016, 121: 57-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Mohamed S, Claes C, Tsang CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Review of small gauge vitrectomy: progress and innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Journal of ophthalmology 2017, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Graphical Abstract: Conceptual illustration of light-driven and light-steered COF microswimmers towards targeted intraocular drug delivery and photothermal therapy applications under optical coherence tomography-based real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Drug Loaded COF Microswimmers Light-driven propulsion Visible Light Laser Optical Trapping & Source Real-time Imaging Targeted Drug Release & Photothermal Therapy Central Retina Eye Optical Coherence Tomography Figure 1: Structural properties of the two types of COF particles used as light-powered microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a-c: Imine-linked TABP-PDA-COF nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: Precursors for synthesis and molecular structure of the 2D covalent organic framework that stacks in the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b: Calculated pore size distribution from nitrogen sorption isotherms at 77 K (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S1, S2 for details), highlighting a fairly uniform pore diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c: SEM image of TABP-PDA COF nanoparticles with a narrow diameter distribution around 450 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d-f: Azo-linked TpAzo-COF microparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d: Precursors for synthesis and molecular structure of the 2D network that stacks in the 3rd dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e: Calculated pore size distribution from nitrogen sorption isotherms at 77 K (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S3, S4 for details), highlighting a relatively uniform pore diameter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' f: SEM images of the TpAzo-COF microparticles with a sponge-like structure and high levels of textural porosity, including macropores and heterogeneous size distribution (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='97 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='62 µm, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' S3, S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 685m²/g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='41nm 200nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='25 NH2 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 (p)^p TAPB PDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='05 PDA H2N NH2 10 20 30 40 TAPB Pore width (nm) d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 TpAzo COF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='54 mm 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='mm OHI HO OH TpAzo 00 H2N OH Azo Tp 80 Pana width (nm) Figure 2: Optical properties and propulsion of TABP-PDA-COF and TpAzo-COF microswimmers in water and ionic media and their phototaxis behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a, f: Absorbance properties and optical band gap extracted from UV-Vis diffuse reflectance spectra of TABP-PDA-COF (a) and TpAzo-COF (f) particles, respectively, measured in the solid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b, g: Mean speeds of the COF microswimmers illuminated in distilled water at different wavelengths under the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The dashed line denotes the local Brownian motion speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Density: 100 µg/ml, N = 50 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Error bar = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c, h: Propulsion in NaCl with increasing concentration and wavelength highlighting strong ionic tolerance for light-driven propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d, i: Comparison of propulsion speed in different commonly used biological media (dPBS, MEM) and MEM modified by removing glucose or adding FBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Density: 100 µg/ml, N = 50 particles (a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Mean ± S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e, j: Phototactic control of diluted COF microswimmer particles following illumination from the side (S=start, E=end of trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Mumination directbion Figure 3: COF microswimmer biocompatibility, drug loading, and triggered release properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a-d: In vitro cell viability results for COF microswimmers a, c: cell viability percentages of HUVEC cells in the presence of increasing TAPB-PDA-COF and TpAzo-COF microswimmer concentrations with/without 470 nm and 630 nm illumination, respectively, for 30 minutes, mean ± S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b, d: Corresponding fluorescence images of live cells (green) and dead cells (red) with 25 μg/ml, 30 minutes, 470 nm and 630 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e-h: DOX uptake & release results for COF microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e: TAPB-PDA-COF loading and release capacity with Doxorubicin (DOX) in MEM at different pH over time, reaching 138% for TABP-PDA-COF loaded in MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' f: Corresponding fluorescence image of DOX (red) loaded TAPB-PDA-COFs at 25 µg/ml concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' g: TpAzo- COF with 75% loading and their subsequent stepwise release at different pH conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' in LiV :Dead Live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='Dead DoX Loaded Particles DOX Loaded Particles nsulin Loaded Particle Insulin Loaded Particlesneutral pH (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2), slightly acidic conditions (pH=5), and acidic (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='3) as encountered around cancer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' h: Corresponding fluorescence image of DOX (red) loaded TpAzo-COFs at 25 µg/ml concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' i-l: Insulin uptake & release results for COF microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' i: Insulin loading of TAPB-PDA-COF with 60 % loading in MEM and release in different pH values over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' j: Corresponding fluorescence images of FITC (green) labeled insulin-loaded TAPB-PDA COFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' k: Insulin loading of TpAzo-COF with 40% loading in MEM and its release at different pH values over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' l: Corresponding fluorescence images of FITC (green) labeled insulin-loaded TpAzo-COFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' All scale bars are 100 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Figure 4: Indocyanine green (ICG) loading, imaging, and hyperthermia functions of both COF microswimmer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: ICG uptake into suitable structural pores (TAPB-PDA-COF) or texturally porous structures (Tp-Azo-COF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b,c: NIR-based heating of 50% and 100% ICG-loaded COF particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d: Intensity of photoacoustic signal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' ICG loading, highlighting high sensitivity regimes at low loading concentrations for TAPB-PDA-COF microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e: The photoacoustic signal intensity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' ICG loading highlights high sensitivity regimes at low loading concentrations for TpAzo COF microswimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='9nm structure texture Figure 5: Real-time imaging of COF motion by photoacoustic and optical coherence tomography imaging modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a-c: Photoacoustic imaging of focused light-driven actuation of ICG-loaded COFs in both intraocular fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' After 30 min, the accumulation of COF microswimmers in the focus of the light with different wavelengths is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d: Mean speeds of COF microswimmer particles illuminated with 470 nm light in intraocular fluids (Video S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' e: Optical coherence images of COFs in aqueous humor (Video S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The COF swimmers’ light-driven movement on the tubing’s light-applied side is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The scale bar is 500 μm on each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' No Light Light No LighitSupporting Information Designing Covalent Organic Framework based Light driven Microswimmers towards Intraocular Theranostic Applications Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TABP-PDA COF structural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: Powder XRD after washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b: FT-IR of the precursors and the COF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c: BET surface area measurement for overall surface area analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TABP-PDA COF particle morphology and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: SEM image illustrating uniform size distribution of the washed COF microparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b: Particle size distribution showing high uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c: TEM image showing a single COF nanoparticle consisting of crystalline domains with a lateral size of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 300 Experimental Simulated N-H Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=') Transmittance mmn C-H =0 C=N 50) -Adsorption TAPB PDA IDesaption TAPB-PDA COF 5 10 15 20 25 30 35 40 4000 3500 3000 2500 2000 1500 1000 500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 20 (Degrees) Wavenumber(cm-1) PAP3μm 20- 15- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 250300350400450500550 100nm Particle size (nm) Figure S3: TpAzo-COF structural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: Powder XRD after washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b: FTIR of the COF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c: BET surface area measurement for overall surface area analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TpAzo-COF particle morphology and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' a: SEM image illustrating the agglomerated structure of TpAzo-COF microparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b: SEM image (zoomed in) showing sponge- like inner structure with macropores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' c: Particle size distribution showing non-uniformity of the particle agglomerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' The particle size is centered around 7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' d: TEM image showing the interconnection of crystalline COF nanosheets with a domain size of approximately 50 nm or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' b a TpAzo exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 100 BET surface TpAzosim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 600 90 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=') area: 635 m2 g 1 80 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' 400 300 70 200 Volume 09 100 a Adsorption Desorption 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='6 4000350030002500200015001000500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content='8 5 10 15 20 25 30 35 40 P/Po 2e(Degree) Wavenumber(cm 1)500nm 20μm 100nmSupporting Videos Video S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Light-driven propulsion of 100 µg/ml TABP-PDA and TpAzo COF microswimmers inside distilled water with a 470-nm wavelength light source Video S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Phototaxis behavior of TABP-PDA and TpAzo COF microswimmers inside MEM using a directional 470-nm wavelength light source Video S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' TABP-PDA COF and TpAzo COF microswimmer propulsion inside the porcine aqueous and porcine vitreous humor fluid Video S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Optical coherence tomography (OCT) imaging and guided trapping of TABP-PDA and TpAzo COF microswimmers inside the aqueous humor fluid Video S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} +page_content=' Optical coherence tomography (OCT) imaging and guided propulsion of TABP-PDA and TpAzo COF microswimmers inside the anterior chambers of the porcine eye' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FST4oBgHgl3EQfYTgs/content/2301.13787v1.pdf'} diff --git a/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/2301.02610v1.pdf.txt b/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/2301.02610v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55a888662ff161eb7dd8e75f4594e5327e222528 --- /dev/null +++ b/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/2301.02610v1.pdf.txt @@ -0,0 +1,1236 @@ +Feedback-Gated Rectified Linear Units +Marco Kemmerling 1 +Abstract +Feedback connections play a prominent role in +the human brain but have not received much +attention in artificial neural network research. +Here, a biologically inspired feedback mecha- +nism which gates rectified linear units is pro- +posed. On the MNIST dataset, autoencoders with +feedback show faster convergence, better perfor- +mance, and more robustness to noise compared +to their counterparts without feedback. +Some +benefits, although less pronounced and less con- +sistent, can be observed when networks with +feedback are applied on the CIFAR-10 dataset. +1. Introduction +The brain has served as inspiration for artificial neural net- +works (ANNs) for decades. While these models are usually +heavily simplified compared to the brain, they have seen +significant successes in areas such as image recognition +(Krizhevsky et al., 2012), speech recognition (Hinton et al., +2012), and machine translation (Sutskever et al., 2014) in +recent times. +Despite successes, it is clear that the average human brain +is vastly more powerful and versatile than any model used +in practice today, and as such it may be useful to investigate +how and where exactly the brain and ANNs differ. +One such discrepancy between ANNs and the brain is the +existence of feedback, or top-down connections. +While +there is clear evidence of prominent feedback connections +in the brain, ANNs have overwhelmingly been designed +based on the feedforward paradigm, although networks that +do not work solely on the feedforward principle exist and +are called recurrent neural networks (RNNs). Most RNNs +used in practice today focus on recurrent connections from +one layer to itself (e.g. +LSTM networks (Hochreiter & +Schmidhuber, 1997)), which, while recurrent, arguably do +not constitute top-down connections. These networks are +1University +of +Maastricht, +Maastricht, +The +Nether- +lands. +Correspondence +to: +Marco +Kemmerling +. +typically applied on problems where the input consists of +sequence data, where the recurrence allows for memory of +previously seen elements of the sequence. +However, the usefulness of recurrent connections or feed- +back is not necessarily restricted to sequence data. If the +input is image data, a first look, or pass, at an image could +be used to construct a rough idea of what the image con- +tains, as well as to identify areas of interest, which can then +be further examined on a second pass. +While the network architectures considered in this paper +feature real top-down connections, the focus is not on the +network topology itself, but on how these top-down con- +nections influence the behaviour of single neurons, i.e. a +mechanism for incorporating feedback. +This feedback mechanism is derived from neuroscience lit- +erature and examined from two broad angles: (1) Whether +the feedback mechanism can in any way improve on stan- +dard methods. Relevant metrics include convergence speed +and performance quality of the trained network. (2) If ex- +amining the feedback’s properties and how it behaves un- +der certain conditions (e.g. noisy signals) can offer any in- +sights into what role the feedback might fulfil in the brain. +Needless to say, care has to be taken when trying to infer +functionality of mechanisms in the brain from simplified +artificial networks. Nevertheless, experimentation on arti- +ficial models offers an intriguing opportunity, as they are +naturally easier to investigate and manipulate than the real +brain. +In the remainder of this paper, some neuroscientific back- +ground is explored in section 2 to serve as context for the +feedback mechanism, followed by a description of the feed- +back mechanism itself as it occurs in the brain (section 2.1). +In section 3 the mechanism is adapted for use in ANNs and +some practical considerations on its use are given in sec- +tion 3.1. The following sections describe a range of experi- +ments with the intention to provide answers to the research +questions posed above. +2. Neuroscientific Background +The neocortex, part of the cerebral cortex, is a part of the +brain that evolved in mammals comparatively recently. It +comprises around 80% of the human brain (Markram et al., +arXiv:2301.02610v1 [cs.NE] 6 Jan 2023 + +Feedback-Gated Rectified Linear Units +2004) and is therefore often speculated to be responsible +for the emergence of higher intelligence. +The most abundant type of neuron in the neocortex is the +pyramidal neuron, constituting between 70-85% of cells. +In contrast to the remaining neurons in the neocortex, so +called interneurons, which are mostly inhibitory, pyramidal +neurons are excitatory (DeFelipe & Fari˜nas, 1992). +As the name suggests, pyramidal neurons have a cell body +roughly shaped like a pyramid, with a base at the bottom +and an apex at the top. Pyramidal neurons have two types +of dendrites: basal dendrites, originating at the base, and +one apical dendrite, originating at the apex. This apical +dendrite terminates in what is called the apical tuft, where +heavy branching of the apical dendrite occurs. (DeFelipe +& Fari˜nas, 1992). +These apical and basal dendrites are not just differently lo- +cated, but also serve different functions. Basal dendrites +receive regular feedforward input, while the apical tuft den- +drites receive feedback input (Larkum, 2013). +The neocortex appears to have a distinct structure which +is characterised by its organisation into layers as well as +columns. The columnar organisation is based on the ob- +servation that neurons stacked on top of each other tend to +be connected and have similar response properties, while +only few connections exist between columns. Columns are +hence hypothesised to be a basic functional unit in the cor- +tex, although this is somewhat debated in the neuroscience +community (Goodhill & Carreira-Perpi˜n´an, 2002). +The further organisation into six layers was proposed by +Brodman in 1909 (Brodmann, 1909). Layers 1 and 6 are +of particular interest here. Layer 1 consists of almost no +cell bodies, but mostly connections between axons and the +apical dendrites of pyramidal neurons (Shipp, 2007), i.e. it +serves as a connection hub for feedback signals. Layer 6 +sends signals to neurons in the thalamus which then in turn +sends signals to layer 1 neurons in the same column (Shipp, +2007), i.e. layers 1 and 6 create a loop where feedback is +sent from layer 6 and received by layer 1. +2.1. Distal Input to Pyramidal Neurons +As described above, apical tuft dendrites receive feedback +input, which appears to modulate the gain of the corre- +sponding neuron (Larkum, 2004). It is hypothesised that +this is a way for the cortex to combine an internal repre- +sentation of the world with external input, i.e. feedback +to a neuron may predict whether this particular neuron +should be firing, and even small feedforward input may +lead the neuron to fire as long as the feedback signal is +strong (Larkum, 2013). +Taking both feedforward and feedback input into account, +the firing rate of a neuron can be modelled as follows +(Larkum, 2004): +f = g(µS + αµD + σ + fβ(µD) − θ) +(1) +where f is the firing rate of the neuron, g the gain, µS the +average somatic current (i.e. feedforward input), µD the +average distal current (i.e. feedback input), α is an atten- +uation factor, σ represents fluctuations in the current, θ is +the firing threshold, and β(µD) is an increasing function of +the dendritic mean current which saturates for values above +some current threshold. +3. Feedback-Gated Rectified Linear Units +The model described in the previous section serves as a +basis to derive an activation function which can replace +the common rectified linear unit (ReLU) (Nair & Hinton, +2010), i.e. f(x) = max(0, x). +To arrive at a more practical activation function, g and θ are +dropped from equation 1, since the threshold is modelled +through the bias unit and the gain (i.e. slope) of a ReLU is +by definition 1 and can thus be safely dropped. Dropping +the summands αµD and σ is less justifiable, but since they +do not contribute to the core property of gain increase, they +will be disregarded here, arriving at the following simpli- +fied relationship: +f = µS + fβ(µD) +(2) +Removing f from the right hand side: +f = +1 +1 − β(µD)µS +(3) +What remains is an exact definition of β(µD), which, ac- +cording to (Larkum, 2004), is “an increasing function of the +dendritic mean current µ which saturates for values above +1000pA“. In other words, the function is bounded, i.e. the +gain cannot be increased to arbitrarily high values. Accord- +ingly, some maximum value βmax the function can produce +and a threshold value η which describes when this maxi- +mum is reached need to be defined. Assuming a piecewise +linear model, β(µD) is thus defined as follows: +β(µD) = min +�βmax +η +µD, βmax +� +(4) +As there are no obvious values to assign to βmax and η, +they are treated as hyperparameters. Since setting βmax to +1 results in a division by 0 and a value of βmax > 1 causes +a negative slope, βmax should be smaller than 1. + +Feedback-Gated Rectified Linear Units +Plugging equation 4 into equation 3 yields: +f = +1 +1 − min( βmax +η +µD, βmax) +µS +(5) +Since negative values for µS are not taken into account in +the above equations, µS is replaced with max(0, µS), i.e. +the classic ReLU function: +f = +max(0, µS) +1 − min( βmax +η +µD, βmax) +(6) +3.1. Feedback-Gated ReLUs in Practice +The feedback path attempts to mimic the top-down path in +the brain. As such, the origin of feedback terminating in a +layer should be a layer that is higher in the (feedforward) +hierarchy. +Since feedback from higher layers can only be computed +if these higher layers have priorly received feedforward in- +put, at least two time steps are needed to incorporate the +modified ReLUs into a network. Concretely, some data, +e.g. an image is fed into the network twice, where the first +pass enables the computation of feedback which can then +be utilised in the second pass. Although more than two +timesteps are not required, it is possible to use an arbitrary +number of timesteps, which is examined in section 4.1.1. +Any layer that receives feedback requires an additional set +of weights to compute µD. Specifically, each layer hi with +size n receiving feedback from layer hj with size m intro- +duces n × m additional parameters. +The resulting networks can then be unrolled to create a +feedforward network, so that for t timesteps, each layer oc- +curs t times, while using the same weights at each timestep +(see figure 1). Since the unrolled network is purely feedfor- +ward, the standard backpropagation is a suitable learning +rule. +In convolutional neural networks (LeCun, 1989), feedback +is implemented on a filter-wise basis, i.e. each neuron does +not receive its own unique feedback signal, but rather ev- +ery filter receives a unique feedback signal that is shared +between all units belonging to that filter. +Dropout (Srivastava et al., 2014) should be used by drop- +ping out the same units in all passes. Otherwise, if e.g. +dropout is only applied on the last pass, the remaining units +will still receive signals from dropped out units in previous +passes, which defeats the purpose of dropout. +4. Experimental Results +The preceding sections describe a feedback mechanism and +how it can be implemented in practice. Here, a range of ex- +Figure 1. Left: autoencoder with (partial) feedback. Right: Un- +rolled autoencoder. +periments is performed to observe how this feedback mech- +anism changes the behaviour of ANNs. Several networks +are applied on two datasets, MNIST (LeCun et al., 2010) +and CIFAR-10 (Krizhevsky et al., 2014). Specifically, the +experiments are designed to answer the research questions +posed in the introduction: (1) whether feedback can im- +prove the performance of ANNs, (2) whether observing +how the feedback works in artificial models can reveal any +clues on what function feedback has in the brain. Sections +4.1.3, 4.1.4, and 4.2.2 serve to answer the latter question, +where section 4.1.3 is more of a general analysis of feed- +back, while sections 4.1.4 and 4.2.2 test whether feedback +might increase the networks robustness to noise. The re- +maining sections are concerned primarily with question (1) +in that they test convergence speed and performance quality +in various configurations. +4.1. MNIST +The MNIST dataset is composed of 28 × 28 pixel binary +images of handwritten digits, split into 60000 training and +10000 test instances (LeCun et al., 2010). Each image is +associated with one of ten classes representing the digits +between 0 and 9. +The models used in the following experiments are based +on a (non-convolutional) autoencoder with two encoding +and two decoding layers. The input layer has dimension +(1 × 784), the first encoding layer (E1) outputs data of di- +mension (1×392), the second (E2) of dimension (1×196), +the first decoding layer (D1) of dimension (1 × 392) and +the second decoding layer (D2) restores the data back to its +original dimension. Except for the final layer, each layer is +followed by a ReLU activation. The final layer makes use +of a sigmoid activation function. +First experiments were performed with only a single feed- +back connection between the first decoder and the first en- +coder (see figure 1). + +D2 +不 +D1 +E2 +不 +E1 +个 +InputD2 +个 +个个个个 +个 +>E1 +InputFeedback-Gated Rectified Linear Units +Figure 2. Test set loss of autoencoders with and without feedback. +The dimension of the second encoding layer is 196. +Figure 3. Test set loss of autoencoders with and without feedback. +The dimension of the second encoding layer is 10. +Optimal values for η and βmax were determined by a grid +search (βmax = 0.95, η = 5). +Figure 2 shows the loss curves for the autoencoder with +and without feedback. While the autoencoder with feed- +back converges noticeably faster, the difference is relatively +small. It is conceivable that feedback might have a greater +effect if the difficulty of the task is increased. While diffi- +culty is not a well defined term, reducing the dimension of +the second encoding layer (i.e. the bottleneck) can arguably +be seen as an increase in difficulty. +The dimension of the second encoding layer is thus reduced +to 10 (this modification will persist in all subsequent ex- +periments) and the experiment is repeated. Indeed, figure +3 shows a much larger gap between the autoencoder with +feedback and the one without it, supporting the hypothe- +sis that feedback may be more beneficial on more difficult +tasks. +Figure 4. Autoencoder performance with varying numbers of +timesteps. Each configuration was trained and evaluated 10 times. +The curves shown are the averaged losses on the test set. +4.1.1. MORE THAN TWO TIMESTEPS +While at least two timesteps are required to incorporate +feedback, it is not clear whether exactly two timesteps +should be used or whether > 2 timesteps can be benefi- +cial. To examine this, autoencoders with 1, 2, 4, 6, and 8 +timesteps are trained. +The results, depicted in figure 4, show that more than two +timesteps yield no or negligible improvement. This may +of course be data and/or task dependent. Since MNIST is +a fairly simple dataset (binary images, clear separation of +background and foreground, etc.), it is not inconceivable +that tasks on other datasets may benefit from more than two +timesteps. +4.1.2. COMPREHENSIVE FEEDBACK +In the previous experiments, feedback is only sent from +one decoding layer to one encoding layer. Naturally, there +are many more possible configurations that incorporate fur- +ther feedback connections. In the following experiment, +each layer receives feedback from every layer above it, i.e. +every possible top-down connection is present in the net- +work. This will be referred to as comprehensive feedback, +whereas the previous approach will be referred to as partial +feedback. +As shown in figure 5, the configuration explained above +does not only converge faster than a standard autoencoder, +but also settles to a smaller loss value, which was not the +case when only partial feedback was applied. +4.1.3. FEEDBACK VS CONSTANT GAIN +In an effort to gain some understanding on how exactly +feedback helps to improve performance, the frequency of +different feedback values is examined. +A distinction is +made between feedback and gain, where feedback refers + +0.250 +Without feedback +With feedback +0.225 +0.200 +0.175 +B80 +0.150 +0.125 +0.D75 +DOEZ +4000 +00 +8400 + babches0.26 - +Without feedback +0.24 - +With feedback +0.22 +0.20 + 0.18 - +0.16 +0.14 +0.12 +0.10 +0 +DOZ +4000 +00 +DOt8 + babches0.26 +1 timestep +2 timesteps +0.24 +4 timesteps +0.22 +6 timesteps +8 timesteps +0.20 + 0.18 +0.16 +0.14 +0.12 +0 +DOEZ +4000 +00 +DOt8 + babchesFeedback-Gated Rectified Linear Units +Figure 5. Loss on the test set of autoencoders without feedback, +partial feedback, and comprehensive feedback. Note that the hori- +zontal axis is different from previous figures, i.e. the training time +is longer. +to µD and gain refers to +1 +1−min( βmax +η +(µD),βmax). +Figure 6 shows the data as collected in a network with a +single feedback connection. +While there are some smaller gain values, the overwhelm- +ing majority of values are the maximum gain the network +can produce. This raises the question whether there is much +benefit to learning feedback or whether it might be simi- +larly beneficial to simply multiply all activation values by +a constant. +This is easily tested by setting the gain of every ReLU in +the affected layer to a constant value of 10. +As can be seen in figure 7, this does lead to a steeper loss +curve than the standard autoencoder, although not quite as +steep as that of the autoencoder with actual learned feed- +back. Further, the performance after training is completed +is worse than that of the standard autoencoder. +Repeating this same experiment for more than one feed- +back connection, i.e. for an autoencoder with comprehen- +sive feedback, yields results as illustrated in figure 8. +In this setup, the simple multiplication by a constant ini- +tially converges even faster than the autoencoder with +learned feedback. While it does not achieve the same per- +formance as the feedback autoencoder in later stages of +training, it is on par with the standard autoencoder’s per- +formance. +Clearly, the effects of feedback cannot be fully explained +by this constant gain, but the idea of a constant gain seems +to have some merit. +Figure 6. Distribution of feedback (top) and gain (bottom) values +collected in a network with partial feedback over the complete +MNIST test set. +Figure 7. Comparison of a standard autoencoder, an autoencoder +with partial feedback, and an autoencoder with partial constant +gain (the gain of all units in the second encoding layer is set to +10) + +0.250 +Without feedback +Partial feedback +0.225 +Comprehensive feedback +0.200 +0.150 +0.125 +0 +2500 +5000 +DOSE +# betchesFeedbadk Distribution +840000 +ODOM +500000 +400000 +ODODE +240000 +0 +40 +20 +0 +21 +40Gain Distribution +COADO +CODOST +0 +2 +t +8 +1f0.26 +Without feedback +0.24 - +Partial feedback +Partial constant gain +0.22 +0.20 +0.16 +0.14 +0.12 +0.10 +0 +2400 +4000 +00 +00 + betchesFeedback-Gated Rectified Linear Units +Figure 8. Comparison of a standard autoencoder, an autoencoder +with comprehensive feedback, and an autoencoder with compre- +hensive gain (the gain of all layers is set to 10). +4.1.4. NOISY ACTIVATIONS +While noisy signals are usually not an issue in artificial net- +works, noise in the brain is very prevalent (Faisal et al., +2008). To see whether feedback makes the model more +robust to noise, gaussian noise with zero mean and vari- +ous standard deviations is added to the (pre-)activations of +both the network with feedback and the one without it. The +networks are only evaluated with added noise, training is +performed without noise. Note that in the network with +feedback, noise is added to the activations in both passes. +h = f(W T x + b + N(0, σ2) ) +(7) +As figure 9 shows, the use of feedback significantly in- +creases the network’s robustness to noise. While this is not +especially useful for machine learning models, it may be +part of the reason why the feedback path exists in the brain. +Figure 9. Gaussian noise with zero mean and standard deviation +σ = 2.0 is added to networks with and without feedback. The +top row shows input instances to the network, the middle and bot- +tom row show reconstructions of the network without and with +feedback (respectively). +4.2. CIFAR-10 +The CIFAR-10 dataset is composed of 32×32 pixel colour +images of various objects, split into 50000 training and +10000 test instances. Each image belongs to one of the fol- +Figure 10. Gaussian noise with zero mean and varying standard +deviations (horizontal) is added to networks with and without +feedback. The quality of the reconstruction, as measured by the +loss function (vertical axis), with respect to the magnitude of the +standard deviation is shown for both networks. +lowing classes: airplane, automobile, bird, cat, deer, dog, +frog, horse, ship, truck (Krizhevsky et al., 2014). +4.2.1. AUTOENCODER +Similarly to the MNIST experiments, an autoencoder is +trained on the CIFAR-10 dataset. Again, the architecture +consists of two encoding and two decoding layers. Con- +trary to MNIST, the encoding/decoding layers used here +are convolutional/transposed convolutional layers with 16 +5 × 5 filters. +As figure 11 shows, the autoencoder with feedback clearly +performs better than the one without it, although the differ- +ence between the two is not as pronounced as it is in the +MNIST experiments. +Curiously, if batch normalisation (Ioffe & Szegedy, 2015) +is used after the activation functions, feedback cannot im- +prove on the performance of the standard autoencoder. This +may suggest that somehow feedback and batch normalisa- +tion are interacting in such a way that the feedback is ren- +dered ineffective. +4.2.2. NOISY ACTIVATIONS +The experiment from section 4.1.4 is repeated on the +CIFAR-10 dataset. The network employed is the autoen- +coder without batch normalisation from the previous ex- +periment. +Since feedback increased the robustness to noise in the +MNIST autoencoder, the same behaviour would be ex- +pected here. However, as apparent in figure 13, the net- +work with feedback is much more sensitive to (even small +amounts of) noise than the one without feedback. +This may be an indication that the feedback learned by + +0.250 +Without feedback +Comprehensive feedback +0.225 +Comprehensive constant gain +0.200 + 0.175 +0.150 +0.125 +0 +2400 +4000 +00 +00 +# betches7210414a5965ahthoez225 +Without Feedback +With Feedback +2D - +15 +B80] +LD +0.5 - +0.D +i +2 +3 +4 +5 +6 +8 +standard deviationFeedback-Gated Rectified Linear Units +Figure 11. Test set loss of autoencoders with and without feed- +back on the CIFAR-10 dataset. Neither model makes use of batch +normalisation. +Figure 12. Test set loss of autoencoders with and without feed- +back on the CIFAR-10 dataset. Both models make use of batch +normalisation. +Figure 13. Gaussian noise with zero mean and varying standard +deviations is added to the CIFAR-10 autoencoders with and with- +out feedback. Although this is not apparent due to the scale of the +plot, the data for the network without feedback follows a similar +shape to the one with feedback. +the network is fundamentally different from the feedback +learned in the MNIST experiments, such that it has a com- +pounding effect on noise, rather than a rectifying one. +4.2.3. CLASSIFICATION +Classification on the CIFAR-10 dataset is performed using +a convolutional neural network. The network consists of +two convolutional layers with 64 filters of size 5 × 5, each +followed by a max pooling (Zhou & Chellappa, 1988) layer +with a 2×2 window and a stride of 2. The convolution and +pooling layers are followed by a fully connected layer (200 +units) and a softmax (Bridle, 1990) layer. Batch normali- +sation is applied after the pooling layers and dropout with +a rate of 0.5 is applied after the pooling and the fully con- +nected layers. +To test whether feedback can improve classification per- +formance, the network is trained with (comprehensive) and +without feedback. Figure 14 shows only a marginal per- +formance difference between the two networks, with the +feedback network being slightly better. At the end of train- +ing, the classification accuracy over the complete test set is +about 0.7% higher for the network with feedback. +Note that the network employed here makes use of batch +normalisation, which, as shown in the previous sec- +tion, may be problematic in combination with feedback. +Whether this is the case here is not clear, since this particu- +lar network does not converge when batch normalisation is +disabled (be it with or without feedback). +5. Conclusion +The feedback mechanism presented here is able to im- +prove performance of conventional networks both in terms + +Without feedback +0.10 +With feedback +0.08 +0.04 +0.02 +0 +1400 +2400 +3000 +4000 +50i00 +# betchesWithout feedback +0.10 - +With feedback +0.08 +0.D2 +0.D0 +DOZ +4000 +8000 +1000 +# babches0.35 +Without feedback +0.30 +With feedback +0.25 +0.20 +B80] +0.15 +0.10 +0.05 +0.00 +0.00 +0.02 +0.04 +0.05 +0.08 +0.10 +standard deviationFeedback-Gated Rectified Linear Units +Figure 14. Classification loss on the CIFAR-10 test set. The train- +ing time of 200000 batches corresponds to 512 epochs. +of convergence speed and performance of the trained net- +work when applied on the MNIST dataset. The benefits +of feedback are less clear, however, when applied on the +CIFAR-10 dataset. In principle, an autoencoder with feed- +back can outperform a corresponding autoencoder without +feedback to a small degree, but this positive effect of feed- +back is negated when batch normalisation is utilised in the +autoencoders. Understanding this unfavourable interaction +between feedback and batch normalisation may be an op- +portunity to gain a deeper understanding on how feedback +works and what role it fulfils. +Feedback appears to have some positive effect when per- +forming classification on CIFAR-10, although this effect +is so small that drawing any firm conclusions seems ill- +advised. +When investigating the networks robustness to noise, an +even larger divide between performance on MNIST and +CIFAR-10 can be observed. On CIFAR-10, feedback is not +only not beneficial, it actually heavily increases the net- +work’s sensitivity to noise, while the MNIST autoencoder +becomes more robust when feedback is present. +A possible explanation for this difference across datasets +could be that the effectiveness of the feedback mechanism +is data-dependent, i.e. it may be leveraging the highly regu- +lar structure of the MNIST dataset and is thus not as useful +on the less regularly structured CIFAR-10 dataset. +A further general difference between the experiments on +the two different datasets is the use of convolutional lay- +ers, which were used in all of the CIFAR-10 experiments, +but not in any of the MNIST experiments. It may be that +providing feedback on a filter-wise basis is too simplistic, +or that some other aspect related to convolution is not con- +ducive to the feedback mechanism. Further research on the +combination of feedback and convolutional networks may +lead to some configuration that allows for more clear bene- +fits of feedback. +Naturally, it might also be the case that the results on +MNIST are merely an outlier, which somehow defies a +more fundamental problem with the usage of feedback in +current ANNs, e.g. it may be that backpropagation is not +an ideal learning algorithm for feedback, or that feedback +relies on more realistic models such as spiking neural net- +works (Ghosh-Dastidar & Adeli, 2009). +Should clear evidence arise that feedback is useful beyond +MNIST, an interesting avenue of future research would be +the creation of feedback based multi-modal models, where +sensory inputs from multiple different sources are com- +bined to perform e.g. a classification task. For instance, +if a network receives both visual and auditory input, the +barking of a dog may result (mediated by feedback) in a +higher expectation to observe a dog in the visual input. +Acknowledgements +I want to thank Kurt Driessens, Mario Senden, and Alexan- +der Kroner for their supervision during this project. +References +Bridle, John S. Probabilistic interpretation of feedforward +classification network outputs, with relationships to sta- +tistical pattern recognition. In Neurocomputing, pp. 227– +236. Springer, 1990. +Brodmann, Korbinian. +Vergleichende Lokalisationslehre +der Grosshirnrinde in ihren Prinzipien dargestellt auf +Grund des Zellenbaues. Barth, 1909. +DeFelipe, Javier and Fari˜nas, Isabel. The pyramidal neu- +ron of the cerebral cortex: Morphological and chemical +characteristics of the synaptic inputs. Progress in Neu- +robiology, 39(6):563–607, 1992. +Faisal, A. Aldo, Selen, Luc P. J., and Wolpert, Daniel M. +Noise in the nervous system. +Nature Reviews Neuro- +science, 9(4):292–303, 2008. +Ghosh-Dastidar, Samanwoy and Adeli, Hojjat. +Spiking +neural networks. International journal of neural systems, +19(04):295–308, 2009. +Goodhill, Geoffrey J and Carreira-Perpi˜n´an, Miguel ´A. +Cortical columns. +Encyclopedia of cognitive science, +2002. +Hinton, Geoffrey, Deng, Li, Yu, Dong, Dahl, George E, +Mohamed, Abdel-rahman, Jaitly, Navdeep, Senior, An- +drew, Vanhoucke, Vincent, Nguyen, Patrick, Sainath, +Tara N, et al. Deep neural networks for acoustic mod- +eling in speech recognition: The shared views of four +research groups. IEEE Signal Processing Magazine, 29 +(6):82–97, 2012. + +225 +Without feedback +With feedback +200 +175 +150 - +125 +1D0 +0.75 +0.50 +0 +25000 50000 75000 140000125000150000175000 240000 +# betchesFeedback-Gated Rectified Linear Units +Hochreiter, Sepp and Schmidhuber, J¨urgen. 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Connectionism in perspective, pp. 143–155, 1989. +LeCun, Yann, Cortes, Corinna, and Burges, Christo- +pher JC. Mnist handwritten digit database. AT&T Labs +[Online]. Available: http://yann. lecun. com/exdb/mnist, +2, 2010. +Markram, Henry, Toledo-Rodriguez, Maria, Wang, Yun, +Gupta, Anirudh, Silberberg, Gilad, and Wu, Caizhi. In- +terneurons of the neocortical inhibitory system. Nature +Reviews Neuroscience, 5(10):793–807, 2004. +Nair, Vinod and Hinton, Geoffrey E. Rectified linear units +improve restricted boltzmann machines. In Proceedings +of the 27th international conference on machine learning +(ICML-10), pp. 807–814, 2010. +Shipp, Stewart. Structure and function of the cerebral cor- +tex. Current Biology, 17(12):R443–R449, 2007. +Srivastava, Nitish, Hinton, Geoffrey E, Krizhevsky, Alex, +Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: a +simple way to prevent neural networks from overfitting. +Journal of machine learning research, 15(1):1929–1958, +2014. +Sutskever, Ilya, Vinyals, Oriol, and Le, Quoc V. +Se- +quence to sequence learning with neural networks. In +Advances in neural information processing systems, pp. +3104–3112, 2014. +Zhou, YT and Chellappa, R. Computation of optical flow +using a neural network. +In IEEE International Con- +ference on Neural Networks, volume 1998, pp. 71–78, +1988. + +Feedback-Gated Rectified Linear Units +6. Appendix +6.1. Hyperparameter Tuning +As mention in section 4.1, optimal values for βmax and η +are determined by a grid search. The initial grid is defined +by η = [5, 10, 15, . . . , 50] and βmax = [0.1, 0.2, . . . , 0.8]. +The highest value for βmax (0.8) consistently shows the +best performance regardless of η’s values, as exemplified +by figure 15. Note that a high constant value of η with +varying values of βmax will generally lead to less spread +between the loss curves, since the activation function will +be more sensitive to βmax when η is low. +Figure 15. Autoencoder performance with varying hyperparame- +ters. Top: η is fixed at 5 and βmax is varied, bottom: η is fixed at +50 and βmax is varied. +While higher values of βmax lead to better performance, +the inverse relationship can be seen with η, i.e. lower values +of η lead to better performance. This is illustrated in figure +16. +Figure 16. Autoencoder performance when βmax is fixed at 0.8 +and η is varied. +A second grid search with η = [1, 2, 3, 4, 5], βmax = +[0.8, 0.85, 0.9, 0.95] is performed to determine whether +even lower/higher values can further improve performance. +Indeed, increasing βmax to 0.95 leads to better perfor- +mance, but further decreasing η is not advantegeous. +6.2. Feedback-Controlled Threshold +Equation 1 describes not only gain modulation through +feedback, but also an adjustment of the activation functions +threshold, i.e. αµD is one of the terms in the summation. +While gain modulation is the main property of interest in +this paper, it is conceivable that the change in threshold +plays a significant part in this mechanism as well. +Incorporating this threshold mechanism into equation 6 +leads to: +f = +max(0, µS + αµD) +1 − min( βmax +η +µD, βmax) +(8) +where α is a parameter to be learned by the network. While +α could also be set to a constant (tuned) value, prior exper- +iments suggest that it is beneficial to let the network adjust +alpha during the course of training. +As can be seen in figure 17, the added threshold mecha- +nism is not able to improve upon the network implementing +the gain mechanism. Although the models with feedback- +controlled threshold both perform better than the standard +autoencoder, the model with only gain and no threshold +mechanism still has the overall best performance. + +Eta: 5 +0.26 - +betamax: 0.1 +0.24 +betamax: 0.2 +betamax: 0.3 +0.22 +-betamax:0.4 +betamax: 0.5 +0.20 +betamax:0.6 +betamax: 0.7 +betamax: 0.8 +0.16 +0.14 +0.12 +OT'O +DOEZ +4000 +DOt8 +# betchesEta: 50 +0.26 - +betamax: 0.1 +0.24 - +betamax: 0.2 +betamax: 0.3 +0.22 +betamax: 0.4 +betamax: 0.5 +0.20 +betamax: 0.6 +betamax: 0.7 +betamax: 0.8 +0.16 +0.14 - +0.12 +O1O +0 +DOEZ +4000 +00 +DOt8 +# betchesBeta max: 0.B +0.26 +Eta: 5 +Eta: 10 +0.24 +Eta: 15 +0.22 +Eta: 20 +Eta: 25 +0.20 +Eta: 30 +Eta: 35 +Eta: 40 +0.16 +Eta: 45 +Eta: 50 +0.14 +0.12 +OT'O +0 +DOZ +4000 +00 +DOt8 +# bebchesFeedback-Gated Rectified Linear Units +Figure 17. Performance of the standard autoencoder, an autoen- +coder with feedback-controlled threshold, an autoencoder with +feedback-controlled gain, and an autoencoder with both feedback- +controlled threshold and gain on the MNIST test set. +6.3. Input With Reduced Contrast +Images with reduced contrast are presented to the trained +(on regular contrast images) network, to see if the second +pass can reconstruct an image that is more akin to a regular +contrast image. To reduce the contrast, each pixel of the +image is multiplied by some contrast factor 0 ≤ c ≤ 1. +Figure 19 shows the absolute difference in mean pixel value +between the first and second pass reconstructions for a +number of different contrast factors. A high contrast in- +put image leads to a larger difference in mean pixel value, +while a low contrast image leads to a smaller difference +between first and second pass reconstructions. +Figure 18. Absolute difference in mean pixel value between first +and second pass reconstructions as a function of different contrast +factors (from 0.0 to 1.0 in 0.1 increments). A contrast factor of +1.0 corresponds to no reduction in contrast, while a contrast factor +of 0.0 means the input images are entirely black. +Figure 19. From top to bottom: original image, contrast reduced +image, first pass reconstruction, second pass reconstruction. The +contrast reduced image was produced by multiplying the original +image with a contrast factor of 0.5, i.e. each pixel in the con- +trast reduced image has values in the range [0.0, 0.5] instead of +[0.0, 1.0] +6.4. Additional Figures +The following figures contain additional data that was col- +lected as part of the experiments in section 4. + +Test loss +Standard AE +0.250 +Only threshold +Only Gain +0.225 +Threshold + Gain +0.200 +0.150 +0.125 +0.100 +0 +2500 +5000 +7500 +10000 1250015000 17500 24000 +# bebches0.030 +2nd pa +0.025 +1st & +0.020 +betw. +0.D15 +Difference +0.D10 +0.D05 +0.0O0 +0.D +t0 +0.6 +0.B +1D +conbrast fectarFeedback-Gated Rectified Linear Units +Figure 20. Visualisation of activations in the MNIST autoencoder +for one particular test instance. The leftmost column corresponds +to the input layer and the remaining columns correspond to the +first encoding layer, the second encoding layer, the first decoding +layer, and the second decoding layer, respectively. The number of +rectangles in each column corresponds to the number of units in +that layer. Larger values are represented by green coloured rect- +angles, and smaller values by white ones. Top: first pass, bottom: +second pass. +Figure 21. T-SNE visualisation of the second encoding layer of +the autoencoder over the whole MNIST test set. From top to bot- +tom: first pass, second pass, first pass with noise (as described in +section 4.1.4), second pass with noise. + +50 +25 +0 +25 +50 +75 +100 +75 +50 +-25 +0 +25 +5050 +25 +0 +25 +50 +75 +60 +40 +-20 +0 +21 +4025 ++ +25 +50 +75 +60 +40 +-20 +0 +24 +40 +824 +0 +-20 +40 +80 +80 +60 +40 +-20 +0 +2 +40 +84Feedback-Gated Rectified Linear Units +Figure 22. Histograms as seen in section 4.1.3, but for the autoencoder with comprehensive feedback. + +1 +Ho +1 - min(e μo.βmax +COIDODE +1250001 +2500000 +Encoder +OADOSE +150000 +ODOS +CODO +2500D0 +500000 +0 +0 +2 +4 +6 +14 +15000 +000 +12500 +Encoder +7500 +40000 +5000 +2500 +DO +5 +15 +253035 +40 +2 +3 +4 +5 +7 +9 +ONDO +CODODE +240000 +DOIDOEL +0 +500 +400 +DOE- +200 +DOL- +0 +0. +0 +4 +6 +8 +1Feedback-Gated Rectified Linear Units +Figure 23. Different gain values are manually fed into the second encoding layer of the network and the resulting reconstruction is +visualised. In each of the above images, one specific input image is presented to the network, but the gain is varied. In row i of each +image, every unit of the second encoding layer receives a gain of 10, except for unit i, which receives a gain between 0 and 10, depending +on the column it is in. When using the MNIST autoencoder with comprehensive feedback, it can be observed that only one unit in the +second encoding layer has any variation in gain (the remaining ones have a constant gain of 10 regardless of the input). This one unit +corresponds to the fourth row from the bottom of each image and seems to be responsible for setting the ‘intensity‘ of the reconstruction. + +77777777 +7777777 +777777 +777777 +7 +7 +7 +7 +7 +Z +7777777 +7 +7 +7777777 +7 +7 +777777777 +7 +7777777777722222271 +222222222 +3333222222 +ZZZZZEEEE1 +3 +222222222 +222222222 +22222222222hhhhhhhhhhh +hhhtttt +4 +44444 +hhhhh +4 +hhhhh +4 +4 +4 +4 +hhhhhbbbt +44444444444G +G +799977 +G6666665555Feedback-Gated Rectified Linear Units +Figure 24. CIFAR-10 classification as seen in section 4.2.3. From +top to bottom: test set accuracy, training set loss, training set accu- +racy, training set loss after applying a moving average filter (win- +dow size 100). + +0.B +0.7 +0.6 - +0.4 +0.3 - +0.2 +Without feedback +0.1 +With feedback +0 +25000 50000 75000 140000125000150000175000240000 +#betches225 +Without feedback +With feedback +200 +175 +150 +LDO - +0.75 +0.50 +0 +25000 50000 75000 140000125000150000175000240000 +# babches0.9 +0.B +0.7 - +0.6 +0.4 +0.3 +0.2 +Without feedback +0.1 +With feedback +0 +25000 50000 75000 140000125000150000175000240000 +# babches13 +Without feedback +12 +With feedback +11 +LD +0.9 +0.B +0.7 +0.6 +0 +75000 140000 125000 150000 175000 +# betches \ No newline at end of file diff --git a/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/load_file.txt b/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fcac5c88cecb26e904541cc5913842339452d2f8 --- /dev/null +++ b/3NE0T4oBgHgl3EQfuwHF/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf,len=632 +page_content='Feedback-Gated Rectified Linear Units Marco Kemmerling 1 Abstract Feedback connections play a prominent role in the human brain but have not received much attention in artificial neural network research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Here, a biologically inspired feedback mecha- nism which gates rectified linear units is pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' On the MNIST dataset, autoencoders with feedback show faster convergence, better perfor- mance, and more robustness to noise compared to their counterparts without feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Some benefits, although less pronounced and less con- sistent, can be observed when networks with feedback are applied on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Introduction The brain has served as inspiration for artificial neural net- works (ANNs) for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' While these models are usually heavily simplified compared to the brain, they have seen significant successes in areas such as image recognition (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=', 2012), speech recognition (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=', 2012), and machine translation (Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=', 2014) in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Despite successes, it is clear that the average human brain is vastly more powerful and versatile than any model used in practice today, and as such it may be useful to investigate how and where exactly the brain and ANNs differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' One such discrepancy between ANNs and the brain is the existence of feedback, or top-down connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' While there is clear evidence of prominent feedback connections in the brain, ANNs have overwhelmingly been designed based on the feedforward paradigm, although networks that do not work solely on the feedforward principle exist and are called recurrent neural networks (RNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Most RNNs used in practice today focus on recurrent connections from one layer to itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' LSTM networks (Hochreiter & Schmidhuber, 1997)), which, while recurrent, arguably do not constitute top-down connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' These networks are 1University of Maastricht, Maastricht, The Nether- lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfuwHF/content/2301.02610v1.pdf'} +page_content=' Correspondence to: Marco Kemmerling enc.ssrc : 0;} +(a) Source rtp sess ssrc function +ulong rtp_sess_ssrc(long param_1){ +uint local_14 ; +if (param_1 == 0){ +local_14 = 0; +} else { +local_14 = * (uint *) (param_1 + 4);} +return (ulong) local_14; +} +(b) Decompiled rtp sess ssrc function +ulong FUN_00100d30 ( long param_1 ){ +uint local_14 ; +if (param_1 == 0) { +local_14 = 0 ; +} else { +local_14 = * (uint *) (param_1 + 4);} +return ( ulong ) local_14 ;} +(c) Stripped decompiled rtp sess ssrc function +Fig. 1: Example source, decompiled and stripped code snippet +many of the variable types and failed to identify the struct +datatype. +Using our trained BinT5 model we can summarise the +decompiled code and generate the following summary: Get +the source for an RTP/RTCP Socket. This summary gives us +an indication of the purpose of the function. Integrating this +generated summary into Ghidra increases the readability of +the entire binary. Keep in mind that a reverse engineer has +to understand not just this function, but hundreds of different +functions in a single binary. +C. Stripping +Aside from compiling with higher optimisation levels, bi- +naries can also be stripped to obfuscate the underlying code +and to resist analysis [24]. Commercial off-the-shelf software +is often stripped to reduce the memory and storage footprint +of the binaries, and to resist analysis to protect the intellectual +property of the creator. Many vulnerable and malicious bina- +ries are, unfortunately, also stripped to resist security analysis +and hide their faults [5]. +Unix and Unix-like operating systems include a strip utility. +The strip utility removes any operands that are not nec- +essary for the execution of the binary while ensuring that +the execution of the binary remains unchanged. The exact +implementation and what constitutes unnecessary operands are +left to the implementor.9 The strip utility as implemented in +GNU/Linux removes the symbol table from the binary. The +symbol table contains each symbol’s location, type and name. +Like higher optimisation levels, the use of stripping can +greatly complicate the efforts to reverse engineer a binary, +as well as reduce the accuracy and effectiveness of reverse +engineering tools [24]. +For example, we compile, strip and decompile the function +in Figure 1a, and the resulting stripped decompiled function +is shown in Figure 1c. In addition to the details lost by the +decompilation process, the stripper removed all symbols, like +the function names. +D. Code Summarisation Task: +Code summarisation (also referred to as source code sum- +marisation) is the task of writing short descriptions from +source code, usually a single-sentence summary of the source +code. The main use is for software documentation, like the +one-sentence JavaDoc description used in Java [19]. This +documentation is important for program comprehension and +maintenance. But the process of writing and maintaining +these descriptions is a labour-intensive and time-consuming +task, which is where the benefits of automating that process +arise. Automatic code summarisation is an active and popular +research problem in the field of software engineering [19]. +E. Transformer-based Models +Transformers were originally proposed by Vaswani et al. +as a sequence-to-sequence architecture [25]. Unlike the Re- +current Neural Networks [26] (RNN), the Long Short-Term +Memory [27] (LSTM) variant of RNNs [26] and Convolutional +Neural Networks [28] (CNN), Transformers only use a mecha- +nism called self-attention to capture dependencies between the +input and output. The current state-of-the-art NLP models for +programming languages such as CodeT5 [14], CodeBERT [15] +and PolyGlotCodeBERT [16] are all based on the Transformer +architecture [25]. +F. Transfer Learning +Pre-trained Transformers-based language models, such as +RoBERTa [29], CodeBERT [15] and CodeT5 [14] utilise +a pre-train then fine-tune paradigm. The bespoke paradigm +was initially introduced by Kenton and Toutanova. In this +paradigm, the models are first trained in an unsupervised +manner on a large unlabelled dataset. These pre-trained models +can then be fine-tuned to perform a more specialised task, +such as summarisation. Transfer learning uses the knowledge +that is obtained in one task to solve a different task. It +allows the creation of general models that are trained once +on massive datasets. These general models, which contain +general domain knowledge can then be fine-tuned for a specific +downstream task. This approach is quicker and requires less +training data than training a model on the downstream task +from scratch [30]. +9strip: https://pubs.opengroup.org/onlinepubs/7908799/xcu/strip.html +3 + +Source +Code +Compilation +Decompilation +Decompiled + +Stripping +Decompilation +Function +Extraction + +Comment +Alignment +Comment +Alignment +Stripped +Comment +Extraction +Demi-Stripped +Comment +Alignment +Demi +Stripping +Fig. 2: Data Collection Pipeline +III. CAPYBARA DATASET +We require a dataset of decompiled functions labelled with +a descriptive summary to create and assess our solution. This +dataset should be relatively large to suit the ‘data-hungry’ +nature of deep-learning models. Furthermore, the dataset needs +to feature a diverse set of data representative of our solution’s +actual real-life use case. +A. Data Collection +To create such a large and diverse dataset we made use +of BinSwarm [7], an existing dataset of aligned decompiled +and stripped decompiled functions10. BinSwarm collects C- +based projects from Github. The projects are filtered to only +include those that are actively being developed, using Travis +CI and built for Ubuntu Linux. The projects are built using +Docker. The resulting binaries are then copied and stripped, +and both the stripped and unstripped binaries are decompiled +using Ghidra. The functions are extracted from the stripped +and unstripped decompiled code and aligned with the source +code. The BinSwarm dataset only contains aligned tuples of +source code and (stripped-) decompiled functions. We extract +documentation from the original source code files to add +descriptive comments to this dataset. To that end, we depend +on the documentation included in the source code by the +original authors in the form of single and multiline comments. +We locate the functions in the unbuilt project files and align the +decompiled functions with the comments in the source code +using srcML11 to extract any documentation located directly +before a function signature. A high-level overview of the entire +process is shown in Figure 2. +A function’s documentation often also contains other details +besides the descriptive summary. We found that C projects +do not follow a single documentation standard. For example, +Javadoc for Java has a short one-line description or summary +for each method at the beginning of the multiline comment +10BinSwarm: https://hub.docker.com/r/binswarm/cbuilds +11srcML: https://www.srcml.org/ +/** @brief Select the source of Microcontroller +Clock Output +�→ +* Exact sources available depend on your target. +* On devices with multiple MCO pins, this function +controls MCO1 +�→ +* @param[in] mcosrc the unshifted source bits +*/ +Fig. 3: Example of documentation from jeanthom/ DirtyJTAG: +rcc set mco +block. In C, there is no singular documentation standard, so +there might not be a single-line summary, and we will need +to locate it in the comment block automatically. +a) Summary Extraction Rules: We observe that the ma- +jority of single-line data are descriptive summaries, so we +extract the first sentence. We identify many documentation +styles in our multi-line data, we define some automated rules +to extract summaries from the documentation: +• @brief or @purpose: If the documentation contains a +‘@brief’ or ‘@purpose’ tag, we extract the first sentence +after the tag. The ‘brief‘ tag is part of the Doxygen docu- +mentation standard12, an example is shown in Figure 313. +• Description: If the documentation contains a line with +‘Description:‘, we extract the following sentence. +• @param or @v: Documentation that contains an ‘@v’ +or ‘@param’ tag, usually has a summary in the sentence +before the tag. We extract that sentence. +b) Filtering Rules: To improve the quality of the dataset +we filter out samples based on the rules used by the Code- +SearchNet dataset [20] included in the CodeXGlue benchmark +for the summarisation task [31]: +• Documentation length: We remove any summaries that +are too long or too short and remove anything shorter +than 3 or longer than 256 tokens. +• Special tokens: We follow the example of the Code- +SearchNet [20] and remove all documentation that con- +tains special tokens. We scan for web tokens (like +‘http://’), HTML tokens (like ‘’), paths (like +‘C://Users/..’), since this documentation usually refers to +external resources. We additionally filter any developer +tokens (like ‘FIXME:’), as these documents do not pro- +vide meaningful information about the function itself, but +contain comments about the development process. +• Language: We filter out any documentation that was not +written in English using the FastText language identifica- +tion algorithm [32]. Around 92.19% of the documentation +is in English. +• Empty documentation: We find that a large number of +functions did not have any documentation associated with +them at all. We simply remove these samples from the +dataset. +12Doxygen:https://doxygen.nl/manual/docblocks.html +13jeanthom/DirtyJTAG:rcc set mco:https://gitlab.com/ +insane-adding-machines/unicore-mx/-/blob/master/lib/stm32/common/ +rcc common all.c#L192 +4 + +GCC000• Abstract Syntax Tree: The authors of the CodeSearch- +Net dataset [20] additionally, remove any samples that do +not parse into an AST. We choose to omit this step since +all of our samples have been successfully compiled and +have thus at one point been parsed into an AST by the +compiler. +B. Dataset Preparation +a) Synthesis of Demi-stripped Code: From the dataset +of decompiled functions, we also create another dataset. We +emulate the process of stripping by removing all the iden- +tifiers from the decompiled code and replacing them with +placeholders. For clarity, we call this demi-stripped data. Like +the stripped dataset, the identifiers are all removed, but this is +only done after the decompilation process. The decompiler still +had access to the identifiers and could use the symbol table +during decompilation. Most importantly, this demi-stripped +dataset still has the same structure and control flow as the +unstripped decompiled dataset and avoids any decompilation +issues arising from stripping. +b) Data Split: The dataset is split into a train, test and +validation set. These sets constitute approximately, 80%, 10% +and 10% +[19] of the complete dataset. As recommended +by Shi et al. and LeClair and McMillan, we prevent leakage +of vocabulary and code patterns between the sets, by sampling +the sets in a cross-project manner [13, 19]. This means that an +entire project gets assigned to one of the sets, and functions +from the same project cannot be assigned to different sets. The +projects in the test and validation set are the same across all +datasets. +c) Duplication: Large corpora of code, like the cor- +pus gathered by BinSwarm, tend to have a high degree +of duplication [19]. As a result, snippets of code that are +relatively unchanged appear in multiple parts of the corpus. +This can be in the form of copied, generic or auto-generated +functions. These functions will appear in multiple repositories +and might be duplicated across the training and testing data. +Besides exact duplicates, near-duplicates can also occur. Near- +duplicates differ in a few minor aspects like additional code +comments or different function names. While removing exact +duplicates is relatively fast and straightforward, removing +near-duplicates is much more challenging and computationally +intensive [33]. The issue with code duplication in classical +code summarisation is that the models and tools are supposed +to be used to generate summaries for new and unseen code. +The evaluation metrics should therefore measure the gener- +alisation of the tool on new samples [33]. Duplicates and +near-duplicates are not defined as new samples. A user of +such a tool could simply look these samples up. Furthermore, +large, high-capacity models like CodeT5 with 220M [14] or +CodeBERT with 128M [15] parameters, have a large capacity +to memorise duplicated code [33]. +However, the use case outlined in this work is more akin +to deobfuscation. As explained by Allamanis, deobfuscation +could be a use case where duplicates are valid and part of the +true distribution of the problem [33]. Compiled code contains +0 +200 +400 +600 +800 +1000 +1200 +1400 +Token count +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +Density +Source code +Decompiled code +Fig. 4: Tokens in source C and decompiled code +a lot of duplicate code, and understanding this code is still +difficult and essential for understanding the binary. While +regular source code allows the reader to look up code snippets, +decompiled binaries have an additional obfuscation applied. +We, therefore, focus on the model’s performance on code +with duplicates as we believe duplicates to be part of the true +distribution of the data, but we also report the deduplicated +results. +C. Dataset Properties +Table I shows the size of the processed dataset. Of the 2.1M +aligned decompiled functions, we extract documentation for +215k of them, and we found that the majority of samples, 1.5M +did not have any documentation at all. Furthermore, BinSwarm +only provided us with 415k aligned stripped samples, and we +can extract documentation for only 14k of these samples. +Dataset +Including duplicates +Deduplicated +C/Demi/Decom +214,587 +79,673 +Stripped +14,245 +7,826 +TABLE I: Number of functions in dataset +The vast majority of documentation is in the form of +multi-line comments as opposed to single-line or double-slash +comments. We found that the documentation and comments +had a mean length of 42.60 and 8.14 tokens, respectively. +Figure 4 shows the distribution of the number of tokens in +source code and decompiled code. The source and decompiled +code have a mean length of 399 and 779 tokens, respectively. +Figure 5, shows that decompiled code also has close to double +the LOC of source code, with means of 30.77 and 53.42 lines +for source and decompiled, respectively. +The majority of decompiled functions are compiled with +optimisation level -O2, with a similar number of -O1 and - +O3 samples and relatively few -O0 samples. Stripped data has +a very even distribution of optimisation levels, with only - +O0 having significantly fewer samples. Note that there are +more optimisation levels than shown in Figure 6, for brevity +the different levels are grouped into their base optimisation +level. -Oa is grouped with -O0, -Of and -Og are grouped +with -O1, -Os is grouped with -O2. We also observe some +5 + +0 +25 +50 +75 +100 +125 +150 +175 +200 +LOC +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Density +Source code +Decompiled code +Fig. 5: LOC in source C and decompiled code +0 +1 +2 +3 +Optimisation level +0 +20000 +40000 +60000 +80000 +100000 +120000 +Decompiled +0 +1000 +2000 +3000 +4000 +Stripped +Decompiled +Stripped +Fig. 6: Distribution of optimisation levels in decompiled (left) +and stripped (right) +samples with an optimisation level higher than -O3 (-O8 and +-O7), as specified by the GCC documentation, these levels are +equivalent to -O314. +IV. BINT5 +We select CodeT5 [14] as the base-model for our experi- +ments since it is the highest-scoring publicly-available model +on the CodeXGLUE [31] Code Summarisation benchmark15. +CodeT5 is a programming language model built on the T5 +(Text-to-text Transfer Transformer) architecture [34] and pre- +trained on a mix of supervised and unsupervised tasks. CodeT5 +employs an encoder-decoder architecture. In contrast to other +models, CodeT5 is trained using both unimodal (PL only) and +bimodal (NL-to-PL) tasks in eight programming languages. +This bimodal training allows CodeT5 to perform strong cross- +modal tasks such as code summarisation and code generation +(PL-to-NL). Many other models only use the data and lan- +guages included in the CodeXGlue dataset [15, 16, 31], while +CodeT5 also uses a mined dataset of C and C++ code for +its pre-training objectives [14]. The inclusion of C training +data should help the model with the CAPYBARA dataset. +There could be some overlap in the training data between +CAPYBARA and the dataset used by Wang et al. which would +cause leakage, we address these concerns in Section VII. +14GCC optimisation levels: https://gcc.gnu.org/onlinedocs/gcc-4.4.2/gcc/ +Optimize-Options.html#Optimize-Options +15CodeXGLUE benchmark: https://microsoft.github.io/CodeXGLUE/ +Fine-Tuning +Base +Model +BinT5 +Evaluation +Results +CAPYBARA +training & validation data +CAPYBARA +test data +Fig. 7: BinT5 fine-tuning pipeline +CodeT5 also utilises the transfer learning paradigm, which +allows us to train the model with relatively little data. In +this case, we make use of the CodeT5-base model, which +was trained on mixed upstream tasks by the authors [14]. An +overview of how we applied the model to create BinT5 is +provided in Figure 7. +V. EXPERIMENTAL SETUP +To assess the effectiveness of our approach, we first evaluate +the performance of the model, we then identify the aspects of +the data that make this task inherently difficult, and we finally +investigate aspects of the datasets and their influence on the +complexity of the task. +A. Research Questions +In the context of the study, we thereby formulate the +Research Questions (RQ) as follows. +RQ1: How effective are fine-tuned Transformer-based models +at decompiled code summarisation? To investigate the +application of existing models to binaries using CAPY- +BARA, we set a baseline by training a model on the code +summarisation task on the source C-code dataset. We then +train a summarisation model on both the decompiled and +the stripped dataset. We use the evaluation metrics to +compare the performance of the different models. +RQ2: Which aspects of the input contribute most to model per- +formance? We investigate which aspects of decompiled +code increase the difficulty of the task. We, therefore, +look at the impact of the symbol table on decompilation, +for this, we fine-tune a model on the demi-stripped dataset +and compare it to the other models. We also investigate +the importance of the function name by removing just the +function name from the decompiled code. Furthermore, +we investigate the impact of the optimisation level by +exploring the performance per optimisation level. +RQ3: What is the impact of dataset properties on model per- +formance? We finally investigate how the construction +of CAPYBARA influences the models. To answer the +final research question we remove the duplicates from +the datasets and retrain the models, after which we +compare the performance to the baselines. Furthermore, +we investigate the impact of dataset size, by incrementally +reducing the size of the training sets. +6 + +"Summarize Python: def inc_value(x): +"increment value" +"Generate Python: increment value' +'def inc_value(x)... +CodeT5 +"Defect: if x=0: x += 1 +"true" +"Refine: if x=0: x += 1' +"if x == 0: x += 1" +"Translate Python to C: if x==O: x += 1 +"if (x==0) {x += 1;}"B. Baselines +To first establish a performance baseline, we train a CodeT5- +base model on the summarisation task on source C. Note +that only samples which are aligned with decompiled code +are included in the source C dataset. The baseline is used to +compare the decompiled C, stripped decompiled C and the +demi-stripped datasets to the source code. +C. Evaluation Metrics +We evaluate the performance between the reference sum- +mary from CAPYBARA and the candidate summary produced +by BinT5 using the EM, BLEU-4 [35], ROUGE-L [36] and, +METEOR [37] metrics. +a) Exact Match (EM): The simplest metric is the EM +which scores a prediction one if it matches its reference exactly +and zero otherwise. +b) BLEU-4: The most widely used metric in the code +summarisation task is the Bilingual Evaluation Understudy +Score (BLEU) [13]. BLEU-4 produces a percentage number +between 0 and 100, which defines the similarity between a +candidate and a set of reference sentences. BLEU-4 calculates +the cumulative 4-gram precision scores, the number of match- +ing 4-grams divided by the total number of 4-grams in the +candidate sentence [35]. The unigrams and bigrams account +for the adequacy of the candidate while the longer three and +4-grams account for fluency. To prevent short sentences the +result is multiplied by a brevity penalty as well. A smoothing +function is applied to prevent sequences with no matching 4- +grams to score zero [38]. While Shi et al. recommend BLEU-4 +with smoothing method 4 [13], we opted to use the Moses [39] +implementation of BLEU-4 which uses smoothing method 2 +since this is also utilised by CodeSearchNet, CodeXGlue and +CodeT5 [14, 20, 31]. +c) ROUGE-L: ROUGE or Recall-Oriented Understudy +for Gisting Evaluation, is a package which includes several +metrics, the most popular among them is ROUGE-L [36]. +ROUGE-L is more recall oriented than BLEU-4. ROUGE-L +simply finds the longest common subsequence (LCS) between +the reference and the candidate. Note that the words do not +need to be consecutive but they have to be in order. +d) METEOR: METEOR or Metric for Evaluation for +Translation with Explicit Ordering [37] uses word lists and +stemming to also take synonyms into account and calculates +the harmonic mean of the unigram precision and recall. Similar +to ROUGE-L, METEOR is more recall-focused. METEOR has +a higher correlation with human judgement than BLEU-4 [19] +at the sentence level. +D. CodeT5 finetuning and testing +The concept of transfer learning, which is utilised in BinT5, +depends on the use of a fine-tuning step to train the pre-trained +model on the downstream task. We fine-tune a pre-trained +CodeT5-base model on the constructed dataset. The model is +trained on the summarisation task as defined in the model. We +train the model on the train set, then evaluate it after every +epoch on the validation set and finally test on the test set. +During training, we measure the model performance using the +BLEU-4 metric. +E. Data deduplication +To create a deduplicated version of the CAPYBARA +dataset we make use of a fork16 of the near-duplicate-code- +detector [33]. We use this tool to compare all the datasets’ +functions and find clusters of near-duplicate functions. We +randomly select one function per cluster and discard the rest +from the dataset. We use the standard tool configuration as +recommended by Allamanis. Of the removed duplicates, we +observe that a relatively large number originates from common +libraries, such as SQLite17, that are packaged with binary +programs. Thus a certain amount of duplication is also likely +to occur “in the wild”. +F. Configuration +We process and visualise the data with Pandas 1.4.3 and +Ghidra 10.0.418. FastText 1.0.3 with the largest lid.176.bin +model is used to detect languages. We train the model using +Transformers version 4.16.2 running on Torch 1.9.0+cu111 in +the nvidia/cuda:11.4.0-base docker container image. We share +a Docker image with all the libraries required to run BinT5 +pre-installed on DockerHub19. +A grid search of the optimal settings was infeasible from a +time perspective, so we performed training mainly using the +recommended settings from the CodeT5-base model [14]. We +double the source length for the decompiled, stripped, and +demi-stripped code to 512 tokens instead of the standard 256 +tokens used for the source code to compensate for the fact +that the average length of decompiled code is almost twice as +long as the source code. We trained the model on a machine +with an NVIDIA GeForce RTX3080 with 10GB of VRAM +and an AMD Ryzen Threadripper 3990X 64-Core Processor +with 192GB of RAM running Ubuntu 20.04.4 LTS. The GPU +is running Nvidia driver version 510.60.02 with Cuda 11.6. +The authors of CodeT5 used an NVIDIA A100 GPU with +40GB of VRAM for fine-tuning [14]. To compensate for the +lack of memory, we reduced the batch size to 2, which was the +maximum length that could still fit in the VRAM, we increase +the ‘gradient accumulation steps’ to 24 to still achieve the +effective standard batch size of 48. +VI. RESULTS +We present the results of our experiments to answer the +research questions, results are grouped per research question. +The metrics are calculated for each sample from the test set, +and the average scores are presented. +A. RQ1: Model Effectiveness +The performance of the CodeT5-base model on each of the +datasets is presented in table II. +16Near +Duplicate +Code +Detector: +https://github.com/SERG-Delft/ +near-duplicate-code-remover +17SQLite: https://www.sqlite.org/index.html +18It is not recommended to use Ghidra versions before 10.1 since these +versions have not been patched against a Log4J RCE +19BinT5 Docker Image: https://hub.docker.com/r/aalkaswan/bint5/tags +7 + +BLEU-4 +EM +METEOR +ROUGE-L +C +60.83 +52.19 +65.33 +66.51 +DecomC +58.82 +48.92 +63.14 +64.51 +Stripped +11.26 +1.85 +14.50 +17.25 +TABLE II: Result of fine-tuning CodeT5-base on mined +datasets +BLEU-4 +EM +METEOR +ROUGE-L +DecomC +58.82 +48.92 +58.4 +60.32 +Demi +44.21 +35.10 +47.89 +49.59 +NoFunName +46.99 +37.12 +45.92 +48.07 +TABLE III: Result of fine-tuning CodeT5-base on synthetic +data +We found that the decompiled code model generally pro- +duced good summaries, evidenced by the BLEU-4 score of +58.82, which is slightly lower than the baseline set by the +source code. The stripped model mainly produced unusable +summaries, as evidenced by the BLEU-4 score of 11. The +high EM score could be an indication of a high duplication +factor. +Initial experiments with GraphCodeBERT [40] and Poly- +glotGraphCodeBERT [16] base models fine-tuned on CAPY- +BARA show performance around 5 and 3 BLEU-4 lower, +respectively. This is a relatively small difference, especially +considering the model size. This shows that the performance of +BinT5 does not heavily depend on the additional pre-training +on C and C# performed by Wang et al.. Furthermore, this result +shows that it is improbable that significant dataset leakage has +taken place. +We found a relatively large difference between the number +of recovered decompiled and stripped decompiled functions. +This can likely be attributed to the fact that Ghidra struggles +a lot more with recovering stripped functions. Recall that the +symbol table commonly contains information regarding the +location and name of functions. When this table is dropped, +the start- and endpoints of functions are hard to infer by +automatic tools, especially since many functions get inlined, +and JUMP instructions replace CALL instructions. Aside from +difficulties in demarcating functions, it is also difficult to +align the associated source code function with the decompiled +function. With unstripped code, the function name remains, +meaning the functions can be aligned using the name. We +attempted to utilise an existing solution by Alves-Foss and +Song called Jima [41] to find function boundaries. Jima is the +current state-of-the-art tool for function boundary detection +in stripped binaries. The tool is implemented as a plugin for +Ghidra, but in our experiments, we find no statistical difference +between the base performance of Ghidra and Jima on our +dataset. The difficulties in extracting stripped functions, make +training and applying a model to stripped binaries challenging. +Opt level +BLEU-4 +EM +METEOR +ROUGE-L +-O0 +72.88 +34.18 +73.19 +74.84 +-O1 +50.30 +59.84 +55.36 +54.84 +-O2 +62.31 +46.23 +64.50 +66.05 +-O3 +54.68 +54.99 +58.25 +59.28 +TABLE IV: Average BLEU-4 score of decompiled code per +optimisation level +B. RQ2: Input Properties +As can be observed in Table III, the summaries produced +by the demi-stripped model were substantially worse than the +decompiled model, but most were still very usable, evident +by the BLEU-4 score above 44. Just removing the function +name gave quite similar results to demi-stripping. We find that +the loss of identifiers significantly lowers the performance of +the model, but stripped code also suffers from decompilation +faults, which seem to have a much larger impact on the model +performance. Hence, the performance of BinT5 on demi- +stripped code can be viewed as more representative of the +actual model and not impacted by faults introduced by Ghidra. +Table IV shows the average score per optimisation level. We +can observe that -O0 and -O2 perform better than -O1 and - +O3. Recall that -O0 is completely unoptimised, and that the +vast majority of our decompiled dataset is compiled with -O2, +which would explain why those optimisation levels perform +better. +C. RQ3: Dataset Properties +The performance of the base model on each of the dedupli- +cated datasets is presented in table V: +BLEU-4 +EM +METEOR +ROUGE-L +∆BLEU-4 +C +45.86 +32.87 +46.06 +47.53 +14.97 +DecomC +42.48 +28.08 +25.23 +27.66 +16.34 +Demi +25.38 +14.51 +42.47 +44.47 +18.83 +Stripped +7.19 +0.00 +4.75 +5.50 +4.07 +TABLE V: Result of fine-tuning CodeT5-base on the dedupli- +cated datasets and the difference with the baseline +We find that the influence of deduplication on our model’s +performance is relatively small on source code, at only 24%. +Duplicates have a relatively large impact on the decompiled +(28%) and demi-stripped (43%) code. Deduplication also +greatly decreases the EM rate across the board. Duplicates +have a relatively large impact on performance, but even with +the duplicates removed the model still produces many high- +quality summaries. The experiments on deduplication show +that the model seems to have a deeper understanding of the +data and is not simply reproducing previously seen samples. +As can be seen in Figure 8, the dataset size does not +have much of an impact, the model can be trained with +half or a quarter of the training samples without suffering +a considerable hit to performance. This could be attributed +to the high duplication factor of our dataset. It could also be +8 + +0 +20 +40 +60 +80 +100 +Fraction of train set +25 +30 +35 +40 +45 +50 +55 +60 +BLEU4 +Decompiled +Deduplicated +Fig. 8: BLEU-4 per trainset size for decompiled code and +deduplicated decompiled code +because the model was already pre-trained well by Wang et al. +and requires very little data for fine-tuning. This is a testament +to the relative ease with which these models could be extended +to decompiled code. +We also performed experiments where we did not apply the +filtering rules provided by CodeXGlue and where we always +mined the first sentence of any type of documentation. While +we were able to collect around 480K decompiled samples, the +model performed substantially worse, only scoring 36.97 and +33.26 BLEU-4 on C and decompiled code, respectively. These +results show that the dataset quality also heavily impacts the +model performance. +VII. DISCUSSION +In the previous section, we found that BinT5 shows con- +siderable performance for decompiled code and demi-stripped +code on both regular as well as deduplicated data. While +this is a promising result, we conduct a small investigation +of the decompiled samples. We will put our observations on +identifiers into the context of the extreme summarisation task. +Based on this we discuss the implications of our work. Finally, +we will close this section by discussing the threats to validity. +A. Exploration of Results +To explore the results of BinT5 we pick 25 high and 25 +low-scoring samples from the test set of the deduplicated +decompiled dataset. High samples have a BLEU-4 score higher +than 75 while low-scoring samples have a score lower than 25. +a) High Samples: With the high-performing samples +BinT5 tends to produce summaries which are very close to +the references. For instance, BinT5 produced Print description +of a datatype in XML against the baseline Dump description of +a datatype in XML. Of the 25 high-scoring samples we found +that all have counterparts with a similar function summary +in the training set. These functions also tend to have similar +names, but their decompiled function body was significantly +different, which is likely why deduplication didn’t remove +these functions. +b) Low Samples: From the low-performing samples we +observe that many summaries produced by BinT5 are seman- +tically very similar to the reference. For instance, the function +vl set simd enabled20, has the reference Toggle usage of +SIMD instructions while BinT5 produced Enable or Disable +the Simd Channel. This sample scores a BLEU-4 score of 0.0, +because of the limitations around the BLEU-4 metric, while +for a human evaluator the output is still very usable. Similarly, +for some samples, BinT5 produces shorter summaries contain- +ing shorthands. The reference Check if the given nickname is +blocked for ”normal client” use against Check whether nick is +blocked, also scores poorly. Of the 25 low-scoring samples +we observe that around 11 are semantically similar to the +reference and likely very useful for understanding the function. +B. Identifiers and Extreme Summarisation +We find a relatively small difference in performance be- +tween source code and decompiled code. This indicates that +in-function comments and variable names are relatively unim- +portant for the model performance. Although Ahmed and +Devanbu observed that identifiers might be more important +than syntax in the code-summarisation task [16], we can +further conclude that the function name is explicitly essential +for model performance. Removing just the function name from +the decompiled samples, as opposed to removing all identifiers +in demi-stripping, results in slightly higher performance than +demi-stripped code, which indicates a very high dependence +on the name of the function in the code summarisation task, +which is a logical finding in the context of the extreme code +summarisation task. +The extreme code summarisation task, as proposed by Al- +lamanis et al. aims to reproduce the function name given +a function body [16, 42]. It is framed as a summarisation +problem where the output is around 3 tokens in length, instead +of the 10+ tokens that regular code summarisation targets. +We found similar results when performing this task with +our dataset, namely, high performance on regular decompiled +code (with function names removed) and low performance on +stripped code. +A manual assessment of the stripped data shows that many +of the aligned functions were not decompiled properly. We +find that many functions are cut-off after a few instructions +because the decompiler did not recover the full control flow. +Other functions are missing side effects, like changes to global +variables. +C. Implications +We propose a novel solution to aid reverse engineers in +their work. Among many use cases, this solution could help +malware analysts to understand novel malware and its weak- +nesses quickly. The software can be analysed to find possible +vulnerabilities and malicious payloads. The source code can +20Colmap/Colmap:vl set simd enabled: +https://github.com/colmap/ +colmap/blob/87b3aa325bd8e5fb913788e29e9ac1e085e28b67/lib/VLFeat/ +generic.c#L1070 +9 + +be reconstructed for old binaries for which the source code is +lost. +If the application of NLP to binaries gets significantly better, +and the limitations around stripping and other obfuscation +techniques get resolved, it would have serious implications +for the cybersecurity domain. On one hand, it would assist +defenders, but on the other hand, attackers can leverage +these same methods to find and exploit vulnerabilities, build +malicious payloads and lift intellectual property from binaries. +CAPYBARA itself could be used to create and assess +neural decompilation, to perform a deeper investigation into +the extreme summarisation task, or to simply train a code +summarisation model on C code. CAPYBARA consists of a +large corpus of C and decompiled C code, which could be used +to pre-train language models, such that these models could +support decompiled code out-of-the-box. +While our work focused on decompiled code, our observa- +tions show some limits of transformer-based models and their +applicability to different data. Our dataset can help and inspire +other researchers to improve upon our work. We hope other +researchers use this dataset to train and evaluate their own +models. Furthermore, the process outlined in Chapter III could +help others construct standardised datasets for other tasks. The +steps outlined for the creation of this dataset can be followed +to create other datasets for other languages as well. +D. Threats to Validity +Internal Validity questions if other factors could have +affected the outcome. The training and evaluation data contains +a significant amount of noise, either in the form of badly de- +compiled functions or incorrect documentation. We carefully +collect and process the data, but we are unable to know to +which extent the documentation matches the original code. +While machine learning models (and specifically NLP models) +should be able to handle noisy data, this might introduce some +bias into the models. CodeT5 was also pre-trained on a C and +C# dataset, this dataset is unpublished and we were unable to +reach the authors. Some data leakage might have taken place, +but it is unlikely that it had much of an impact. The data +was only used for pre-training and would only have included +source code. To prevent this threat from arising in any future +studies, we make CAPYBARA publicly available. +External Validity refers to the generalisability of our +results. This work only focuses on stripping and compiler +optimisations as a means of resisting binary analysis, other +techniques like control flow obfuscation and packing are +also used to prevent reverse engineering. Other works focus +on unpacking and deobfuscation, so we consider our work +orthogonal to theirs. The data gathered for CAPYBARA were +exclusively from open-source projects. Decompiling closed- +source projects is explicitly forbidden by some EULAs and the +lack of source code documentation makes it difficult to evalu- +ate using reference summaries. However, reverse engineering +open-source software is not very useful in practice, since the +source code is readily available. Closed-source software might +have different data distribution and will present other chal- +lenges like obfuscation. Finally, only functions that decompile +(Ghidra produces any output) and that are documented, are +represented in CAPYBARA. This is most apparent in the +stripped dataset, where we can only recover a small fraction +of the total number of functions. A deeper investigation into +new decompilation techniques for stripped code, specifically +into the aspect of function boundary detection is left as future +work. +Construct Validity relates to the adequacy of the theoretical +constructs and the use of appropriate evaluation metrics. The +leading metric in our evaluations does not capture semantic +meaning. While BLEU-4 is the most popular metric for this +task, its reliability has been called into question [43, 44]. We, +therefore, included other metrics, which do take semantics into +account, in our evaluation. Finally, our entire approach hinges +on the assumption that function summaries, as they are used +for source code, are useful for binary analysis. Whether or not +this is actually the case, should be further investigated with a +qualitative study. +VIII. RELATED WORK +Binary reverse engineering and the use of NLP for software +engineering are vast and active fields, so we select and discuss +the closest state-of-the-art works in the field. We categorise the +studies into identifier recovery and binary translation. Finally, +we will discuss the open challenges and the relation of our +own work to these challenges. +a) Recovering Identifiers from Stripped Binaries: De- +bin [5] aims to recover debug information from stripped +binaries. The authors use a tree-based classification and a +probabilistic graph-based model. All the variable names and +types are jointly recovered using a maximum a posteriori +probability inference. VarBERT [45] uses a Transformer- +based NLP model for the task of variable name recovery. The +authors pre-trained a BERT model which is then fine-tuned to +predict the names and types from unstripped binaries. +FUNCRE +[7] +uses +a +pre-trained +and +fine-tuned +ROBERTA [29] model to predict usages of inlined library +functions. Recall that compilers with optimisations enabled +can inline functions in the binary (Chapter II). The authors +use indelible markers, which do not get destroyed by the +compiler, to mark usages of library functions and to construct +a dataset and train a model. +b) Binary Translation: Neutron [10] frames decompila- +tion as a neural machine translation problem and utilises an +Attention-LSTM-based neural translation network to translate +disassembled binaries back to C source code. The binaries +are not stripped and do not have any optimisations enabled. +The translations created by Neutron can contain syntax errors, +so the authors apply regular expressions to create a tailor- +made syntax checker. Neutron achieves high accuracy on the +translation task, but only on unstripped and non-optimised +code. +10 + +c) Our Novelty: Several aspects have not been properly +addressed and investigated. The application of code summari- +sation methods to decompiled code has not been addressed +by any work at all. Furthermore, some works on binary code +fail to take compiler optimisations into account [10]. We, +therefore, investigate the application of code summarisation +methods to decompiled code and we enable compiler optimi- +sations. +IX. CONCLUSION +In this paper, we proposed a new automatic binary code +summarisation task. With this new task, we also introduce +CAPYBARA, a novel dataset to train and evaluate models +on this task, with both mined as well as synthetic data. +Paired with this dataset, we train BinT5, a Transformer- +based code summarisation model to show the effectiveness of +CAPYBARA. We used BinT5 to further explore the datasets, +outlining the inherent difficulties in the data. +We found that while BinT5 shows considerable performance +on regular decompiled code, but its performance is being +hampered by the decompiler on stripped code, evidenced by +BinT5s strong performance on demi-stripped code. Further- +more, we found that while duplicates have a large impact +on the model, their presence is not paramount to the model’s +performance. 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[Online]. +Available: https://doi.org/10.1145/3359789.3359825 +[42] M. Allamanis, H. Peng, and C. Sutton, “A convolutional +attention network for extreme summarization of source +code,” +in +Proceedings +of +The +33rd +International +Conference on Machine Learning, ser. Proceedings of +Machine Learning Research, M. F. Balcan and K. Q. +Weinberger, Eds., vol. 48. +New York, New York, USA: +PMLR, 20–22 Jun 2016, pp. 2091–2100. [Online]. +Available: https://proceedings.mlr.press/v48/allamanis16. +html +[43] D. Roy, S. Fakhoury, and V. Arnaoudova, Reassessing +Automatic Evaluation Metrics for Code Summarization +Tasks. New York, NY, USA: Association for Computing +Machinery, 2021, p. 1105–1116. [Online]. Available: +https://doi.org/10.1145/3468264.3468588 +[44] S. Haque, Z. Eberhart, A. Bansal, and C. McMillan, +“Semantic similarity metrics for evaluating source code +summarization,” arXiv e-prints, pp. arXiv–2204, 2022. +[45] P. Banerjee, K. K. Pal, F. Wang, and C. Baral, “Vari- +able name recovery in decompiled binary code using +constrained masked language modeling,” arXiv preprint +arXiv:2103.12801, 2021. +13 + diff --git a/5tAzT4oBgHgl3EQfu_3x/content/tmp_files/load_file.txt b/5tAzT4oBgHgl3EQfu_3x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f1df7cc9d45a36cf9bd675dd5ce52ebe0c8446b --- /dev/null +++ b/5tAzT4oBgHgl3EQfu_3x/content/tmp_files/load_file.txt @@ -0,0 +1,1029 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf,len=1028 +page_content='Extending Source Code Pre-Trained Language Models to Summarise Decompiled Binaries Ali Al-Kaswan Delft University of Technology Delft, The Netherlands a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='al-kaswan@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='nl Toufique Ahmed University of California, Davis Davis, California, USA tfahmed@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='edu Maliheh Izadi Delft University of Technology Delft, The Netherlands m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='izadi@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='nl Anand Ashok Sawant University of California, Davis Davis, California, USA asawant@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='edu Prem Devanbu University of California, Davis Davis, California, USA ptdevanbu@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='edu Arie van Deursen Delft University of Technology Delft, The Netherlands arie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='vandeursen@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='nl Abstract—Binary reverse engineering is used to understand and analyse programs for which the source code is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Decompilers can help, transforming opaque binaries into a more readable source code-like representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Still, reverse engineering is difficult and costly, involving considering effort in labelling code with helpful summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While the automated summarisation of decompiled code can help reverse engineers understand and analyse binaries, current work mainly focuses on summarising source code, and no suitable dataset exists for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In this work, we extend large pre-trained language models of source code to summarise de-compiled binary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Further- more, we investigate the impact of input and data properties on the performance of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our approach consists of two main components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We first build CAPYBARA, a dataset of 214K decompiled function-documentation pairs across various compiler optimisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We extend CAPYBARA further by removing identifiers, and deduplicating the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Next, we fine-tune the CodeT5 base model with CAPYBARA to create BinT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BinT5 achieves the state-of-the-art BLEU-4 score of 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='83, 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='82 and, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='21 for summarising source, decompiled, and obfuscated decompiled code, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This indicates that these models can be extended to decompiled binaries successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, we found that the performance of BinT5 is not heavily dependent on the dataset size and compiler optimisation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We recommend future research to further investigate transferring knowledge when working with less expressive input formats such as stripped binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Index Terms—Decompilation, Binary, Reverse Engineering, Summarization, Deep Learning, Pre-trained Language Models, CodeT5, Transformers I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' INTRODUCTION Reverse engineering binary programs has many applica- tions, in particular, software security [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Binary reverse engineering is a hard task, requiring highly skilled reverse engineers [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Disassemblers and decompilers can help in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Disassemblers transform the binary into a low-level intermediate representation, and decompilers lift the representation to a high-level programming language-like representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' But the output of decompilers is still difficult to read and understand [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Much of the work that goes into reverse engineering a binary is spent labelling functions with semantic descriptions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Current approaches [4–10] mainly focus on recovering aspects lost in the compilation and decompilation process, such as names and types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Existing works fail to address the inherent difficulties in binary code comprehensibility, namely, the need for a high-level overview of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For source code, methods exist to automatically generate summaries from code [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Source code summarisation is used to automatically generate short natural language de- scriptions of code, which support program comprehension and aid maintenance [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While these methods have been successfully applied to programming languages such as Python, Java and PHP [14–16], using pre-trained language models [14–16], none of these methods has been applied to the relatively syntactically-poor output of decompilers (see Figures 1a and 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Being able to quickly determine the context and application of a function, can save valuable analysis time, and greatly benefit reverse engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Function and variable names alone, are inadequate representations of the source code [12], which is why having descriptive summaries of binaries is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Following [17], source code can be described as having two information channels: the algorithmic channel and the natural language channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The algorithmic channel specifies the execution of a program (semantics), while the natural language channel explains the purpose and context of the program to humans [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The natural channel includes function and variable names, code comments and the specific human- readable structure of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Processors only consider the algorithmic channel to execute a program, while humans use both the algorithmic channel and the natural channel to under- stand a piece of code [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, code is very regular and predictable, even more so than natural languages [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The compilation process, which transforms readable code into executable binaries, removes much of the information contained in the natural channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Especially stripped binaries — binaries of which the symbol table is removed — are challenging, since they have almost no identifiers at all as arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='01701v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='CR] 4 Jan 2023 can be observed in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The goal of this paper is to advance the field of binary reverse engineering by exploring the application of code summarisation to decompiled binaries by taking advantage of source code pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' However, there exists no dataset of aligned binaries and source code summaries since this is a new and unexplored task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' As pointed out by LeClair and McMillan, the lack of standardised datasets is a major barrier to ongoing research, which we will address for this task [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In this paper, we create a dataset containing pairs of decompiled and stripped- decompiled functions and summaries of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Dur- ing the creation of this dataset, we conform to the current best practices for dataset construction [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We apply this dataset to an existing pre-trained language model using transfer learning, by fine-tuning this pre-trained model on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For this task, we selected a pre-trained CodeT5 model, which was only trained on source code [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We perform experiments on this model to explore the impact of decompilation, and the importance of identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, we explore the impact of compiler optimisation levels, the dataset size and the level of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our findings are that the decompilation and alignment of stripped functions has a very high failure rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' and the resulting stripped model has low performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' But, we found that the model shows state-of-the-art performance with both decompiled code as well as demi-stripped stripped code, code of which the identifiers were removed after decompilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our experiments on data duplication and dataset size further show that these models can be trained with few data, and that while duplicates have a high impact on performance, their presence is not paramount to model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our key result: language models pre-trained on source code can be fine-tuned on binaries, opening up a range of new possibilities for the automated analysis of binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To summarise, the main contributions of this paper are: CAPYBARA1, a dataset of Combined Aligned de- comPiled BinarY code And Related Annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A novel dataset of aligned, C, decompiled, stripped-decompiled and demi-stripped summary pairs2 (Section III);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BinT53, a Binary summarisation CodeT5 model, a simple and straightforward adaptation of a source code trained code summarisation model to decompiled code using CAPYBARA (Section IV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' An empirical investigation on the impact of the properties of decompiled code and the properties of CAPYBARA (Sections V and VI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The materials, including the processed and raw data, the trained model checkpoints and steps to replicate our exper- iments, are openly available in our replication package4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 1CAPYBARA: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='7229809 2Decompiled code with strip-like obfuscation applied 3BinT5: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='7229913 4Replication package: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/AISE-TUDelft/Capybara-BinT5 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BACKGROUND In this section, we introduce the background of compilers, binary reverse engineering, transfer learning and the code summarisation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Compilers and Optimisation Levels Compilers are programs that convert source code from one programming language to another, but generally, and in the context of this work, the term is used to refer to programs that translate high-level code, like C, to a lower-level language such as machine code or bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For our work, we focus on the GNU Compiler Collection (GCC)5 and Clang/LLVM (Clang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='6 Compilers feature optimisation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Generally, the goal of optimisations is the improvement of runtime performance or program size at the expense of compilation time and the ability to debug [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Compilers use optimisation flags, grouped into optimisation levels, where each level uses a different set of optimisation flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' By default, if GCC is invoked without any optimisation options, the program will be compiled with -O0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' -O1, -O2 and -O3 incrementally apply more optimisation to the binary at the expense of a higher compilation time [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Optimisations can restructure and transform the program in relation to the source code, by changing the control flow or the data of the program [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This obfuscation can complicate the reverse engineering process by reducing the accuracy of tools [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Ghidra Ghidra7 is a free and open-source reverse engineering toolkit developed by the US National Security Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Ghidra contains many separate analysis modules that allow a reverse engineer to analyse binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Ghidra features a disassembler, which assembles binaries back into an intermediate represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In the case of x86-x64 binaries like the binaries this work focuses on, the intermediate representation will be the Assembly language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The decompiler, on the other hand, is a processor language-agnostic transformation engine that takes the disassembled code and creates a source code representa- tion, namely pseudo-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Pseudo-C follows the general language conventions of C, but it cannot be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Observe the relatively simple rtp sess ssrc function from creytiv/re8 shown in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We compile the project using the -O3 compiler level as defined in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We decompile the binaries using Ghidra’s decompiler using the standard configuration, the resulting pseudo-code is shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We observe that aside from the function name, almost the entire natural channel has been destroyed by the compilation and decompilation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The parameter and variable names are gone, any documentation is removed and the relatively simple logic has been unrolled to a much more difficult- to-understand representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Ghidra also incorrectly labelled 5GCC: https://gcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='gnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/ 6Clang: https://clang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/ 7Ghidra: https://ghidra-sre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/ 8re: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/creytiv/re 2 /** Get the Synchronizing source for an RTP/RTCP Socket �→ @param rs RTP Socket @return Synchronizing source / uint32_t rtp_sess_ssrc(const struct rtp_sock *rs){ return rs ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' rs -> enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='ssrc : 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='} (a) Source rtp sess ssrc function ulong rtp_sess_ssrc(long param_1){ uint local_14 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' if (param_1 == 0){ local_14 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' } else { local_14 = * (uint *) (param_1 + 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='} return (ulong) local_14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' } (b) Decompiled rtp sess ssrc function ulong FUN_00100d30 ( long param_1 ){ uint local_14 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' if (param_1 == 0) { local_14 = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' } else { local_14 = * (uint *) (param_1 + 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='} return ( ulong ) local_14 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='} (c) Stripped decompiled rtp sess ssrc function Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 1: Example source, decompiled and stripped code snippet many of the variable types and failed to identify the struct datatype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Using our trained BinT5 model we can summarise the decompiled code and generate the following summary: Get the source for an RTP/RTCP Socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This summary gives us an indication of the purpose of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Integrating this generated summary into Ghidra increases the readability of the entire binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Keep in mind that a reverse engineer has to understand not just this function, but hundreds of different functions in a single binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Stripping Aside from compiling with higher optimisation levels, bi- naries can also be stripped to obfuscate the underlying code and to resist analysis [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Commercial off-the-shelf software is often stripped to reduce the memory and storage footprint of the binaries, and to resist analysis to protect the intellectual property of the creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Many vulnerable and malicious bina- ries are, unfortunately, also stripped to resist security analysis and hide their faults [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Unix and Unix-like operating systems include a strip utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The strip utility removes any operands that are not nec- essary for the execution of the binary while ensuring that the execution of the binary remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The exact implementation and what constitutes unnecessary operands are left to the implementor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='9 The strip utility as implemented in GNU/Linux removes the symbol table from the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The symbol table contains each symbol’s location, type and name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Like higher optimisation levels, the use of stripping can greatly complicate the efforts to reverse engineer a binary, as well as reduce the accuracy and effectiveness of reverse engineering tools [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For example, we compile, strip and decompile the function in Figure 1a, and the resulting stripped decompiled function is shown in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In addition to the details lost by the decompilation process, the stripper removed all symbols, like the function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Code Summarisation Task: Code summarisation (also referred to as source code sum- marisation) is the task of writing short descriptions from source code, usually a single-sentence summary of the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The main use is for software documentation, like the one-sentence JavaDoc description used in Java [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This documentation is important for program comprehension and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' But the process of writing and maintaining these descriptions is a labour-intensive and time-consuming task, which is where the benefits of automating that process arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Automatic code summarisation is an active and popular research problem in the field of software engineering [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Transformer-based Models Transformers were originally proposed by Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' as a sequence-to-sequence architecture [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Unlike the Re- current Neural Networks [26] (RNN), the Long Short-Term Memory [27] (LSTM) variant of RNNs [26] and Convolutional Neural Networks [28] (CNN), Transformers only use a mecha- nism called self-attention to capture dependencies between the input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The current state-of-the-art NLP models for programming languages such as CodeT5 [14], CodeBERT [15] and PolyGlotCodeBERT [16] are all based on the Transformer architecture [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Transfer Learning Pre-trained Transformers-based language models, such as RoBERTa [29], CodeBERT [15] and CodeT5 [14] utilise a pre-train then fine-tune paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The bespoke paradigm was initially introduced by Kenton and Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In this paradigm, the models are first trained in an unsupervised manner on a large unlabelled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These pre-trained models can then be fine-tuned to perform a more specialised task, such as summarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Transfer learning uses the knowledge that is obtained in one task to solve a different task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' It allows the creation of general models that are trained once on massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These general models, which contain general domain knowledge can then be fine-tuned for a specific downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This approach is quicker and requires less training data than training a model on the downstream task from scratch [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 9strip: https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='opengroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/onlinepubs/7908799/xcu/strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='html 3 Source Code Compilation Decompilation Decompiled Stripping Decompilation Function Extraction Comment Alignment Comment Alignment Stripped Comment Extraction Demi-Stripped Comment Alignment Demi Stripping Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 2: Data Collection Pipeline III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CAPYBARA DATASET We require a dataset of decompiled functions labelled with a descriptive summary to create and assess our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This dataset should be relatively large to suit the ‘data-hungry’ nature of deep-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, the dataset needs to feature a diverse set of data representative of our solution’s actual real-life use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Data Collection To create such a large and diverse dataset we made use of BinSwarm [7], an existing dataset of aligned decompiled and stripped decompiled functions10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BinSwarm collects C- based projects from Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The projects are filtered to only include those that are actively being developed, using Travis CI and built for Ubuntu Linux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The projects are built using Docker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The resulting binaries are then copied and stripped, and both the stripped and unstripped binaries are decompiled using Ghidra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The functions are extracted from the stripped and unstripped decompiled code and aligned with the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The BinSwarm dataset only contains aligned tuples of source code and (stripped-) decompiled functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We extract documentation from the original source code files to add descriptive comments to this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To that end, we depend on the documentation included in the source code by the original authors in the form of single and multiline comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We locate the functions in the unbuilt project files and align the decompiled functions with the comments in the source code using srcML11 to extract any documentation located directly before a function signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A high-level overview of the entire process is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A function’s documentation often also contains other details besides the descriptive summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We found that C projects do not follow a single documentation standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For example, Javadoc for Java has a short one-line description or summary for each method at the beginning of the multiline comment 10BinSwarm: https://hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='docker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/r/binswarm/cbuilds 11srcML: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='srcml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/ /** @brief Select the source of Microcontroller Clock Output �→ Exact sources available depend on your target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' On devices with multiple MCO pins, this function controls MCO1 �→ @param[in] mcosrc the unshifted source bits / Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 3: Example of documentation from jeanthom/ DirtyJTAG: rcc set mco block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In C, there is no singular documentation standard, so there might not be a single-line summary, and we will need to locate it in the comment block automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' a) Summary Extraction Rules: We observe that the ma- jority of single-line data are descriptive summaries, so we extract the first sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We identify many documentation styles in our multi-line data, we define some automated rules to extract summaries from the documentation: @brief or @purpose: If the documentation contains a ‘@brief’ or ‘@purpose’ tag, we extract the first sentence after the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The ‘brief‘ tag is part of the Doxygen docu- mentation standard12, an example is shown in Figure 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Description: If the documentation contains a line with ‘Description:‘, we extract the following sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' @param or @v: Documentation that contains an ‘@v’ or ‘@param’ tag, usually has a summary in the sentence before the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We extract that sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' b) Filtering Rules: To improve the quality of the dataset we filter out samples based on the rules used by the Code- SearchNet dataset [20] included in the CodeXGlue benchmark for the summarisation task [31]: Documentation length: We remove any summaries that are too long or too short and remove anything shorter than 3 or longer than 256 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Special tokens: We follow the example of the Code- SearchNet [20] and remove all documentation that con- tains special tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We scan for web tokens (like ‘http://’), HTML tokens (like ‘’), paths (like ‘C://Users/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='.’), since this documentation usually refers to external resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We additionally filter any developer tokens (like ‘FIXME:’), as these documents do not pro- vide meaningful information about the function itself, but contain comments about the development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Language: We filter out any documentation that was not written in English using the FastText language identifica- tion algorithm [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Around 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='19% of the documentation is in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Empty documentation: We find that a large number of functions did not have any documentation associated with them at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We simply remove these samples from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 12Doxygen:https://doxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='nl/manual/docblocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='html 13jeanthom/DirtyJTAG:rcc set mco:https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/ insane-adding-machines/unicore-mx/-/blob/master/lib/stm32/common/ rcc common all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='c#L192 4 GCC000• Abstract Syntax Tree: The authors of the CodeSearch- Net dataset [20] additionally, remove any samples that do not parse into an AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We choose to omit this step since all of our samples have been successfully compiled and have thus at one point been parsed into an AST by the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Dataset Preparation a) Synthesis of Demi-stripped Code: From the dataset of decompiled functions, we also create another dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We emulate the process of stripping by removing all the iden- tifiers from the decompiled code and replacing them with placeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For clarity, we call this demi-stripped data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Like the stripped dataset, the identifiers are all removed, but this is only done after the decompilation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The decompiler still had access to the identifiers and could use the symbol table during decompilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Most importantly, this demi-stripped dataset still has the same structure and control flow as the unstripped decompiled dataset and avoids any decompilation issues arising from stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' b) Data Split: The dataset is split into a train, test and validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These sets constitute approximately, 80%, 10% and 10% [19] of the complete dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' As recommended by Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' and LeClair and McMillan, we prevent leakage of vocabulary and code patterns between the sets, by sampling the sets in a cross-project manner [13, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This means that an entire project gets assigned to one of the sets, and functions from the same project cannot be assigned to different sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The projects in the test and validation set are the same across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' c) Duplication: Large corpora of code, like the cor- pus gathered by BinSwarm, tend to have a high degree of duplication [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' As a result, snippets of code that are relatively unchanged appear in multiple parts of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This can be in the form of copied, generic or auto-generated functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These functions will appear in multiple repositories and might be duplicated across the training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Besides exact duplicates, near-duplicates can also occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Near- duplicates differ in a few minor aspects like additional code comments or different function names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While removing exact duplicates is relatively fast and straightforward, removing near-duplicates is much more challenging and computationally intensive [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The issue with code duplication in classical code summarisation is that the models and tools are supposed to be used to generate summaries for new and unseen code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The evaluation metrics should therefore measure the gener- alisation of the tool on new samples [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Duplicates and near-duplicates are not defined as new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A user of such a tool could simply look these samples up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, large, high-capacity models like CodeT5 with 220M [14] or CodeBERT with 128M [15] parameters, have a large capacity to memorise duplicated code [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' However, the use case outlined in this work is more akin to deobfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' As explained by Allamanis, deobfuscation could be a use case where duplicates are valid and part of the true distribution of the problem [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Compiled code contains 0 200 400 600 800 1000 1200 1400 Token count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='005 Density Source code Decompiled code Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 4: Tokens in source C and decompiled code a lot of duplicate code, and understanding this code is still difficult and essential for understanding the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While regular source code allows the reader to look up code snippets, decompiled binaries have an additional obfuscation applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We, therefore, focus on the model’s performance on code with duplicates as we believe duplicates to be part of the true distribution of the data, but we also report the deduplicated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Dataset Properties Table I shows the size of the processed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='1M aligned decompiled functions, we extract documentation for 215k of them, and we found that the majority of samples, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='5M did not have any documentation at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, BinSwarm only provided us with 415k aligned stripped samples, and we can extract documentation for only 14k of these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Dataset Including duplicates Deduplicated C/Demi/Decom 214,587 79,673 Stripped 14,245 7,826 TABLE I: Number of functions in dataset The vast majority of documentation is in the form of multi-line comments as opposed to single-line or double-slash comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We found that the documentation and comments had a mean length of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='60 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='14 tokens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Figure 4 shows the distribution of the number of tokens in source code and decompiled code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The source and decompiled code have a mean length of 399 and 779 tokens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Figure 5, shows that decompiled code also has close to double the LOC of source code, with means of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='77 and 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='42 lines for source and decompiled, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The majority of decompiled functions are compiled with optimisation level -O2, with a similar number of -O1 and - O3 samples and relatively few -O0 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Stripped data has a very even distribution of optimisation levels, with only - O0 having significantly fewer samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Note that there are more optimisation levels than shown in Figure 6, for brevity the different levels are grouped into their base optimisation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' -Oa is grouped with -O0, -Of and -Og are grouped with -O1, -Os is grouped with -O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We also observe some 5 0 25 50 75 100 125 150 175 200 LOC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='06 Density Source code Decompiled code Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 5: LOC in source C and decompiled code 0 1 2 3 Optimisation level 0 20000 40000 60000 80000 100000 120000 Decompiled 0 1000 2000 3000 4000 Stripped Decompiled Stripped Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 6: Distribution of optimisation levels in decompiled (left) and stripped (right) samples with an optimisation level higher than -O3 (-O8 and O7), as specified by the GCC documentation, these levels are equivalent to -O314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BINT5 We select CodeT5 [14] as the base-model for our experi- ments since it is the highest-scoring publicly-available model on the CodeXGLUE [31] Code Summarisation benchmark15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CodeT5 is a programming language model built on the T5 (Text-to-text Transfer Transformer) architecture [34] and pre- trained on a mix of supervised and unsupervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CodeT5 employs an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In contrast to other models, CodeT5 is trained using both unimodal (PL only) and bimodal (NL-to-PL) tasks in eight programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This bimodal training allows CodeT5 to perform strong cross- modal tasks such as code summarisation and code generation (PL-to-NL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Many other models only use the data and lan- guages included in the CodeXGlue dataset [15, 16, 31], while CodeT5 also uses a mined dataset of C and C++ code for its pre-training objectives [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The inclusion of C training data should help the model with the CAPYBARA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' There could be some overlap in the training data between CAPYBARA and the dataset used by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' which would cause leakage, we address these concerns in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 14GCC optimisation levels: https://gcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='gnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/onlinedocs/gcc-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='2/gcc/ Optimize-Options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='html#Optimize-Options 15CodeXGLUE benchmark: https://microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='io/CodeXGLUE/ Fine-Tuning Base Model BinT5 Evaluation Results CAPYBARA training & validation data CAPYBARA test data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 7: BinT5 fine-tuning pipeline CodeT5 also utilises the transfer learning paradigm, which allows us to train the model with relatively little data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' In this case, we make use of the CodeT5-base model, which was trained on mixed upstream tasks by the authors [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' An overview of how we applied the model to create BinT5 is provided in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' EXPERIMENTAL SETUP To assess the effectiveness of our approach, we first evaluate the performance of the model, we then identify the aspects of the data that make this task inherently difficult, and we finally investigate aspects of the datasets and their influence on the complexity of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Research Questions In the context of the study, we thereby formulate the Research Questions (RQ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ1: How effective are fine-tuned Transformer-based models at decompiled code summarisation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To investigate the application of existing models to binaries using CAPY- BARA, we set a baseline by training a model on the code summarisation task on the source C-code dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We then train a summarisation model on both the decompiled and the stripped dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We use the evaluation metrics to compare the performance of the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ2: Which aspects of the input contribute most to model per- formance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We investigate which aspects of decompiled code increase the difficulty of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We, therefore, look at the impact of the symbol table on decompilation, for this, we fine-tune a model on the demi-stripped dataset and compare it to the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We also investigate the importance of the function name by removing just the function name from the decompiled code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, we investigate the impact of the optimisation level by exploring the performance per optimisation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ3: What is the impact of dataset properties on model per- formance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We finally investigate how the construction of CAPYBARA influences the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To answer the final research question we remove the duplicates from the datasets and retrain the models, after which we compare the performance to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, we investigate the impact of dataset size, by incrementally reducing the size of the training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 6 "Summarize Python: def inc_value(x): "increment value" "Generate Python: increment value\' \'def inc_value(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CodeT5 "Defect: if x=0: x += 1 "true" "Refine: if x=0: x += 1\' "if x == 0: x += 1" "Translate Python to C: if x==O: x += 1 "if (x==0) {x += 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='}"B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Baselines To first establish a performance baseline, we train a CodeT5- base model on the summarisation task on source C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Note that only samples which are aligned with decompiled code are included in the source C dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The baseline is used to compare the decompiled C, stripped decompiled C and the demi-stripped datasets to the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Evaluation Metrics We evaluate the performance between the reference sum- mary from CAPYBARA and the candidate summary produced by BinT5 using the EM, BLEU-4 [35], ROUGE-L [36] and, METEOR [37] metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' a) Exact Match (EM): The simplest metric is the EM which scores a prediction one if it matches its reference exactly and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' b) BLEU-4: The most widely used metric in the code summarisation task is the Bilingual Evaluation Understudy Score (BLEU) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BLEU-4 produces a percentage number between 0 and 100, which defines the similarity between a candidate and a set of reference sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' BLEU-4 calculates the cumulative 4-gram precision scores, the number of match- ing 4-grams divided by the total number of 4-grams in the candidate sentence [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The unigrams and bigrams account for the adequacy of the candidate while the longer three and 4-grams account for fluency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To prevent short sentences the result is multiplied by a brevity penalty as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A smoothing function is applied to prevent sequences with no matching 4- grams to score zero [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' recommend BLEU-4 with smoothing method 4 [13], we opted to use the Moses [39] implementation of BLEU-4 which uses smoothing method 2 since this is also utilised by CodeSearchNet, CodeXGlue and CodeT5 [14, 20, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' c) ROUGE-L: ROUGE or Recall-Oriented Understudy for Gisting Evaluation, is a package which includes several metrics, the most popular among them is ROUGE-L [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' ROUGE-L is more recall oriented than BLEU-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' ROUGE-L simply finds the longest common subsequence (LCS) between the reference and the candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Note that the words do not need to be consecutive but they have to be in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' d) METEOR: METEOR or Metric for Evaluation for Translation with Explicit Ordering [37] uses word lists and stemming to also take synonyms into account and calculates the harmonic mean of the unigram precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Similar to ROUGE-L, METEOR is more recall-focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' METEOR has a higher correlation with human judgement than BLEU-4 [19] at the sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CodeT5 finetuning and testing The concept of transfer learning, which is utilised in BinT5, depends on the use of a fine-tuning step to train the pre-trained model on the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We fine-tune a pre-trained CodeT5-base model on the constructed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The model is trained on the summarisation task as defined in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We train the model on the train set, then evaluate it after every epoch on the validation set and finally test on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' During training, we measure the model performance using the BLEU-4 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Data deduplication To create a deduplicated version of the CAPYBARA dataset we make use of a fork16 of the near-duplicate-code- detector [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We use this tool to compare all the datasets’ functions and find clusters of near-duplicate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We randomly select one function per cluster and discard the rest from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We use the standard tool configuration as recommended by Allamanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Of the removed duplicates, we observe that a relatively large number originates from common libraries, such as SQLite17, that are packaged with binary programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Thus a certain amount of duplication is also likely to occur “in the wild”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Configuration We process and visualise the data with Pandas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='3 and Ghidra 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' FastText 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='3 with the largest lid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='bin model is used to detect languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We train the model using Transformers version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='2 running on Torch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='0+cu111 in the nvidia/cuda:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='0-base docker container image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We share a Docker image with all the libraries required to run BinT5 pre-installed on DockerHub19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A grid search of the optimal settings was infeasible from a time perspective, so we performed training mainly using the recommended settings from the CodeT5-base model [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We double the source length for the decompiled, stripped, and demi-stripped code to 512 tokens instead of the standard 256 tokens used for the source code to compensate for the fact that the average length of decompiled code is almost twice as long as the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We trained the model on a machine with an NVIDIA GeForce RTX3080 with 10GB of VRAM and an AMD Ryzen Threadripper 3990X 64-Core Processor with 192GB of RAM running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='4 LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The GPU is running Nvidia driver version 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='02 with Cuda 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The authors of CodeT5 used an NVIDIA A100 GPU with 40GB of VRAM for fine-tuning [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To compensate for the lack of memory, we reduced the batch size to 2, which was the maximum length that could still fit in the VRAM, we increase the ‘gradient accumulation steps’ to 24 to still achieve the effective standard batch size of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RESULTS We present the results of our experiments to answer the research questions, results are grouped per research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The metrics are calculated for each sample from the test set, and the average scores are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ1: Model Effectiveness The performance of the CodeT5-base model on each of the datasets is presented in table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 16Near Duplicate Code Detector: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/SERG-Delft/ near-duplicate-code-remover 17SQLite: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='sqlite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='org/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='html 18It is not recommended to use Ghidra versions before 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='1 since these versions have not been patched against a Log4J RCE 19BinT5 Docker Image: https://hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='docker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/r/aalkaswan/bint5/tags 7 BLEU-4 EM METEOR ROUGE-L C 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='83 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='19 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='33 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='51 DecomC 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='82 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='92 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='14 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='51 Stripped 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='85 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='25 TABLE II: Result of fine-tuning CodeT5-base on mined datasets BLEU-4 EM METEOR ROUGE-L DecomC 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='82 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='92 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='32 Demi 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='21 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='10 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='89 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='59 NoFunName 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='99 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='12 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='92 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='07 TABLE III: Result of fine-tuning CodeT5-base on synthetic data We found that the decompiled code model generally pro- duced good summaries, evidenced by the BLEU-4 score of 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='82, which is slightly lower than the baseline set by the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The stripped model mainly produced unusable summaries, as evidenced by the BLEU-4 score of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The high EM score could be an indication of a high duplication factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Initial experiments with GraphCodeBERT [40] and Poly- glotGraphCodeBERT [16] base models fine-tuned on CAPY- BARA show performance around 5 and 3 BLEU-4 lower, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This is a relatively small difference, especially considering the model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This shows that the performance of BinT5 does not heavily depend on the additional pre-training on C and C# performed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='. Furthermore, this result shows that it is improbable that significant dataset leakage has taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We found a relatively large difference between the number of recovered decompiled and stripped decompiled functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This can likely be attributed to the fact that Ghidra struggles a lot more with recovering stripped functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Recall that the symbol table commonly contains information regarding the location and name of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' When this table is dropped, the start- and endpoints of functions are hard to infer by automatic tools, especially since many functions get inlined, and JUMP instructions replace CALL instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Aside from difficulties in demarcating functions, it is also difficult to align the associated source code function with the decompiled function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' With unstripped code, the function name remains, meaning the functions can be aligned using the name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We attempted to utilise an existing solution by Alves-Foss and Song called Jima [41] to find function boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Jima is the current state-of-the-art tool for function boundary detection in stripped binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The tool is implemented as a plugin for Ghidra, but in our experiments, we find no statistical difference between the base performance of Ghidra and Jima on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The difficulties in extracting stripped functions, make training and applying a model to stripped binaries challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Opt level BLEU-4 EM METEOR ROUGE-L O0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='88 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='18 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='19 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='84 O1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='30 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='84 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='36 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='84 O2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='31 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='23 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='50 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='05 O3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='68 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='99 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='25 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='28 TABLE IV: Average BLEU-4 score of decompiled code per optimisation level B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ2: Input Properties As can be observed in Table III, the summaries produced by the demi-stripped model were substantially worse than the decompiled model, but most were still very usable, evident by the BLEU-4 score above 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Just removing the function name gave quite similar results to demi-stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We find that the loss of identifiers significantly lowers the performance of the model, but stripped code also suffers from decompilation faults, which seem to have a much larger impact on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Hence, the performance of BinT5 on demi- stripped code can be viewed as more representative of the actual model and not impacted by faults introduced by Ghidra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Table IV shows the average score per optimisation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We can observe that -O0 and -O2 perform better than -O1 and - O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Recall that -O0 is completely unoptimised, and that the vast majority of our decompiled dataset is compiled with -O2, which would explain why those optimisation levels perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RQ3: Dataset Properties The performance of the base model on each of the dedupli- cated datasets is presented in table V: BLEU-4 EM METEOR ROUGE-L ∆BLEU-4 C 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='86 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='87 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='06 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='53 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='97 DecomC 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='48 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='08 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='23 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='66 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='34 Demi 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='38 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='51 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='47 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='47 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='83 Stripped 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='07 TABLE V: Result of fine-tuning CodeT5-base on the dedupli- cated datasets and the difference with the baseline We find that the influence of deduplication on our model’s performance is relatively small on source code, at only 24%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Duplicates have a relatively large impact on the decompiled (28%) and demi-stripped (43%) code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Deduplication also greatly decreases the EM rate across the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Duplicates have a relatively large impact on performance, but even with the duplicates removed the model still produces many high- quality summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The experiments on deduplication show that the model seems to have a deeper understanding of the data and is not simply reproducing previously seen samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' As can be seen in Figure 8, the dataset size does not have much of an impact, the model can be trained with half or a quarter of the training samples without suffering a considerable hit to performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This could be attributed to the high duplication factor of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' It could also be 8 0 20 40 60 80 100 Fraction of train set 25 30 35 40 45 50 55 60 BLEU4 Decompiled Deduplicated Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 8: BLEU-4 per trainset size for decompiled code and deduplicated decompiled code because the model was already pre-trained well by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' and requires very little data for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This is a testament to the relative ease with which these models could be extended to decompiled code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We also performed experiments where we did not apply the filtering rules provided by CodeXGlue and where we always mined the first sentence of any type of documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While we were able to collect around 480K decompiled samples, the model performed substantially worse, only scoring 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='97 and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='26 BLEU-4 on C and decompiled code, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These results show that the dataset quality also heavily impacts the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' DISCUSSION In the previous section, we found that BinT5 shows con- siderable performance for decompiled code and demi-stripped code on both regular as well as deduplicated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While this is a promising result, we conduct a small investigation of the decompiled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We will put our observations on identifiers into the context of the extreme summarisation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Based on this we discuss the implications of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, we will close this section by discussing the threats to validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Exploration of Results To explore the results of BinT5 we pick 25 high and 25 low-scoring samples from the test set of the deduplicated decompiled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' High samples have a BLEU-4 score higher than 75 while low-scoring samples have a score lower than 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' a) High Samples: With the high-performing samples BinT5 tends to produce summaries which are very close to the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For instance, BinT5 produced Print description of a datatype in XML against the baseline Dump description of a datatype in XML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Of the 25 high-scoring samples we found that all have counterparts with a similar function summary in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' These functions also tend to have similar names, but their decompiled function body was significantly different, which is likely why deduplication didn’t remove these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' b) Low Samples: From the low-performing samples we observe that many summaries produced by BinT5 are seman- tically very similar to the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' For instance, the function vl set simd enabled20, has the reference Toggle usage of SIMD instructions while BinT5 produced Enable or Disable the Simd Channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This sample scores a BLEU-4 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='0, because of the limitations around the BLEU-4 metric, while for a human evaluator the output is still very usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Similarly, for some samples, BinT5 produces shorter summaries contain- ing shorthands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The reference Check if the given nickname is blocked for ”normal client” use against Check whether nick is blocked, also scores poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Of the 25 low-scoring samples we observe that around 11 are semantically similar to the reference and likely very useful for understanding the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Identifiers and Extreme Summarisation We find a relatively small difference in performance be- tween source code and decompiled code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This indicates that in-function comments and variable names are relatively unim- portant for the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Although Ahmed and Devanbu observed that identifiers might be more important than syntax in the code-summarisation task [16], we can further conclude that the function name is explicitly essential for model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Removing just the function name from the decompiled samples, as opposed to removing all identifiers in demi-stripping, results in slightly higher performance than demi-stripped code, which indicates a very high dependence on the name of the function in the code summarisation task, which is a logical finding in the context of the extreme code summarisation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The extreme code summarisation task, as proposed by Al- lamanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' aims to reproduce the function name given a function body [16, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' It is framed as a summarisation problem where the output is around 3 tokens in length, instead of the 10+ tokens that regular code summarisation targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We found similar results when performing this task with our dataset, namely, high performance on regular decompiled code (with function names removed) and low performance on stripped code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A manual assessment of the stripped data shows that many of the aligned functions were not decompiled properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We find that many functions are cut-off after a few instructions because the decompiler did not recover the full control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Other functions are missing side effects, like changes to global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Implications We propose a novel solution to aid reverse engineers in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Among many use cases, this solution could help malware analysts to understand novel malware and its weak- nesses quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The software can be analysed to find possible vulnerabilities and malicious payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The source code can 20Colmap/Colmap:vl set simd enabled: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='com/colmap/ colmap/blob/87b3aa325bd8e5fb913788e29e9ac1e085e28b67/lib/VLFeat/ generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content='c#L1070 9 be reconstructed for old binaries for which the source code is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' If the application of NLP to binaries gets significantly better, and the limitations around stripping and other obfuscation techniques get resolved, it would have serious implications for the cybersecurity domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' On one hand, it would assist defenders, but on the other hand, attackers can leverage these same methods to find and exploit vulnerabilities, build malicious payloads and lift intellectual property from binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CAPYBARA itself could be used to create and assess neural decompilation, to perform a deeper investigation into the extreme summarisation task, or to simply train a code summarisation model on C code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CAPYBARA consists of a large corpus of C and decompiled C code, which could be used to pre-train language models, such that these models could support decompiled code out-of-the-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While our work focused on decompiled code, our observa- tions show some limits of transformer-based models and their applicability to different data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our dataset can help and inspire other researchers to improve upon our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We hope other researchers use this dataset to train and evaluate their own models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, the process outlined in Chapter III could help others construct standardised datasets for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The steps outlined for the creation of this dataset can be followed to create other datasets for other languages as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Threats to Validity Internal Validity questions if other factors could have affected the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The training and evaluation data contains a significant amount of noise, either in the form of badly de- compiled functions or incorrect documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We carefully collect and process the data, but we are unable to know to which extent the documentation matches the original code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While machine learning models (and specifically NLP models) should be able to handle noisy data, this might introduce some bias into the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CodeT5 was also pre-trained on a C and C# dataset, this dataset is unpublished and we were unable to reach the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Some data leakage might have taken place, but it is unlikely that it had much of an impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The data was only used for pre-training and would only have included source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' To prevent this threat from arising in any future studies, we make CAPYBARA publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' External Validity refers to the generalisability of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This work only focuses on stripping and compiler optimisations as a means of resisting binary analysis, other techniques like control flow obfuscation and packing are also used to prevent reverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Other works focus on unpacking and deobfuscation, so we consider our work orthogonal to theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The data gathered for CAPYBARA were exclusively from open-source projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Decompiling closed- source projects is explicitly forbidden by some EULAs and the lack of source code documentation makes it difficult to evalu- ate using reference summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' However, reverse engineering open-source software is not very useful in practice, since the source code is readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Closed-source software might have different data distribution and will present other chal- lenges like obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, only functions that decompile (Ghidra produces any output) and that are documented, are represented in CAPYBARA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This is most apparent in the stripped dataset, where we can only recover a small fraction of the total number of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' A deeper investigation into new decompilation techniques for stripped code, specifically into the aspect of function boundary detection is left as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Construct Validity relates to the adequacy of the theoretical constructs and the use of appropriate evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The leading metric in our evaluations does not capture semantic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' While BLEU-4 is the most popular metric for this task, its reliability has been called into question [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We, therefore, included other metrics, which do take semantics into account, in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, our entire approach hinges on the assumption that function summaries, as they are used for source code, are useful for binary analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Whether or not this is actually the case, should be further investigated with a qualitative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' RELATED WORK Binary reverse engineering and the use of NLP for software engineering are vast and active fields, so we select and discuss the closest state-of-the-art works in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We categorise the studies into identifier recovery and binary translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, we will discuss the open challenges and the relation of our own work to these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' a) Recovering Identifiers from Stripped Binaries: De- bin [5] aims to recover debug information from stripped binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The authors use a tree-based classification and a probabilistic graph-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' All the variable names and types are jointly recovered using a maximum a posteriori probability inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' VarBERT [45] uses a Transformer- based NLP model for the task of variable name recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The authors pre-trained a BERT model which is then fine-tuned to predict the names and types from unstripped binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' FUNCRE [7] uses a pre-trained and fine-tuned ROBERTA [29] model to predict usages of inlined library functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Recall that compilers with optimisations enabled can inline functions in the binary (Chapter II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The authors use indelible markers, which do not get destroyed by the compiler, to mark usages of library functions and to construct a dataset and train a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' b) Binary Translation: Neutron [10] frames decompila- tion as a neural machine translation problem and utilises an Attention-LSTM-based neural translation network to translate disassembled binaries back to C source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The binaries are not stripped and do not have any optimisations enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The translations created by Neutron can contain syntax errors, so the authors apply regular expressions to create a tailor- made syntax checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Neutron achieves high accuracy on the translation task, but only on unstripped and non-optimised code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' 10 c) Our Novelty: Several aspects have not been properly addressed and investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' The application of code summari- sation methods to decompiled code has not been addressed by any work at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Furthermore, some works on binary code fail to take compiler optimisations into account [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We, therefore, investigate the application of code summarisation methods to decompiled code and we enable compiler optimi- sations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a new automatic binary code summarisation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' With this new task, we also introduce CAPYBARA, a novel dataset to train and evaluate models on this task, with both mined as well as synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Paired with this dataset, we train BinT5, a Transformer- based code summarisation model to show the effectiveness of CAPYBARA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We used BinT5 to further explore the datasets, outlining the inherent difficulties in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' We found that while BinT5 shows considerable performance on regular decompiled code, but its performance is being hampered by the decompiler on stripped code, evidenced by BinT5s strong performance on demi-stripped code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Further- more, we found that while duplicates have a large impact on the model, their presence is not paramount to the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Finally, we observe that BinT5 could be trained with just a fraction of the samples in CAPYBARA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our work has shown that a well-known and well-studied task from the source code domain [13], namely source code summarisation, can be applied to binary code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' This is only one of the many different applications of NLP for code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAzT4oBgHgl3EQfu_3x/content/2301.01701v1.pdf'} +page_content=' Our paper constitutes the first 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Calmon 4 +Abstract +Machine learning (ML) models can underperform +on certain population groups due to choices made +during model development and bias inherent in +the data. We categorize sources of discrimina- +tion in the ML pipeline into two classes: aleatoric +discrimination, which is inherent in the data distri- +bution, and epistemic discrimination, which is due +to decisions during model development. We quan- +tify aleatoric discrimination by determining the +performance limits of a model under fairness con- +straints, assuming perfect knowledge of the data +distribution. We demonstrate how to characterize +aleatoric discrimination by applying Blackwell’s +results on comparing statistical experiments. We +then quantify epistemic discrimination as the gap +between a model’s accuracy given fairness con- +straints and the limit posed by aleatoric discrimi- +nation. We apply this approach to benchmark ex- +isting interventions and investigate fairness risks +in data with missing values. Our results indi- +cate that state-of-the-art fairness interventions are +effective at removing epistemic discrimination. +However, when data has missing values, there is +still significant room for improvement in handling +aleatoric discrimination. +1. Introduction +Algorithmic discrimination may occur in different stages of +the machine learning (ML) pipeline. For example, histori- +cal biases in the data-generating process can propagate to +downstream tasks; human biases can influence a ML model +through inductive bias; optimizing solely for accuracy can +lead to disparate model performance across groups in the +data (Suresh & Guttag, 2019; Mayson, 2019). The past +years have seen a rapid increase in algorithmic interventions +that aim to mitigate biases in ML models (see e.g., Zemel +1MIT-IBM +Watson +AI +Lab +2Harvard +College +3UT- +Austin +4Harvard University. +Hao Wang , +Luxi +He +, +Rui +Gao +, +Flavio +P. +Calmon +<flavio@seas.harvard.edu>. +et al., 2013; Feldman et al., 2015; Calmon et al., 2017; +Menon & Williamson, 2018; Zhang et al., 2018; Zafar et al., +2019; Friedler et al., 2019; Bellamy et al., 2019; Kim et al., +2019; Celis et al., 2019; Yang et al., 2020; Jiang & Nachum, +2020; Jiang et al., 2020; Martinez et al., 2020; Lowy et al., +2021; Alghamdi et al., 2022). A recent survey (Hort et al., +2022) found nearly 400 fairness-intervention algorithms, +including 123 pre-processing, 212 in-processing, and 56 +post-processing algorithms introduced in the past decade. +Which sources of biases are (the hundreds of) existing fair- +ness interventions trying to control? In order to create effec- +tive strategies for reducing algorithmic discrimination, it is +critical to disentangle where biases in model performance +originate. For instance, if the training set contains few sam- +ples from a given population group, then increasing sample +diversity is a more effective strategy than selecting a more +complex model class or training strategy. Conversely, if a +model class does not accurately represent the underlying +distribution of a certain population group, then increasing +sample size for that group will not resolve performance +disparities. +We divide algorithmic discrimination into two categories: +aleatoric and epistemic discrimination.1 Aleatoric discrimi- +nation captures inherent biases in the data distribution that +can lead to unfair decisions in downstream tasks. Epistemic +discrimination, in turn, is due to algorithmic choices made +during model development and lack of knowledge about the +optimal “fair” predictive model. +In this paper, we provide methods for measuring aleatoric +and epistemic discrimination in classification task for group +fairness metrics. Since aleatoric discrimination only de- +pends on properties of the data distribution and the fairness +measure of choice, we quantify it by asking a fundamental +question: +For a given data distribution, what would be the best achiev- +able performance (e.g., accuracy) under a group fairness +constraint? +1We borrow this notion from ML uncertainty literature (see +H¨ullermeier & Waegeman, 2021, for a survey). Therein, aleatoric +uncertainty refers to the variability in the outcome of an experiment +resulting from inherently random effects; epistemic uncertainty +refers to uncertainty caused by a lack of knowledge about the best +predictive model. +arXiv:2301.11781v1 [cs.LG] 27 Jan 2023 + +Aleatoric and Epistemic Discrimination in Classification +We refer to the answer as the fairness Pareto frontier. This +frontier delineates the optimal performance achievable by +a classifier when unlimited data and computing power are +available. For a fixed data distribution, the fairness Pareto +frontier represents the ultimate, information-theoretic limit +for accuracy and group fairness beyond which no model can +achieve. Characterizing this limit enables us to (i) separate +sources of discrimination and create strategies to control +them accordingly; (ii) evaluate the effectiveness of existing +fairness interventions for reducing epistemic discrimination; +and (iii) inform the development of data collection methods +that promote fairness in downstream tasks. +At first, computing the fairness Pareto frontier can appear to +be an intractable problem since it requires searching over all +possible classifiers—even if the data distribution is known +exactly. Our main technical contribution is to provide a pre- +cise characterization of this frontier by solving a sequence of +optimization problems. Our main proof technique is based +on Blackwell’s seminal results (Blackwell, 1953), which +proposed the notion of comparisons of statistical experi- +ments and inspired a line of works introducing alternative +comparison criteria (see e.g., Shannon, 1958; Cam, 1964; +Torgersen, 1991; Cohen et al., 1998; Raginsky, 2011). Here, +we apply these results to develop an algorithm that itera- +tively refines the achievable fairness Pareto frontier. We also +prove convergence guarantees for our algorithm and demon- +strate how it can be used to benchmark existing fairness +interventions. +We quantify epistemic discrimination by comparing a clas- +sifier’s performance with the information-theoretic optimal +given by the fairness Pareto frontier. Our experiments indi- +cate that given sufficient data, state-of-the-art fairness inter- +ventions are effective at reducing epistemic discrimination +as their gap to the information-theoretic limit is small (see +Figure 1). Existing interventions do not eliminate aleatoric +discrimination as this type of discrimination is not caused +by choice of learning algorithm or model class, and is due +to the data distribution. +We further analyze the fairness Pareto frontier to show that +factors such as data missing values can significantly con- +tribute to aleatoric discrimination. We observe that when +population groups have disparate missing patterns, aleatoric +discrimination escalates, leading to a sharp decline in the +effectiveness of fairness intervention algorithms (see Fig- +ure 2). +Related Work +There is significant work analyzing the tension between +group fairness measures and model performance metrics +(Kleinberg et al., 2016; Chouldechova, 2017; Corbett- +Davies et al., 2017; Chen et al., 2018; Dutta et al., 2020; +Wang et al., 2021). We recast the fairness Pareto frontier +in terms of the conditional distribution PˆY|Y,S of predicted +outcome ˆY given true label Y and group attributes S. This +conditional distribution is related to confusion matrices2 +conditioned on each subgroup (see Remark 1 for detailed +discussion). In this regard, our work is related to Verma & +Rubin (2018); Alghamdi et al. (2020); Kim et al. (2020); +Yang et al. (2020); Berk et al. (2021), which observed that +many group fairness metrics can be written in terms of the +confusion matrices for each subgroup. Among them, the +closest work to ours is Kim et al. (2020), which optimized +accuracy and fairness objectives over these confusion matri- +ces and proposed a post-processing technique for training +fair classifiers. However, they only imposed marginal sum +constraints for the confusion matrices. We demonstrate +that the feasible region of confusion matrices can be much +smaller (see Remark 2 for an example), leading to a tighter +approximation of the fairness Pareto frontier. +Recently, many strategies have been proposed to reduce +the tension between group fairness and model performance +by investigating properties of the data distribution. For ex- +ample, Blum & Stangl (2019); Suresh & Guttag (2019); +Fogliato et al. (2020); Wang et al. (2020); Mehrotra & +Celis (2021); Fernando et al. (2021); Wang & Singh (2021); +Zhang & Long (2021); Kallus et al. (2022); Jeong et al. +(2022) studied how noisy or missing data affect fairness and +model accuracy. Dwork et al. (2018); Ustun et al. (2019); +Wang et al. (2021) considered training a separate classifier +for each subgroup when their data distributions are differ- +ent. Another line of research introduces data pre-processing +techniques that manipulate data distribution for reducing its +bias (e.g., Calmon et al., 2017; Kamiran & Calders, 2012). +Among all these works, the closest one to ours is Chen +et al. (2018), which decomposed group fairness measures +into bias, variance, and noise (see their Theorem 1) and pro- +posed strategies for reducing each term. The main difference +compared with Chen et al. (2018) is that we characterize +a fairness Pareto frontier that depends on not only fairness +metrics but also on a performance measure. This effort gives +a complete picture of how data distribution influences the +fairness-accuracy tension and is more technically involved. +Also, the fairness Pareto frontier only depends on the data +distribution and fairness metrics of choice so it cannot be +improved by adding more data or altering the model class. +2. Preliminaries +Next, we introduce notation, overview the key results in +Blackwell (1953) on comparisons of experiments, and out- +line the fair classification setup considered in this paper. +2A confusion matrix (Kulkarni et al., 2020) is a table that +measures the performance of a given ML model. In binary classi- +fication, a confusion matrix reports the number of true positives, +false negatives, false positives, and true negatives. + +Aleatoric and Epistemic Discrimination in Classification +Notation. +For a positive integer n, let [n] ≜ {1, · · · , n}. +We denote all probability distributions on the set X by +P(X). Moreover, we define the probability simplex ∆m ≜ +P([m]). When random variables A, X, Z form a Markov +chain, we write A → X → Z. We write the mutual infor- +mation between A, X as I(A; X) ≜ EPA,X +� +log +PA,X(A,X) +PA(A)PX(X) +� +. +Since I(A; X) is determined by the marginal distribution +PA and the conditional distribution PX|A, we also write +I(A; X) as I(PA; PX|A). When A, X are independent, we +write A +|= +X. +If a random variable A ∈ [n] has finite support, the condi- +tional distribution PX|A : [n] → P(X) can be equivalently +written as P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈ +P(X). Additionally, if X is a finite set [m], then PX|A +can be fully characterized by a transition matrix. We use +T (m|n) to denote all transition matrices from [n] to [m]: +� +� +�P ∈ Rn×m ��� 0 ≤ Pi,j ≤ 1, +m +� +j=1 +Pi,j = 1, ∀i ∈ [n] +� +� +� . +Comparisons of Experiments +Given two statistical experiments (i.e., conditional distribu- +tions) P and Q, is there a way to decide which one is more +informative? Here P and Q have the common input alpha- +bet [n] and potentially different output spaces. Blackwell +gave an answer in his seminal work (Blackwell, 1953) from +a decision-theoretic perspective. We review these results +next. +Let A be a closed, bounded, convex subset of Rn. A deci- +sion function f(x) = (a1(x), · · · , an(x)) is any mapping +from X to A. It is associated a loss vector: +v(f) = +�� +a1(x)dP1(x), · · · , +� +an(x)dPn(x) +� +. (1) +The collection of all v(f) is denoted by B(P , A). Black- +well defined that P is more informative than Q if for every +A, B(P , A) ⊇ B(Q, A). Intuitively, this result means any +risk achievable with Q is also achievable with P . Moreover, +Blackwell considered the standard measure P ∗ which is the +probability distribution of p(¯X) where p(x) : X → ∆n is a +function defined as +� +dP1 +dP1 + · · · + dPn +, · · · , +dPn +dP1 + · · · + dPn +� +. +(2) +and ¯X follows the probability distribution P1+···+Pn +n +. One +of the most important findings by Blackwell in his paper is +to discover the following equivalent conditions. +Lemma 1 (Blackwell (1951; 1953)). The following three +conditions are equivalent: +• P is more informative than Q; +• for any continuous and convex function φ : ∆n → R, +� +φ(p)dP ∗(p) ≥ +� +φ(p)dQ∗(p); +• there is a stochastic transformation T such that TPi = +Qi. In other words, there exists a Markov chain A → +X → Z for any distributions on A such that P = PX|A +and Q = PZ|A. +Additionally, if P and Q can be characterized by transition +matrices, the above conditions are also equivalent to: +• there exists a transition matrix M such that Q = +P M. +If P = PX|A is more informative than Q = PZ|A, by the +third condition of Lemma 1 and the data processing inequal- +ity, +I(PA; PX|A) ≥ I(PA; PZ|A) +(3) +holds for any marginal distribution PA. However, the con- +verse does not hold in general—even if (3) holds for any PA, +P is not necessarily more informative than Q (see Rauh et al. +(2017) for a counter-example). In this regard, Blackwell’s +conditions are “stronger” than mutual information-based +form of the data processing inequality. +Fair Classification +Consider a multi-class classification task, where the goal is +to train a probabilistic classifier h : X → ∆C that uses input +features X to predict their true label Y ∈ [C]. Additionally, +assume the classifier produces a predicted outcome ˆY ∈ [C] +and let S ∈ [A] represent group attributes (e.g., race and +sex). Our framework can be easily extended to the setting +where multiple subgroups overlap (Kearns et al., 2018). +Throughout this paper, we focus on three standard group +fairness measures: statistical parity (SP) (Feldman et al., +2015), equalized odds (EO) (Hardt et al., 2016; Pleiss et al., +2017), and overall accuracy equality (OAE) (Berk et al., +2021) (see Table 1 for their definitions) but our analysis can +be extended to many other group fairness metrics, including +the ones in Table 1 of Kim et al. (2020). +We use Blackwell’s conditions to provide a precise charac- +terization of fairness Pareto frontier in multi-class classifica- +tion. In brief, Blackwell’s conditions allow us to approxi- +mate the set of achievable joint distributions PˆY|S,Y across +all classifiers h. Since both accuracy and the group fairness +criteria in Table 1 can be cast in terms of PˆY|S,Y, this ap- +proximation can then be used to bound the best achievable +accuracy under a group fairness constraint. We will develop +this procedure in detail in the next section. + +Aleatoric and Epistemic Discrimination in Classification +FAIRNESS METRIC +ABBR. +DEFINITION +EXPRESSION W.R.T. P +Statistical Parity +SP ≤ αSP +| Pr(ˆY = ˆy|S = s) − Pr(ˆY = ˆy|S = s′)| ≤ αSP +��� +�C +y=1 +� +µs,y +µs P(s,y),ˆy − +µs′,y +µs′ P(s′,y),ˆy +���� ≤ αSP +Equalized Odds +EO ≤ αEO +| Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)| ≤ αEO +��P(s,y),ˆy − P(s′,y),ˆy +�� ≤ αEO +Overall Accuracy Equality +OAE ≤ αOAE +| Pr(ˆY = Y |S = s) − Pr(ˆY = Y |S = s′)| ≤ αOAE +����C +y=1 +� +µs,y +µs P(s,y),y − +µs′,y +µs′ P(s′,y),y +���� ≤ αOAE +Table 1. Standard group fairness metrics under multi-group and multi-class classification task. Here αSP, αEO, αOAE, ∈ [0, 1] are threshold +parameters, ˆy, y ∈ [C], s, s′ ∈ [A], and µs,y, µs are defined in Proposition 1. Our analysis can be extended to many other group fairness +metrics (see e.g., Table 1 in Kim et al., 2020). +3. Fairness Pareto Frontier +In this section, we introduce our main concept—fairness +Pareto frontier (FairFront). We use it to measure aleatoric +discrimination and quantify epistemic discrimination by +comparing a classifier’s performance to the FairFront. We +recast FairFront in terms of the conditional distribution +PˆY|S,Y and apply Blackwell’s conditions to characterize the +feasible region of this conditional distribution. This effort +converts a functional optimization problem into a convex +program with a small number of variables. However, this +convex program may involve infinitely many constraints. +Hence, we introduce a greedy improvement algorithm that +iteratively computes FairFront and tightens the feasible re- +gion of PˆY|S,Y. We end this section by establishing a con- +vergence guarantee for our algorithm. +Recall that we refer to aleatoric discrimination as the inher- +ent biases of the data distribution that can lead to an unfair +or inaccurate classifier. As its definition suggests, aleatoric +discrimination only relies on properties of the data distri- +bution and fairness metric of choice—it does not depend +on the hypothesis class nor optimization method. Below +we introduce FairFront that delineates a curve of optimal +accuracy over all probabilistic classifiers under certain fair- +ness constraints. We use FairFront to quantify aleatoric +discrimination. +Definition 1. For αSP, αEO, αOAE +∈ +[0, 1], we define +FairFront(αSP, αEO, αOAE) as the solution of the following +optimization problem: +max +h +E +� +IˆY=Y +� +(4a) +s.t. SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE +(4b) +where ˆY is produced by applying the classifier h to X; the +maximum is taken over all measurable h; and the definitions +of SP, EO, and OAE are in Table 1. +Solving this functional optimization problem is difficult +since it optimizes over all measurable classifiers. There is a +line of works that proposed different fairness-intervention al- +gorithms for training group-fair classifiers (see e.g., Menon +& Williamson, 2018; Zhang et al., 2018; Zafar et al., 2019; +Celis et al., 2019; Yang et al., 2020; Wei et al., 2021; Al- +ghamdi et al., 2022). They restrict the model class and +vary loss functions and optimizers to find classifiers that +approach FairFront as close as possible. However, these +algorithms only describe a lower bound for FairFront. They +do not determine what is the best achievable accuracy for a +given fairness constraint. +We circumvent the above-mentioned challenges by rewrit- +ing FairFront in terms of the conditional distribution PˆY|S,Y. +The caveat is that although each classifier yields a PˆY|S,Y, +not every conditional distribution corresponds to a valid +classifier (see an example in Remark 2). Hence, we intro- +duce the following definition which characterizes all feasible +PˆY|S,Y. +Definition 2. Given PX|S,Y, we define C as the set of all +conditional distributions PˆY|S,Y where ˆY is produced by +some probabilistic classifier h. In other words, +C ≜ +� +PˆY|S,Y | (S, Y) → X → ˆY +� +. +(5) +Throughout this paper, we write PˆY|S,Y or its corresponding +transition matrix P interchangeably. +Remark 1. We demonstrate the connection between the +conditional distribution PˆY|S,Y and confusion matrices in the +setting of binary classification with binary subgroups. We +define ˆC as the counterpart of C when we replace PX|S,Y with +an empirical distribution ˆPX|S,Y computed from a dataset. +The confusion matrix for group s ∈ {0, 1} consists of four +numbers: True Positive (TPs), False Positive (FPs), False +Negative (FNs), True Negative (TNs). Assume that the num- +ber of positive-label data n+ +s = TPs + FNs and negative- +label data n− +s = TNs + FPs are given—these numbers do +not depend on the classifier. Then there is a one-to-one + +Aleatoric and Epistemic Discrimination in Classification +mapping from each element in ˆC to a confusion matrix: +ˆPˆY|S=s,Y=+(+) = 1 +n+ +s +TPs, +ˆPˆY|S=s,Y=−(+) = 1 +n− +s +FPs, +ˆPˆY|S=s,Y=+(−) = 1 +n+ +s +FNs, +ˆPˆY|S=s,Y=−(−) = 1 +n− +s +TNs. +Hence, ˆC essentially characterizes all feasible confusion +matrices and C is the population counterpart of ˆC. Note +that C is determined by the data distribution while ˆC (and +confusion matrices) are tailored to a specific dataset. +We +establish +basic +properties +of +C +and +FairFront(αSP, αEO, αOAE) in the following lemma. Then we +demonstrate how to use C for characterizing the fairness +Pareto frontier. +Lemma 2. C is a convex subset of T (C|AC) and +FairFront(αSP, αEO, αOAE) is a concave function w.r.t. +αSP, αEO, αOAE. +Proposition 1. FairFront(αSP, αEO, αOAE) in (4) is equal to +the solution of the following convex optimization: +max +P ∈RAC×C +A +� +s=1 +C +� +y=1 +µs,yP(s,y),y +(6a) +s.t. SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE +(6b) +P ∈ C. +(6c) +Here the constants µs,y ≜ Pr(S = s, Y = y) and µs ≜ +Pr(S = s) for s ∈ [A], y ∈ [A] and P(s,y),ˆy denotes the +(C(s − 1) + y)-th row, ˆy-th column of P . +For example, in the setting of binary classification with +binary group attribute, the above optimization only has 8 +variables, 14 linear constraints + a single convex constraint +P ∈ C. Hence, its optimal value can be directly computed +by standard convex optimization solvers as long as we know +how to characterize C. Next, we discuss two special cases— +X is independent of (S, Y) or X is discrete—under which C +has a simple characterization. +Remark 2. Note that Kim et al. (2020) investigated fairness +Pareto frontiers via confusion matrices. The main difference +is that Definition 1 in Kim et al. (2020) relaxed the constraint +(6c) to P ∈ T (C|AC) where T (C|AC) represents all +transition matrices from [AC] to [C]. This leads to a loose +approximation of the frontier because C is often a strict +subset of T (C|AC). To demonstrate this point, consider +the scenario where X +|= +(S, Y). Then ˆY +|= +(S, Y) by data +processing inequality so +C = {P ∈ T (C|AC) | each row of P is the same} . (7) +Optimizing over C rather than T (C|AC) can significantly +tighten the fairness Pareto frontier. +Remark 3. If X is a discrete variable with a finite sup- +port [D], we can write PX|S,Y as a transition matrix Φ ∈ +T (D|AC). By introducing an auxiliary variable M ∈ +T (C|D), we can write P ∈ C equivalently as linear con- +straints: P = ΦM by using the last condition of Lemma 1. +Consequently, Proposition 1 boils down to a linear program. +However, this characterization fails to generalize to continu- +ous data because Φ and M will have an infinite dimension; +for categorical data, this characterization suffers from the +curse of dimensionality since the support size of X grows +exponentially fast w.r.t. the number of features. +The above two remarks provide precise characterizations of +C under specific assumptions. In what follows, we consider +a more general setting by leveraging Blackwell’s conditions +(Section 2). Before diving into the analysis, we first intro- +duce a function g : X → ∆AC defined as +g(x) = +� +PS,Y|X(1, 1|x), · · · , PS,Y|X(A, C|x) +� +. +(8) +To obtain this function in practice, one can train a probabilis- +tic classifier that uses input features X to predict (S, Y). We +use this classifiers’ output probability as an approximation +of the function g. +The following theorem is the main theoretical result in this +paper. It provides a precise characterization of the set C +through a series of convex constraints. +Theorem 1. The set C is the collection of all transition +matrices P ∈ T (C|AC) such that the following condition +holds: +For any k ∈ N and any {ai | ai ∈ [−1, 1]AC, i ∈ [k]}, +C +� +ˆy=1 +max +i∈[k] +� +aT +i ΛΛΛµpˆy +� +≤ E +� +max +i∈[k]{aT +i g(X)} +� +, +(9) +where pˆy +is the +ˆy-th column of P +and ΛΛΛµ += +diag(µ1,1, · · · , µA,C). +Intuitively, (9) uses piece-wise linear functions to approx- +imate the boundary of C where k represents the num- +ber of linear pieces. +Unfortunately, replacing P ∈ C +with this series of constraints in (6) may result in an in- +tractable problem since standard duality-based approaches +will lead to infinitely many dual variables. +To resolve +this issue, we first fix k and let Ck be the set of P such +that (9) holds under this fixed k. Accordingly, we define +FairFrontk(αSP, αEO, αOAE) as the optimal value of (6) when +replacing C with Ck. Since C1 ⊇ C2 ⊇ · · · ⊇ C, we have +FairFront1(αSP, αEO, αOAE) ≥ FairFront2(αSP, αEO, αOAE) ≥ +· · · +≥ +FairFront(αSP, αEO, αOAE). However, computing +FairFrontk(αSP, αEO, αOAE) still involves infinitely many con- +straints. +Next, we introduce a greedy improvement algorithm that +consists of solving a sequence of tractable optimization + +Aleatoric and Epistemic Discrimination in Classification +Algorithm 1 Approximate the fairness Pareto frontier. +Input: D = {(xi, yi, si)}N +i=1, maximum number of iterations +T; maximum pieces k; classifier g(x); threshold parameters +αSP, αEO, αOAE. +Initialize: set A = ∅; µs,y = |{i|si=s,yi=y}| +N +; t = 1. +Repeat: +Solve a convex program: +max +P +A +� +s=1 +C +� +y=1 +µs,yP(s,y),y +s.t. P ∈ T (C|AC) +SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE +C +� +ˆy=1 +max +i∈[k] +� +aT +i ΛΛΛµpˆy +� +≤ E +� +max +i∈[k]{aT +i g(X)} +� +∀(a1, · · · , ak) ∈ A. +Let vt and P t be the optimal value and optimal solution. +Solve a DC program: +min +ai∈[−1,1]AC +i∈[k] +E +� +max +i∈[k]{aT +i g(X)} +� +− +C +� +ˆy=1 +max +i∈[k] +� +aT +i ΛΛΛµpt +ˆy +� +. +If the optimal value is ≥ 0 or t = T, +stop; +otherwise, +add the optimal (a1, · · · , ak) to A and t = t + 1. +return: vt, P t, A. +problems for approximating FairFrontk(αSP, αEO, αOAE). We +use A to collect the constraints of P and set A = ∅ initially. +At each iteration, our algorithm solves a convex program +to find an optimal P that maximizes the accuracy while +satisfying the desired group fairness constraints and the con- +straints in A; then we verify if this P is within the set Ck by +solving a DC (difference of convex) program (Shen et al., +2016; Horst & Thoai, 1999). If P ∈ Ck, then the algo- +rithm stops; otherwise, the algorithm will find the constraint +that is mostly violated by P and add this constraint to A. +We describe our algorithm in Algorithm 1 and establish a +convergence guarantee in the following theorem. +Theorem 2. Let T = ∞. If Algorithm 1 stops, its output P t +is an optimal solution of FairFrontk(αSP, αEO, αOAE). Other- +wise, any convergent sub-sequence of {P t}∞ +t=1 converges +to an optimal solution of FairFrontk(αSP, αEO, αOAE). +Note that the output vt from Algorithm 1 is always an up- +per bound for FairFront(αSP, αEO, αOAE), assuming the es- +timation error is sufficiently small. The tightness of this +upper bound is determined by k (i.e., how well the piece- +wise linear functions approximate the boundary of C), T +(i.e., the total number of iterations). On the other hand, +running off-the-shelf in-processing and post-processing +fairness interventions can only yield lower bounds for +FairFront(αSP, αEO, αOAE). In the next section, we compare +our upper bound given by Algorithm 1 with the lower +bounds given by some state-of-the-art methods to demon- +strate the tightness of our algorithm. +4. Numerical Experiments +Recall that FairFront in Definition 1 characterizes the high- +est achievable accuracy under a fairness constraint. We use +it to quantify aleatoric discrimination and measure epistemic +discrimination by comparing a classifier’s accuracy and fair- +ness violation with FairFront. In this section, we apply +FairFront to analyze the performance of existing fairness in- +terventions and how data biases, specifically missing values, +impact their effectiveness. We find that given sufficient data, +state-of-the-art fairness interventions are successful at reduc- +ing epistemic discrimination as their gap to the FairFront is +small. However, we also discover that when different popu- +lation groups have varying missing data patterns, aleatoric +discrimination increases, which diminishes the performance +of fairness intervention algorithms. We provide additional +experimental results and details in Appendix C. +4.1. Benchmark Fairness Interventions +Setup. +We evaluate our results on the UCI Adult dataset +(Bache & Lichman, 2013), the ProPublica COMPAS dataset +(Angwin et al., 2016), and the German Credit dataset (Bache +& Lichman, 2013).3 We consider a binary classification +problem with binary groups since most existing fairness +interventions are designed for this scenario. For the Adult +dataset, we choose sex (female or male) as the group at- +tribute and income (> 50K or <= 50K) as the target for pre- +diction; for the COMPAS dataset, we choose race (African- +American or Caucasian) as the group attribute and is recid +(recid. or no recid.) as the target for prediction. The details +about how we pre-process these datasets are deferred to Ap- +pendix C. We measure fairness violations via Max equalized +odds: +max | Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)| +where the max is taken over y, ˆy, s, s′. We run Algorithm 1 +with k = 6 pieces, 20 iterations, and varying αEO to get +FairFront on each dataset. We compute the expectations +and the g function from the empirical distributions and solve +the DC program by using the DCCP package provided by +Shen et al. (2016). +Fairness +interventions. +We +consider +five +existing +fairness-intervention algorithms: +Reduction (Agar- +wal +et +al., +2018), +EqOdds +(Hardt +et +al., +2016), +CalEqOdds (Pleiss et al., 2017), LevEqOpp (Chzhen +3We defer the experimental results on the German Credit +dataset to Appendix C. + +Aleatoric and Epistemic Discrimination in Classification +Figure 1. Comparing existing fairness interventions with FairFront on the Adult (Left) and COMPAS (Right) datasets. We use FairFront +to quantify aleatoric discrimination and measure epistemic discrimination by comparing a classifier’s accuracy and fairness violation +with FairFront. As shown, SOTA fairness interventions are effective at reducing epistemic discrimination as their gap to the FairFront is +small. +et al., 2019), and FairProjection (Alghamdi et al., +2022). +Among them, Reduction is an in-processing +method and the rest are all post-processing methods. For +the first three benchmarks, we use the implementations +from IBM AIF360 library (Bellamy et al., 2018); for +LevEqOpp and FairProjection, we use the Python +implementations from the Github repo in Alghamdi et al. +(2022). For Reduction and FairProjection, we +can vary their tolerance of fairness violations to produce a +fairness-accuracy curve; for EqOdds, CalEqOdds, and +LevEqOpp, each of them produces a single point since +they only allow hard equality constraint. We note that +FairProjection is optimized for transforming prob- +abilistic classifier outputs (see also Wei et al., 2021), but +here we threshold the probabilistic outputs to generate bi- +nary predictions which may limit its performance. Finally, +we train a random forest as the Baseline classifier. In +order to emulate the setting where the data distribution is +known exactly, we train models using the entire dataset and +resample 30% data as the test set. +Results. +We benchmark fairness interventions against +FairFront in Figure 1. First, we observe that if we run +Algorithm 1 for a single iteration, which is equivalent to +solving Proposition 1 without (6c), its solution is very close +to 1 for all αEO. This demonstrates the benefits of incorporat- +ing Blackwell’s conditions into the fairness Pareto frontier. +Second, we observe that fairness-accuracy curves given +by state-of-the-art (SOTA) fairness interventions are very +close to the fairness Pareto frontier. This result not only +demonstrates the tightness of our approximation (recall +that Algorithm 1 gives an upper bound of FairFront and +benchmarks give a lower bound) but also shows that SOTA +fairness interventions have already achieved near-optimal +fairness-accuracy curves—their epistemic discrimination +is small since they approach the FairFront limit. In what +follows, we demonstrate how missing values in data can +increase aleatoric discrimination and dramatically reduce +the effectiveness of fairness interventions. +4.2. Fairness Risks in Missing Values +Real-world data often have missing values and the missing +patterns can be different across different protected groups +(see Jeong et al., 2022, for some examples). There is a grow- +ing line of research (see e.g., Jeong et al., 2022; Fernando +et al., 2021; Wang & Singh, 2021; Subramonian et al., 2022; +Caton et al., 2022; Zhang & Long, 2021; Schelter et al., +2019) studying the fairness risks of data with missing val- +ues. In this section, we apply our result to demonstrate how +disparate missing patterns influence the fairness-accuracy +curves. +Setup. +We choose sex (group 0: female, group 1: male) as +the group attribute for the Adult dataset, and race (group 0: +African-American, group 1: Caucasian) for the COMPAS +dataset. To investigate the impact of disparate missing pat- +terns on aleatoric discrimination, we artificially generate +missing values in both datasets. This is necessary as the +datasets do not contain sufficient missing data. The miss- +ing values are generated according to different probabilities +for different population groups. For each data point from +group 0, we erase each input feature with a varying proba- +bility p0 ∈ {10%, 50%, 70%}, while for group 1, we erase + +Adult +Aleatoric +84.5 +Epistemic +discrimination, +discrimination +84.0 +(%) +83.5 +Accuracy ( +83.0 +FairFront +Baseline +82.5 +FairProjection +Reduction +82.0 +LevEqOpp +CalEqOdds +EqOdds +81.5 +0 +2.5 +5.0 +7.50 +10.0 +12.5 +Max equalized odds (%)COMPAS +77.0 +76.0 +(%) +Accuracy +75.0 +74.0 +FairFront +Baseline +FairProjection +Reduction +73.0 +LevEqOpp +CalEqOdds +EqOdds +72.0 +0 +5.0 +10.0 +15.0 +20.0 +Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification +Figure 2. We demonstrate the fairness risks of disparate missing patterns. We vary missing probability of group 0 (female in Adult/African- +American in COMPAS) among {10%, 50%, 70%} and let the missing probability of group 1 (male in Adult/Caucasian in COMPAS) be +10%. We use mode imputation to pre-process missing data. We apply Reduction to the imputed data and plot its fairness-accuracy +curve against the FairFront with the level of transparency representing the degree of disparity in the missing patterns. +each input feature with a fixed probability p1 = 10%. We +then apply mode imputation to the missing values, replacing +them with the mode of non-missing values for each feature. +Finally, we apply Algorithm 1 along with Reduction and +Baseline to the imputed data. The experimental results +are shown in Figure 2. +Results. +As we increase the missing probability of +group 0, FairFront decreases since it becomes more difficult +to accurately predict outcomes for group 0. This in turn +affects the overall model performance, since the fairness +constraint requires that the model performs similarly for +both groups. We also observe the fairness-accuracy curves +of Reduction decrease as the missing data for group 0 +become more prevalent. In other words, as the missing data +for group 0 increase, it becomes more difficult to maintain +both high accuracy and fairness in the model’s prediction. +5. Final Remarks and Limitations +The past years have witnessed a growing line of research in- +troducing various fairness-intervention algorithms. Most of +these interventions focus on optimizing model performance +subject to group fairness constraints. Though comparing +and benchmarking these methods on various datasets is valu- +able (e.g., see benchmarks in Friedler et al., 2019; Bellamy +et al., 2019; Wei et al., 2021), this does not reveal if there is +still room for improvement in their fairness-accuracy curves, +or if existing methods approach the information-theoretic +optimal limit when infinite data is available. Our results +address this gap by introducing the fairness Pareto frontier, +which measures the highest possible accuracy under a group +fairness constraint. We precisely characterize the fairness +Pareto frontier using Blackwell’s conditions and present a +greedy improvement algorithm that approximates it from +data. Our results show that the fairness-accuracy curves +produced by state-of-the-art fairness interventions are very +close to the fairness Pareto frontier on standard datasets. +Additionally, we demonstrate that when data are biased +due to missing values, the fairness Pareto frontier degrades. +Although existing fairness interventions can still reduce per- +formance disparities, they come at the cost of significantly +lowering overall model accuracy. The methods we present +for computing the fairness Pareto frontier can also be ap- +plied to analyze other sources of aleatoric discrimination, +such as when individuals may misreport their data or when +there are measurement errors. Overall, the fairness Pareto +frontier can serve as a valuable framework for guiding data +collection and cleaning. +Our results indicate that existing fairness interventions can +be effective in reducing epistemic discrimination, and there +are diminishing returns in developing new fairness inter- +ventions focused solely on optimizing accuracy for a given +group fairness constraint on pristine data. However, existing +fairness interventions have yet to effectively provide both +fair and accurate classification when additional sources of +aleatoric discrimination are present (such as missing values +in data). This suggests that there is still significant need for +research on handling aleatoric sources of discrimination that +appear throughout the data collection process. + +Adult +84.5 +84.0 +Accuracy (%) +83.5 +83.0 +82.5 +82.0 +FairFront +Baseline +Reduction +81.5 +0 +10.0 +20.0 +30.0 +40.0 +50.0 +Max equalized odds (%)COMPAS +76.0 +74.0 +72.0 +%) +70.0 +Accuracy ( +68.0 +65.9 +63.9 +62.0 +FairFront +Baseline +60.0 +Reduction +0 +20.0 +40.0 +60.0 +Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification +References +Agarwal, A., Beygelzimer, A., Dud´ık, M., Langford, J., and +Wallach, H. A reductions approach to fair classification. +In International Conference on Machine Learning, pp. +60–69. PMLR, 2018. +Alghamdi, W., Asoodeh, S., Wang, H., Calmon, F. P., Wei, +D., and Ramamurthy, K. N. Model projection: Theory +and applications to fair machine learning. In 2020 IEEE +International Symposium on Information Theory (ISIT), +pp. 2711–2716. 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A conditional distribution PX|A : [n] → P(X) can be equivalently written as +P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈ P(X). Let A be a closed, bounded, convex subset of Rn. A decision +function is a mapping f : X → A, which can also be written as f(x) = (a1(x), · · · , an(x)). A decision function is +associated a loss vector: +v(f) = +�� +a1(x)dP1(x), · · · , +� +an(x)dPn(x) +� +. +(11) +The collection of all v(f) is denoted by B(PX|A, A) or B(P , A). +For a vector λλλ ∈ ∆n such that λλλ > 0, we define a function pλλλ(x) : X → ∆n: +pλλλ(x) = +� +λ1dP1 +λ1dP1 + · · · + λndPn +, · · · , +λndPn +λ1dP1 + · · · + λndPn +� +. +(12) +Note that pλλλ(X) is a sufficient statistic for X, considering A as the parameter (it can be proved by Fisher-Neyman factorization +theorem). In other words, two Markov chains hold: A → pλλλ(X) → X and A → X → pλλλ(X) for any distribution on A. +Consider a new set of probability distributions P ∗ +λλλ ≜ (L(pλλλ(X1)), · · · , L(pλλλ(Xn))) where L(Xi) = Pi. Here P ∗ +λλλ can be +viewed as a conditional distribution from [n] to P(∆n) since each L(pλλλ(Xi)) is a probability distribution over ∆n. The +following lemma follows from the sufficiency of pλλλ(X). +Lemma 3 (Adaptation of Theorem 3 in Blackwell (1951)). For any A, B(P , A) = B(P ∗ +λλλ, A). +Proof. Suppose that f ∗(p) = (a∗ +1(p), · · · , a∗ +n(p)) is a decision function for (P ∗ +λλλ, A). Accordingly, we define f(x) = +(a∗ +1(pλλλ(x)), · · · , an(pλλλ(x))) where the function pλλλ is defined in (12). Then it is clear that f is a decision function for +(P , A). By the law of unconscious statistician, we have +� +a∗ +i (p)dP ∗ +λ +λ +λ,i(p) = E [a∗ +i (pλλλ(Xi))] = +� +a∗ +i (pλλλ(x))dPi(x). +(13) +Hence, v(f ∗) = v(f), which implies B(P ∗ +λλλ, A) ⊆ B(P , A). For the other direction, suppose f(x) = (a1(x), · · · , an(x)) +is a decision function for (P , A). Let f ∗(p) = (a∗ +1(p), · · · , a∗ +n(p)) where a∗ +i (p) ≜ E [ai(Xi) | pλλλ(Xi) = p]. Since pλλλ(X) +is a sufficient statistics, for any i ∈ [n] +L(Xi|pλλλ(Xi) = p) = L(X1|pλλλ(X1) = p). +(14) +Therefore, f ∗(p) = E [f(X1)|pλλλ(X1) = p]. Since A is a convex set, f ∗ is a decision function for (P ∗, A). By the law of +total expectation, we have +� +a∗ +i (p)dP ∗ +λλλ,i(p) = +� +ai(x)dPi(x). +(15) +Hence, v(f) = v(f ∗), which implies B(P , A) ⊆ B(P ∗ +λλλ, A). +For a vector λλλ ∈ ∆n such that λλλ > 0, the condition distribution PX|A induces a weighted standard measure P ∗ +λλλ ≜ L (pλλλ(¯X)) +where L(¯X) = λ1P1 + · · · + λnPn. +Theorem 3 (Adaptation of Theorem 4 in Blackwell (1951)). For any two conditional distributions PX|A and QY|A, let +P ∗ +λλλ and Q∗ +λλλ be their weighted standard measures, respectively. Then B(PX|A, A) ⊇ B(QY|A, A) for any closed, bounded, +convex set A if and only if for any continuous convex φ : ∆n → R, +� +φ(p)dP ∗ +λλλ(p) ≥ +� +φ(p)dQ∗ +λλλ(p) +Proof. First, by Lemma 3, we know B(PX|A, A) = B(P ∗ +λλλ, A) and B(QY|A, A) = B(Q∗ +λλλ, A). +We denote ΛΛΛ = +diag(λ1, · · · , λn). Consider any A = conv(a1, · · · , ak). Let +f ∗(p) = argmin +a∈A +pTΛΛΛ−1a. +(16) + +Aleatoric and Epistemic Discrimination in Classification +Note that f ∗(p) ∈ {a1, · · · , ak} since this set contains all the extreme points of A.4 By definition, for any decision function +w.r.t. (P ∗ +λλλ, A), we have +pTΛΛΛ−1f(p) ≥ pTΛΛΛ−1f ∗(p), +∀p. +(17) +Let v = v(f). By the same reason with (13), we have +vj = +� +aj(pλλλ(x))dPj(x) +(18) += 1 +λj +� +aj(pλλλ(x)) +λjdPj +λ1dP1 + · · · + λndPn +(x)(λ1dP1 + · · · + λndPn)(x) +(19) += 1 +λj +� +aj(pλλλ(x))[pλλλ(x)]j(λ1dP1 + · · · + λndPn)(x) +(20) += 1 +λj +E [aj(pλλλ(¯X))[pλλλ(¯X)]j] +(21) += 1 +λj +� +aj(p)pjdP ∗ +λλλ(p), +(22) +where the last step is due to the law of unconscious statistician. Therefore, +n +� +j=1 +vj = +� +pTΛΛΛ−1f(p)dP ∗ +λλλ(p) +(23) +≥ +� +pTΛΛΛ−1f ∗(p)dP ∗ +λλλ(p) +(24) += +� +min +i {pTΛΛΛ−1ai}dP ∗ +λλλ(p). +(25) +The equality is achieved by v(f ∗). Hence, for any A = conv(a1, · · · , ak) +min +v∈B(PX|A,A) +n +� +j=1 +vj = +� +min +i {aT +i ΛΛΛ−1p}dP ∗ +λλλ(p). +(26) +Recall that Theorem 2.(3) in Blackwell (1951) states +B(PX|A, A) ⊇ B(PY|A, A) +for every closed, bounded, convex A +⇔ +min +v∈B(PX|A,A) +n +� +j=1 +vj ≤ +min +v∈B(PY|A,A) +n +� +j=1 +vj +for every closed, bounded, convex A. +By approximation theory, the second condition can be relaxed to any A that is a convex hull of a finite set. By (26), this +relaxed condition is equivalent to +� +φ(p)dP ∗ +λλλ(p) ≥ +� +φ(p)dQ∗ +λλλ(p) +(27) +for all φ(p) that are the maximum of finitely many linear functions. By approximation theory again, the above condition is +equivalent to the one holding for any continuous convex function φ. +B. Omitted Proofs +B.1. Proof of Lemma 2 +Proof. Clearly, C is a subset of T (C|AC). Let λ ∈ (0, 1) and PˆY0|S,Y, PˆY1|S,Y ∈ C. Now we introduce a Bernoulli random +variable B such that Pr(B = 0) = λ. Finally, we define ˆYλ = BˆY1 +(1−B)ˆY0. By definition, we have (S, Y) → X → ˆYλ +4If (16) has multiple optimal solutions, we always choose the one from {a1, · · · , ak}. + +Aleatoric and Epistemic Discrimination in Classification +so PˆYλ|S,Y ∈ C. Moreover, +PˆYλ|S,Y = λPˆY0|S,Y + (1 − λ)PˆY1|S,Y. +Hence, C is convex. +Let λ ∈ (0, 1). Assume P and ¯P achieve the maximal values of Proposition 1 under (αSP, αEO, αOAE) and (¯αSP, ¯αEO, ¯αOAE), +respectively. We define Pλ = λP + (1 − λ) ¯P , which satisfies the constraints of Proposition 1 with thresholds (λαSP + (1 − +λ)¯αSP, λαEO + (1 − λ)¯αEO, λαOAE + (1 − λ)¯αOAE). Finally, since the objective function of Proposition 1 is a linear function, it +is equal to λFairFront(αSP, αEO, αOAE) + (1 − λ)FairFront(¯αSP, ¯αEO, ¯αOAE) under Pλ. +B.2. Proof of Theorem 1 +Proof. The proof relies on Theorem 3 and Lemma 1. For simplicity, we write the conditional PˆY|S,Y as its corresponding +transition matrix P . Let µµµ = (Pr(S = 1, Y = 1), · · · , Pr(S = A, Y = C)). The function (12) in our setting can be written +as: +pµµµ(ˆy) = +� +µ1,1P(1,1),ˆy +� +s,y µs,yP(s,y),ˆy +, · · · , +µA,CP(A,C),ˆy +� +s,y µs,yP(s,y),ˆy +� +. +(28) +pµµµ(x) = +� +µ1,1dPX|S=1,Y=1 +� +s,y µs,ydPX|S=s,Y=y +(x), · · · , +µA,CdPX|S=A,Y=C +� +s,y µs,ydPX|S=s,Y=y +(x) +� +. +(29) +Note that pµµµ(x) = g(x) due to Bayes’ rule (see (8) for the definition of g). By Lemma 1, we can rewrite C in Definition 2 as +C = +� +P | PˆX|S,Y is more informative than P +� +. +(30) +By Lemma 1 and Theorem 3, the above set is further equivalent to all transition matrices P ∈ T (C|AC) satisfying +C +� +ˆy=1 +φ +� +µ1,1P(1,1),ˆy +� +s,y µs,yP(s,y),ˆy +, · · · , +µA,CP(A,C),ˆy +� +s,y µs,yP(s,y),ˆy +� � +s,y +µs,yP(s,y),ˆy ≤ E [φ(g(X))] +(31) +for any function φ : ∆AC → R which is the maximum of finitely many linear functions. Now we can write φ(p) = +maxi∈[k] +� +aT +i p +� +—we ignore the bias term because aT +i p+bi = (ai +bi1)T p. Then the inequality in (31) can be simplified +as +C +� +ˆy=1 +max +i∈[k] +� +aT +i ΛΛΛµpˆy +� +≤ E +� +max +i∈[k]{aT +i g(X)} +� +, +(32) +where pˆy is the ˆy-th column of P and ΛΛΛµ = diag(µ1,1, · · · , µA,C). Finally, we can always normalize the above inequality +so that each ai ∈ [−1, 1]AC. +B.3. Proof of Theorem 2 +Proof. We denote +f(P ) ≜ +A +� +s=1 +C +� +y=1 +µs,yP(s,y),y, +g(P ; a1, · · · , ak) ≜ +C +� +ˆy=1 +max +i∈[k] +� +aT +i ΛΛΛµpˆy +� +− E +� +max +i∈[k]{aT +i g(X)} +� +, +F ≜ Ck ∩ {P ∈ T (C|AC) | SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE} . +Let Ft be the constraint set of P at the t-th iteration of our algorithm. Note that F ⊆ Ft by definition. If the algorithm +stops at the t-th iteration, then for any {ai | ai ∈ [−1, 1]AC, i ∈ [k]}, P t satisfies +g(P t; a1, · · · , ak) ≤ 0, + +Aleatoric and Epistemic Discrimination in Classification +which implies P t ∈ F. Consequently, +f(P t) = max +P ∈Ft f(P ) ≥ max +P ∈F f(P ) ≥ f(P t). +As a result, f(P t) = maxP ∈F f(P ) so P t is an optimal solution of FairFrontk(αSP, αEO, αOAE). +If the algorithm never stops, consider any convergent sub-sequence of P t that converges to a limit point P ∗ ∈ T (C|AC). +To simplify our notation, we assume P t → P ∗ as t → ∞. Since {Ft}t≥1 is non-increasing and they all contain F, there +exists a set F∗ such that +lim +t→∞ Ft = F∗, +F ⊆ F∗. +Therefore, we have +f(P ∗) = lim +t→∞ f(P t) = lim +t→∞ max +P ∈Ft f(P ) = max +P ∈F∗ f(P ). +Since F ⊆ F∗, we have +f(P ∗) = max +P ∈F∗ f(P ) ≥ max +P ∈F f(P ). +If P ∗ ̸∈ F, then there exists a (¯a1, · · · , ¯ak), such that g(P ∗; ¯a1, · · · , ¯ak) > 0. Let (a1,t, · · · , ak,t) be the output of Step +2 at t-th iteration. Since P ∗ ∈ Ft for all t, we have +g(P ∗; a1,t, · · · , ak,t) ≤ 0. +(33) +By the optimality of (a1,t, · · · , ak,t), we have +g(P t; a1,t, · · · , ak,t) ≥ g(P t; ¯a1, · · · , ¯ak). +(34) +Suppose that some sub-sequence of (a1,t, · · · , ak,t) converges to a vector (a∗ +1, · · · , a∗ +k). For the sake of simplicity, we +assume (a1,t, · · · , ak,t) → (a∗ +1, · · · , a∗ +k) as t → ∞. On the one hand, taking limit of t → ∞ on both sides of (34) leads to +g(P ∗; a∗ +1, · · · , a∗ +k) ≥ g(P ∗; ¯a1, · · · , ¯ak). +On the other hand, taking limit of t → ∞ on both sides of (33) leads to +g(P ∗; a∗ +1, · · · , a∗ +k) ≤ 0. +Therefore, +0 ≥ g(P ∗; a∗ +1, · · · , a∗ +k) ≥ g(P ∗; ¯a1, · · · , ¯ak) > 0, +which is impossible. Therefore, P ∗ ∈ F and, as a result, we have +f(P ∗) = max +P ∈F∗ f(P ) ≥ max +P ∈F f(P ) ≥ f(P ∗) =⇒ max +P ∈F f(P ) = f(P ∗). +C. Details on the Experimental Results +C.1. Additional Experiments +In this section, we present additional experimental results to further support our findings. First, we reproduce our ex- +perimental results on the German Credit dataset (Bache & Lichman, 2013). We compare existing fairness interventions +with FairFront in Figure 3. Our observation is consistent with those on the previous two datasets—the fairness- +accuracy curves given by SOTA fairness interventions, such as Reduction and FairProjection, are close to the +information-theoretic limit. +As previously demonstrated in Figure 1 and 3, the fairness-accuracy curves generated by Reduction and +FairProjection are close to FairFront. To further evaluate the performance of these methods, we train a classifier that +approximates the Bayes optimal and feed it to Reduction and FairProjection. The results are shown in Figure 4, +which demonstrates that the (training) accuracy-fairness curves generated by Reduction and FairProjection can +approach FairFront when using the Bayes optimal baseline classifier. + +Aleatoric and Epistemic Discrimination in Classification +Figure 3. Comparing existing fairness interventions with FairFront on the German Credit dataset. +C.2. Dataset +Adult. +We use sex (female or male) as the group attribute and income (> 50K or <= 50K) as the target for +prediction. +We use sex, hours-per-week, education-num, age, marital status, relationship status (husband or wife) +as the input features—we include the group attribute as an input feature. +We group age into a total of 12 dis- +joint intervals: [0, 20), [20, 25), · · · , [65, 70), [70, ∞); we group hours-per-week into a total of 14 disjoint intervals: +[0, 10), [10, 15), · · · , [65, 70), [70, ∞). +COMPAS. +We use race (African-American or Caucasian) as the group attribute and is recid (recid. or no recid.) as the +target for prediction. We use race, age, c charge degree, sex, priors count, c jail in, c jail out as the input features—we +include the group attribute as an input feature. We use the last two features by taking their difference to be their length of stay. +We remove entries where COMPAS case could not be found (is recid = -1) and entries with inconsistent arrest information. +We also binarize sex and remove traffic offenses. We quantize age the same way we do in the Adult dataset and quantize +length of stay by every 30 days and let 0 be a separate category. +German Credit. +We use age (below or above 25 years old) as the group attribute and the credit column, which represents +whether the loan was a good decision, as the target for prediction. We use loan duration in month, credit amount, age, +number of existing credits at this bank, sex, credit history, savings, and length of present employment as input features. We +include the group attribute age as an input feature. We group credit amount into three disjoint intervals: [0, 5000), [5000, +10000),[10000,∞). We group duration of loan into two categories: under 36 months and over 36 months. +C.3. Benchmark +Each benchmark method’s hyper-parameter values are provided below. Each point in Figure 1 for Baseline, EqOdds, +CalEqOdds, Reduction, LevEqOpp, and FairProjection is obtained by applying the obtained classifier to 10 +different test sets. For the Adult dataset, we use Random Forest with n estimators=15, min samples leaf=3, criterion = +log loss, bootstrap = False as our baseline classifier; for the COMPAS dataset, we use Random Forest with n estimators = 17 +as our baseline classifier. For the German Credit dataset, we use Random Forest with n estimators=100,min samples split +=2,min samples leaf=1 as our baseline classifier. They are all implemented by Scikit-learn (Pedregosa et al., 2011). +EqOdds (Hardt et al., 2016). +We use AIF360 implementation of EqOddsPostprocessing and the default hyper- +parameter setup. +CalEqOdds (Pleiss et al., 2017). +We use AIF360 implementation of CalibratedEqOddsPostprocessing and +the default hyper-parameter setup. + +German Credit +86.0 +85.5 +85.0 +%) +84.5 +Accuracy ( +84.0 +83.5 +FairFront +Baseline +FairProjection +83.0 +Reduction +LevEqOpp +82.5 +CalEqOdds +EqOdds +0 +5.0 +10.0 +15.0 +20.0 +Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification +Figure 4. Comparing Reduction and FairProjection with FairFront on the Adult (Left), COMPAS (Middle), and German Credit +(Right) datasets. We train a baseline classifier that approximates the Bayes optimal and feed it into the two fairness interventions. As +shown, their fairness-accuracy curves are very close to the FairFront in this case. +Reduction (Agarwal et al., 2018). +We use AIF360 implementation of ExponentiatedGradientReduction. +We vary the allowed fairness constraint violation ϵ ∈ {0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15} for Adult dataset and ϵ ∈ +{0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15} for Adult with missing values. We vary ϵ ∈ {0.001, 2, 5, 10, 15, 20, 25, 30, 35, 40} for +COMPAS to obtain a fairness-accuracy curve, and ϵ ∈ {0.001, 0.1, 0.5, 1, 2, 7, 8, 10, 15, 20, 25, 30} for COMPAS with 50% +missing values in the minority group. We use ϵ ∈ {20, 50, 80, 95} for German Credit dataset and ϵ ∈ {5, 8, 10, 20, 23} +when using Bayes Optimal classifier. +LevEqOpp (Chzhen et al., 2019). +We use the Python implementation of LevEqopp from the Github repo in Alghamdi +et al. (2022). We follow the same hyperparameters setup as in the original method. +FairProjection (Alghamdi et al., 2022). +We use the implementation from the Github repo in Alghamdi et al. (2022) +and set use protected = True. We use Random Forest with n estimators = 17 as the baseline classifier to predict S from +(X, Y). We set the list of fairness violation tolerance to be {0.07, 0.075, 0.08, 0.085, 0.09, 0.095, 0.1, 0.5, 0.75, 1.0} for +Adult dataset and {0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.1, 0.5, 1.0} for COMPAS dataset to obtain a fairness-accuracy +curve. We set the list of fairness violation tolerance to be {0.005, 0.01, 0.02, 0.07, 0.1, 0.15} on the German Credit dataset +experiment, and {0.0001, 0.001, 0.005, 0.01, 0.015, 0.02, 0.05} when using a Bayes optimal baseline classifier. + +Adult +84.5 +84.2 +%) +84.0 +Accuracy +83.7 +83.5 +83.2 +FairFront +83.0 +Baseline +FairProjection +Reduction +82.7 +0 +2.5 +5.00 +7.50 +10.0 +12.5 +Max equalized odds (%)COMPAS +77.0 +76.8 +Accuracy +76.6 +76.4 +FairFront +Baseline +FairProjection +76.2 +Reduction +0 +5.0 +10.0 +15.0 +20.0 +Max equalized odds (%)German Credit +86.0 +85.9 +85.9 +(%) +85.8 +Accuracy ( +85.8 +85.7 +85.7 +FairFront +85.6 +Baseline +FairProjection +85.6 +Reduction +0 +5.0 +10.0 +15.0 +20.0 +25.0 +Max equalized odds (%) \ No newline at end of file diff --git a/6tFKT4oBgHgl3EQfTy2d/content/tmp_files/load_file.txt b/6tFKT4oBgHgl3EQfTy2d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f2400351883c78eaca0dbcbc98e9e6ae0e94e83 --- /dev/null +++ b/6tFKT4oBgHgl3EQfTy2d/content/tmp_files/load_file.txt @@ -0,0 +1,1294 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf,len=1293 +page_content='Aleatoric and Epistemic Discrimination in Classification Hao Wang 1 Luxi He 2 Rui Gao 3 Flavio P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' Calmon 4 Abstract Machine learning (ML) models can underperform on certain population groups due to choices made during model development and bias inherent in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' We categorize sources of discrimina- tion in the ML pipeline into two classes: aleatoric discrimination, which is inherent in the data distri- bution, and epistemic discrimination, which is due to decisions during model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' We quan- tify aleatoric discrimination by determining the performance limits of a model under fairness con- straints, assuming perfect knowledge of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' We demonstrate how to characterize aleatoric discrimination by applying Blackwell’s results on comparing statistical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' We then quantify epistemic discrimination as the gap between a model’s accuracy given fairness con- straints and the limit posed by aleatoric discrimi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' We apply this approach to benchmark ex- isting interventions and investigate fairness risks in data with missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' Our results indi- cate that state-of-the-art fairness interventions are effective at removing epistemic discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' However, when data has missing values, there is still significant room for improvement in handling aleatoric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' Introduction Algorithmic discrimination may occur in different stages of the machine learning (ML) pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' For example, histori- cal biases in the data-generating process can propagate to downstream tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' human biases can influence a ML model through inductive bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' optimizing solely for accuracy can lead to disparate model performance across groups in the data (Suresh & Guttag, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' Mayson, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' The past years have seen a rapid increase in algorithmic interventions that aim to mitigate biases in ML models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=', Zemel 1MIT-IBM Watson AI Lab 2Harvard College 3UT- Austin 4Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFKT4oBgHgl3EQfTy2d/content/2301.11781v1.pdf'} +page_content=' Hao Wang 1 the +dimensionless angular frequency is a real quantity and the perturbations propagate as harmonic waves +in time. On the other hand, for big wavelengths λJ/λ < 1 the angular frequency becomes a pure +imaginary quantity and the perturbations will grow or decay in time, which will depend on the sign +of the solution (45). The perturbations which grow in time are referred as Jeans instability, which is +associated with the gravitational collapse of self-gravitating gas clouds. +The analysis of Jeans instability within the first and second post-Newtonian approximation by +considering the Eulerian hydrodynamic equations were investigated in [21–23] and [24], respectively. +Here if we consider a collisionless Boltzmann equation where ν∗ → ∞ (44) reduces to +ω3 +∗ + +� +1 − κ2 +∗ + +�33 +10 + 2 +κ2∗ ++ 3κ2 +∗ +2 +− 2U0 +c2s +(1 − 2κ2 +∗) +�c2 +s +c2 +� +ω∗ = 0, +(46) + +7 +which is the dispersion relation in the first post-Newtonian approximation where dissipative effects +are not considered. There is a difference of this expression with the one in [8], since here the constant +value is 33/10 while there is 9/2. The reason of this difference is that here we have considered the +mass, mass-energy and momentum densities hydrodynamic equations while in the former work only +the mass and momentum densities hydrodynamic equations were taken into account. +For big wavelengths with respect to Jeans wavelength λJ/λ < 1 three different values associated with +the dimensionless angular frequencies can be obtained from (44) which correspond to the growth/decay +of the perturbations: +ω∗ = − i +ν∗ +λ2 +J +λ2 +� +1 − 7c2 +s +5c2 +� ++ . . . , +(47) +ω∗ = i +� +1 − 1 +2 +λ2 +J +λ2 +� +1 + +4 +5ν∗ +� ++ +�43 +20 − U0 +c2s ++ λ2 +λ2 +J +− +6 +5ν∗ +� c2 +s +c2 +� ++ . . . , +(48) +ω∗ = −i +� +1 − 1 +2 +λ2 +J +λ2 +� +1 − +4 +5ν∗ +� ++ +�43 +20 − U0 +c2s ++ λ2 +λ2 +J ++ +6 +5ν∗ +� c2 +s +c2 +� ++ . . . . +(49) +On the other hand, if we expand the dimensionless wavenumber in power series of the reduced +angular frequency κ∗ = a0 + a1ω∗ + . . . we get from the dispersion relation (44) the solution where +the perturbations propagate as harmonic waves +κ∗ = +� +5 +3 +� +1 + +�27 +10 + U +� c2 +s +c2 +� ++ 2i +3ν∗ +� +5 +3 +� +1 + 3ν2 +∗ +10 + +�24 +5 + 2U − ν2 +∗ +�3U +10 + 36 +25 +�� c2 +s +c2 +� +ω∗+. . . . (50) +V. +CONSTITUTIVE EQUATIONS +As was previously said the thermodynamic theory of a single relativistic fluid is characterized by +the fields of particle four-flow N µ and energy-momentum tensor T µν whose hydrodynamic equations +are the conservation laws (26). +The representation of the particle four-flow and energy-momentum tensor in terms of non-relativistic +quantities makes use of the four-velocity U µ –where U µUµ = c2 – and of the projector ∆µν = +gµν − U µU ν/c2 – where gµν denotes the metric tensor. The projector has the properties ∆µνUν = 0, +∆µν∆νσ = ∆µσ and in a local Minkowski rest frame where U µ = (c, 0) it reduces to ∆µν = +diag(0, −1, −1, −1). +Two representations for the particle four-flow and energy-momentum tensor in terms of non- +relativistic quantities are the Eckart [17] and the Landau-Lifshitz [25] decompositions. Here we shall +use the Eckart decomposition where the particle four-flow and energy-momentum tensor are written +as +N µ = nU µ, +(51) +T µν = p⟨µν⟩ − (p + ̟) ∆µν + ǫ +c2 U µU ν + 1 +c2 +� +U µq(ν) + U νq(µ) +� +. +(52) +Above n is the particle number density, p the hydrostatic pressure, ̟ the non-equilibrium pressure, +p⟨µν⟩ the pressure deviator, q(µ) the heat flux and ǫ the energy density. The energy density is a sum +of two terms one related with the internal energy density ρε while the other with the mass density ρ, +namely ǫ = ρc2(1 + ε/c2). The following projections of the particle four-flow and energy-momentum +tensor define the non-relativistic quantities (see e.g [16]): +n = 1 +c2 N µUµ, +ǫ = 1 +c2 UµT µνUν, +(p + ̟) = −1 +3∆µνT µν +(53) +p⟨µν⟩ = +� +∆µ +σ∆ν +τ − 1 +3∆µν∆στ +� +T στ, +q(µ) = ∆µ +νUσT νσ, +(54) +In the first post-Newtonian approximation the components of the four-velocity read [2, 3, 8] +U 0 = c +� +1 + 1 +c2 +�V 2 +2 + U +�� +, +U i = ViU 0 +c +, +(55) +where V denotes the hydrodynamic three velocity. + +8 +From the knowledge of the components of the metric tensor in the first post-Newtonian approxima- +tion +g00 = 1 − 2U +c2 + 2 +c4 +� +U 2 − 2Φ +� +, +g0i = Πi +c3 , +gij = − +� +1 + 2U +c2 +� +δij, +(56) +and of the four-velocity components (55) we can determine the components of the projector, which +read +∆00 = −V 2 +c2 − 1 +c4 +� +6UV 2 + V 4 − 2ΠiVi +� +, +∆0i = −Vi +c − 1 +c3 +� +2UVi + V 2Vi − Πi +� +, +(57) +∆ij = − +� +1 − 2U +c2 +� +δij − ViVj +c2 . +(58) +Now we introduce the non-relativistic pressure deviator +pij = pij − pkkδij/3 +whit +δijpij = 0, +(59) +so that the components of the pressure deviator p⟨µν⟩ become [12] +p⟨ij⟩ = pij + 1 +2c2 (pikVkVj + pjkVkVi) , +(60) +p⟨00⟩ = pij +ViVj +c2 , +p⟨0i⟩ = pij +Vj +c . +(61) +In terms of the non-relativistic heat flux vector qi the components of the heat flux q(µ) are +q(i) = qi, +q(0) = qi +Vi +c . +(62) +In the five field thermodynamic theory – where the basic fields are the mass density, momentum +density and internal energy density – the pressure deviator, the dynamic pressure and the heat flux +vector are given by constitutive equations. +Here we can obtain the desired constitutive equations +from the components of the energy-momentum tensor (19) – (24) combined with the decomposition +expressions (53) and (54) and the components of the projection (57) and (58). Hence it follows the +constitutive equations for the non-relativistic heat flux vector and pressure deviator +qi = − 5kp +2mν +� +1 − c2 +s +c2 +U +c2s +� ∂T +∂xi + +p +νc2 ∆ijkl +∂Vk +∂xl +��5kT +2m + 3U + V 2 +2 +� +Vj − Πj +� ++ p +νc2 (V 2δij − ViVj) +� +Vk +∂Vk +∂xj − ∂T +∂xj +� ++ +p +νc2 +� +V 2δij + ViVj +3 +�� ∂U +∂xj − 1 +ρ +∂p +∂xj +� +, +(63) +pij = − p +ν +� +1 + c2 +s +c2 +�3 +2 − U +c2s +�� +∆ijkl +∂Vk +∂xl + +2p +3νc2 +∂Vk +∂xk +� +ViVj − 1 +3V 2δij +� +− p +νc2 ∆ijkl +�1 +2 +∂V 2Vk +∂xl ++ Vk +� ∂U +∂xl − 1 +ρ +∂p +∂xl +�� +. +(64) +The constitutive equation for the dynamic pressure ̟ does not show up in the first post-Newtonian +approximation and it is known that in the kinetic theory of relativistic gases the coefficient of bulk +viscosity – which relates the dynamic pressure with the velocity divergent – is of order O(c−4) (see +e.g. [16]). +Let us fix our attention in the underlined linearized terms in (63) and (64). Without the relativistic +corrections they reduce to the non-relativistic constitutive equations of a viscous and heat conducting +gas, namely +qi = − 5kp +2mν +∂T +∂xi , +pij = − p +ν ∆ijkl +∂Vk +∂xl , +(65) +where the thermal conductivity λ and the shear viscosity µ coefficients are those of the non-relativistic +BGK model +λ = 5kp +2mν , +µ = p +ν . +(66) + +9 +With the first post-Newtonian correction these coefficients read +λ = 5kp +2mν +� +1 − c2 +s +c2 +U +c2s +� +, +µ = p +ν +� +1 + c2 +s +c2 +�3 +2 − U +c2s +�� +. +(67) +We note that the coefficients of shear viscosity and thermal conductivity do depend on the Newtonian +gravitational potential. On the basis of a non-relativistic kinetic theory the influence the gravity on the +thermal coefficient was first reported in [26, 27]. Within the framework of a relativistic kinetic theory +the transport coefficients of shear viscosity, thermal conductivity and bulk viscosity were obtained by +considering a Schwarzschild metric in [28] and the diffusion coefficient in [29]. +VI. +CONCLUSIONS +In this work we have examined a relaxation-time model for the post-Newtonian Boltzmann equation +and determined the non-equilibrium distribution function by using the Chapman-Enskog method and +the equilibrium post-Newtonian Maxwell-J¨uttner distribution function. The components of the energy- +momentum tensor were calculated by using the equilibrium and non-equilibrium distribution functions. +From the conservation laws of the particle four-flow and energy-momentum tensor the linearized field +equations for the mass, momentum and internal energy densities were determined. +A plane wave +solution of these linearized field equations coupled with the three post-Newtonian Poisson equations was +found. By using the Eckart decomposition of the energy-momentum tensor the constitutive equations +for the viscous stress and heat flux vector were obtained and it was shown that the transport coefficients +of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential. +ACKNOWLEDGMENTS +This work was supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), +grant No. 304054/2019-4. +[1] A. Einstein, L. Infeld and B. Hoffmann, The gravitational equations and the problem of motion, Ann. of +Math. 39, 65 (1938). +[2] S. Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Ap. J. 142, 1488 +(1965). +[3] S. Weinberg, Gravitation and cosmology. Principles and applications of the theory of relativity (Wiley, New +York, 1972). +[4] S. Chandrasekhar and Y. Nutku, The second post-Newtonian equations of hydrodynamics in general +relativity. Ap. J. 158, 55 (1969). +[5] C. A. Ag´on, J. F. Pedraza and J. Ramos-Caro, Kinetic theory of collisionless self-gravitating gases: Post- +Newtonian polytropes, Phys. Rev. D 83, 123007 (2011). +[6] V. Rezania and Y. Sobouti, Liouville’s equation in post Newtonian approximation I. Static solutions, +Astron. Astrophys. 354, 1110 (2000). +[7] G.M. Kremer, Post-Newtonian kinetic theory, Ann. Phys. 426, 168400 (2021). +[8] G. M. Kremer, Post-Newtonian hydrodynamics: theory and applications, (Cambridge Scholars Publishing, +Newcastle upon Tyne, 2022). +[9] G. M. Kremer, M. G. Richarte and K. Weber, Self-gravitating systems of ideal gases in the 1PN approxi- +mation, Phys. Rev. D 93, 064073 (2016). +[10] P. J. Greenberg, The post-Newtonian equations of hydrodynamics for a thermally conducting, viscous, +compressible fluid in general relativity, Ap. J. 164, 569 (1971). +[11] J.-C. Hwang and H. Noh, Special relativistic hydrodynamics with gravitation, Ap. J. 833, 180 (2016). +[12] G.M. Kremer, Post-Newtonian non-equilibrium kinetic theory, Ann. Phys. 441, 168865 (2022). +[13] S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases 3rd. (Cambridge University +Press, Cambridge, 1970). +[14] G. M. Kremer, An introduction to the Boltzmann equation and transport processes in gases (Springer, +Berlin, 2010). +[15] C. Marle, Mod`ele cin´etique pour l’´etablissement des lois de la conduction de la chaleur et de la viscosit´e +en th´eorie de la relativit´e, C. R. Acad. Sc. Paris 260, 6539 (1965). + +10 +[16] C. Cercignani and G. M. Kremer, The relativistic Boltzmann equation: +theory and applications +(Birkh¨auser, Basel, 2002) +[17] C. Eckart, The thermodynamics of irreversible processes, III. Relativistic theory of a simple fluid, Phys. +Rev. 58, 919 (1940). +[18] J. H. Jeans, The stability of a spherical nebula. Philos. Trans. R. Soc. A, 199, 1 (1902). +[19] P. Coles and F. Lucchin, Cosmology. The origin and evolution of cosmic structures, 2nd, edn. (John Wiley, +Chichester, 2002). +[20] J. Binney and S. Tremaine, Galactic Dynamics, 2nd. edn. (Princeton University Press, Princeton, 2008). +[21] E. Nazari, A. Kazemi, M. Roshan and S. Abbassi, Post-Newtonian Jeans analysis. Ap. J. 839, 75 (2017). +[22] H. Noh and J.-C. Hwang, Gravitomagnetic instabilities of relativistic magnetohydrodynamics. Ap. J. 906, +22 (2021). +[23] G. M. Kremer, Jeans instability from post-Newtonian Boltzmann equation. Eur. Phys. J. C 81, 927 +(2021). +[24] G. M. Kremer, Plane wave analysis of the second post-Newtonian hydrodynamic equations, Int. J. Geom. +Methods Mod. Phys. 2350039 (2023). +[25] L. D. Landau and E. M. Lifshitz, Fluid mechanics, 2nd ed. (Pergamon Press, Oxford, 1987). +[26] T. Doi T, A. Santos and M. Tij M, Numerical study of the influence of gravity on the heat conductivity on +the basis of kinetic theory Phys. Fluids 11, 3553 (1999). +[27] M. Tij, V. Garz´o and A. Santos, On the influence of gravity on the thermal conductivity, in Rarefied Gas +Dynamics, R. Brun , R. Campargue, R. Gatignol and J.-C. Lengrand , eds. 1999 (Toulouse: C´epadu`es) p. +239 +[28] G. M. Kremer, Relativistic gas in a Schwarzschild metric, J. Stat. Mech. P04016 (2013). +[29] G. M. Kremer, Diffusion of relativistic gas mixtures in gravitational field, Physica A 393 76 (2014). + diff --git a/AtE4T4oBgHgl3EQfEwyK/content/tmp_files/load_file.txt b/AtE4T4oBgHgl3EQfEwyK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aad3365bb803b2db505e1b812b5f944fe59cc904 --- /dev/null +++ b/AtE4T4oBgHgl3EQfEwyK/content/tmp_files/load_file.txt @@ -0,0 +1,359 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf,len=358 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='04880v1 [gr-qc] 12 Jan 2023 Relaxation-Time Model for the Post-Newtonian Boltzmann Equation Gilberto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Kremer1, ∗ 1Departamento de F´ısica, Universidade Federal do Paran´a, Curitiba 81531-980, Brazil The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are deter- mined from a relaxation-time model of the post-Newtonian Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The Chapman- Enskog method is used to calculate the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The components of the energy-momentum tensor are found from the knowledge of the non-equilibrium and the post- Newtonian equilibrium Maxwell-J¨uttner distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The linearized field equations for the mass, momentum and internal energy densities coupled with the three Poisson equations of the post-Newtonian approximation are investigated by considering a plane wave representation of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The constitutive equations for the viscous stress and heat flux vector are obtained and it is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' INTRODUCTION In the seminal work of Einstein, Infeld and Hoffman [1] it was proposed a method of successive approximations in powers of 1/c2 for the solution of Einstein’s field equations, which become the basis of the post-Newtonian approximation for the determination of the energy-momentum tensor components as well as the Eulerian hydrodynamic equations in the first [2, 3] and second [4] approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The post-Newtonian version of the Boltzmann equation in the first and in the second approximations were determined in [5, 6] and [7, 8], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In [7, 8] the energy-momentum tensor components were obtained from the equilibrium Maxwell-J¨uttner distribution function [9] in the first and second post-Newtonian approximations and the Eulerian hydrodynamic equations from a collisionless post- Newtonian Boltzmann equation were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The inclusion of non-equilibrium terms in the post-Newtonian theory was investigated in [10, 11] within the framework of a phenomenological theory of a viscous, heat conducting and compressible fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' On the other hand, the inclusion of non-equilibrium terms in the hydrodynamic equations which follow from the post-Newtonian Boltzmann equation was considered in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In this work the hydrodynamic equations resulted from a post-Newtonian Maxwell-Enskog transfer equation together with a post-Newtonian Grad’s distribution function which takes into account the non-equilibrium fields of viscous stress and heat flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' One interesting subject to be investigate is the determination of the post-Newtonian hydrodynamic equations for a viscous and heat conducting fluid from the post-Newtonian Boltzmann equation where the particle collisions are taken into account through the collision operator of the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Here we shall adopt a relaxation-time model for the collision operator which is known in the non- relativistic framework as the Bhatnagar-Gross-Krook (BGK) model (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' [13, 14]) and in the relativistic one as the Marle model [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' We use the Chapman-Enskog method to determine the non-equilibrium distribution function from the post-Newtonian BGK (Marle) model of the Boltzmann equation and the post-Newtonian Maxwell- J¨uttner distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' From the knowledge of the non-equilibrium distribution function the non-equilibrium contributions to the energy-momentum tensor are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The linearized field equations for the mass, momentum and internal energy densities are determined from the particle four- flow and energy-momentum tensor conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' These linearized field equations are coupled with three Poisson equations from the post-Newtonian approximation and a solution of the coupled system of equations is found in terms of a plane wave representation of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Furthermore, the constitutive equations for the viscous stress and heat flux vector – which correspond to the Navier-Stokes and Fourier laws, respectively – are obtained from the Eckart decomposition [17] of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' It is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The paper is structured as follows: in Section II we introduce the relaxation-time model of the post-Newtonian Boltzmann equation and determine the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The ∗ kremer@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='br 2 particle four-flow and the energy-momentum tensor components are calculated on the basis of the equilibrium Maxwell-J¨uttner and non-equilibrium distribution functions in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The linearized field equations are determined in Section IV and a plane wave solution of the linearized field equations coupled with the three Poisson equations of the post-Newtonian approximation is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In Section V the constitutive equations for the viscous stress and heat flux vector are obtained and the transport coefficients of shear viscosity and thermal conductivity are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the last section the conclusions of the work are stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' RELAXATION-TIME MODEL In the phase space spanned by the space coordinates x and velocity of the particles v a state of a monatomic gas is characterized by the one-particle distribution function f(x, v, t) and its space- time evolution is governed by Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the first post-Newtonian approximation the Boltzmann equation is given by [5, 7, 8] �∂f ∂t + vi ∂f ∂xi + ∂f ∂vi ∂U ∂xi �� 1 + 1 c2 �v2 2 + U � � + 1 c2 ∂f ∂vi � vj �∂Πi ∂xj − ∂Πj ∂xi � −3vi ∂U ∂t + ∂Πi ∂t + 2 ∂Φ ∂xi − 4U ∂U ∂xi − 4vivj ∂U ∂xj + v2 ∂U ∂xi � = Q(f, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (1) Here Q(f, f) denotes the collision operator of the Boltzmann equation which takes into account the binary collisions of the particles and refers to an integral of the product of two particle distribution functions at collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Furthermore, the Newtonian gravitational potential U, the scalar gravitational potential Φ and the vector gravitational potential Πi satisfy Poisson equations, which are obtained from the first post-Newtonian approximation of Einstein’s field equations and read [2, 8] ∇2U = −4πGρ, ∇2Φ = −4πGρ � V 2 + U + ε 2 + 3p 2ρ � , (2) ∇2Πi = −16πGρVi + ∂2U ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (3) Above V denotes the hydrodynamic three-velocity, G the universal gravitational constant and ε, p the specific internal energy and hydrostatic pressure of the gas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The gauge condition 3∂U/∂t + ∂Πi/∂xi = 0 for the gravitational potentials U and Πi holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the BGK (Marle) model the collision operator is replaced by the difference between the one- particle distribution function and its equilibrium value multiplied by a frequency ν which is of order of the collision frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The one-particle distribution function at equilibrium is determined from the relativistic Boltzmann equation by considering that the collision operator vanishes at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the relativistic theory the equilibrium distribution function is the Maxwell-J¨uttner distribution function (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g [16]) and its first post-Newtonian approximation was determined in [9] and reads fMJ = f0 � 1 − 1 c2 �15kT 8m + m(ViVi)2 2kT + 2mUV2 kT + 3mV4 8kT + mV 2V2 2kT + m(ViVi)V2 kT �� , (4) where f0 denotes the non-relativistic Maxwellian distribution function, namely f0 = ρ e− mV2 2kT (2πm 5 3 kT ) 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (5) In the above equation ρ is the mass density, T the absolute temperature, m the rest mass of a gas particle and k the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Furthermore, Vi = vi − Vi is the so-called peculiar velocity which is the difference of the particle velocity vi and the hydrodynamic velocity Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' By considering that the relativistic equilibrium distribution function is the Maxwell-J¨uttner distri- bution fMJ, the collision operator is written as Q(f, f) = −ν(f − fMJ) = −νfNE, (6) where fNE is the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' 3 For the determination of the non-equilibrium distribution function we shall rely on the Chapman- Enskog method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' [13, 14] and insert the equilibrium Maxwell-J¨uttner distribution function (4) into the left-hand side of the Boltzmann equation (1) and compute the non-equilibrium distribution function by considering the BGK (Marle) model (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Hence it follows � 1 + 1 c2 �v2 2 + U � ��∂fMJ ∂ρ �dρ dt + Vi ∂ρ ∂xi � + ∂fMJ ∂T �dT dt + Vi ∂T ∂xi � + ∂fMJ ∂Vi �dVi dt + Vj ∂Vi ∂xj � +∂fMJ ∂U �dU dt + Vi ∂U ∂xi � + ∂fMJ ∂vi ∂U ∂xi � + 1 c2 ∂fMJ ∂vi � vj �∂Πi ∂xj − ∂Πj ∂xi � − 3vi ∂U ∂t + ∂Πi ∂t +2 ∂Φ ∂xi − 4U ∂U ∂xi − 4vivj ∂U ∂xj + v2 ∂U ∂xi � = −νfNE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (7) where d/dt = ∂/∂t + Vi∂/∂xi denotes the material time derivative and ∂fMJ ∂ρ = fMJ ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' ∂fMJ ∂U = −f0 2mV2 kT c2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (8) ∂fMJ ∂T = f0 T �mV2 2kT − 3 2 + 1 c2 �15kT 16m � 1 − mV2 kT + m2V4 k2T 2 � + 5m 2kT � 2UV2 + (ViVi)2 2 +V 2V2 2 + (ViVi)V2 � − m2 2k2T 2 � 2UV4 + (ViVi)2V2 2 + V 2V4 2 + (ViVi)V4 + 3V6 8 ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (9) ∂fMJ ∂V i = mf0 kT � Vi + 1 c2 � 4UVi � 1 − mV2 2kT � − 15kT 8m Vi + (VjVj)Vi + ViV 2 � 1 − mV2 2kT � +(VjVj)Vi � 1 − mV2 kT � + ViV2 2 � 1 − 3mV2 4kT � − m(VjVj)2 2kT Vi �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (10) ∂fMJ ∂vi = −mf0 kT � Vi + 1 c2 � 4UVi � 1 − mV2 2kT � + Vi(V2 + VjVj) − 15kT 8m Vi +Vi � V 2 + 2VjVj + 3V2 2 � − mVi kT �(VjVj)2 2 + V 2V2 2 + (VjVj)V2 + 3V4 8 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (11) As usual in the Chapman-Enskog method the material time derivatives are eliminated from the non-equilibrium distribution function by using the Eulerian balance equations for the mass density ρ, hydrodynamic velocity Vi and absolute temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The Eulerian mass density and the momentum density balance equations in the first post-Newtonian approximation are [2, 8] dρ � 1 + 1 c2 � V 2 2 + 3U �� dt + ρ � 1 + 1 c2 �V 2 2 + 3U �� ∂Vi ∂xi = 0, (12) ρdVi dt + ∂p ∂xi � 1 − 1 c2 � V 2 + 4U + ε + p ρ �� − ρ ∂U ∂xi � 1 + 1 c2 (V 2 − 4U) � + ρ c2 ��1 ρ ∂p ∂t − ∂U ∂t + 4dU dt � Vi − 2 ∂Φ ∂xi − dΠi dt + Vj ∂Πj ∂xi � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (13) For the determination of the Eulerian internal energy density balance equation ρε in the first post- Newtonian approximation one has to go to the second post-Newtonian approximation, since within the framework of the first post-Newtonian approximation one recover only its Newtonian expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The Eulerian internal energy density balance equation reads1 dε dt + p ρ ∂Vi ∂xi + 3p ρc2 dU dt + pVi ρc2 � ∂U ∂xi − 1 ρ ∂p ∂xi � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (14) From the above equation follows the expression for the material time derivative of the absolute temper- ature, if we take into account the relationship for the specific internal energy in the first post-Newtonian approximation which comes from the relativistic kinetic theory of gases (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' [16]) ε = 3kT 2m � 1 + 5kT 4mc2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (15) 1 This equation corrects some misprints in [7, 8] 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' PARTICLE FOUR-FLOW AND ENERGY-MOMENTUM TENSOR COMPONENTS In the relativistic kinetic theory of gases the particle four-flow N µ and the energy-momentum tensor T µν are given in terms of the one-particle distribution function f(x, v, t) by [8, 16] N µ = m4c � uµf √−g d3u u0 , T µν = m4c � uµuνf √−g d3u u0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (16) Here uµ = pµ/m (with uµuµ = c2) denotes the gas particle four-velocity whose components in the first post-Newtonian approximation read [2, 3, 8] u0 = c � 1 + 1 c2 �v2 2 + U �� , ui = vi u0 c , (17) where v is the particle three-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Furthermore, √−g d3u/u0 is an invariant integration element whose first post-Newtonian approximation was determined in [9] and is given by √−g d3u u0 = � 1 + 1 c2 � 2v2 + 6U �� d3v c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (18) Once the one-particle distribution function f = fMJ + fNE and the invariant integration element are known, one can determine the components of the particle four-flow N µ and energy-momentum tensor T µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Indeed, if we insert (4), (7), (17) and (18) into (16) and integrate the resulting equations we get N 0 = ρc m � 1 + 1 c2 �V 2 2 + U �� , N i = N 0 Vi c , (19) T 00 = ρc2 � 1 + 1 c2 � V 2 + 3kT 2m + 2U � + O(c−4) � , (20) T i0 = cρVi � 1 + 1 c2 � V 2 + 2U + 5kT 2m �� + T i0 NE, (21) T ij = ρViVj � 1 + 1 c2 � V 2 + 2U + 5kT 2m �� + p � 1 − 2U c2 � δij + T ij NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (22) Note that there are no non-equilibrium contributions to the components of the particle four-flow (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The non-equilibrium contribution to T 00 is of order O(c−4) (the order of the nth inverse power of light speed is denoted by O(c−n)) while the non-equilibrium contributions to the energy-momentum tensor components T 0i NE and T ij NE are associate with terms related with the collision frequency ν and read T i0 NE = −p νc � 5k 2m ∂T ∂xi + ∆ijklVj ∂Vk ∂xl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (23) T ij NE = − p ν �� 1 + 1 c2 �5kT 2m − U + V 2 2 �� ∆ijklVj ∂Vk ∂xl + 1 c2 ∆ijklVk � ∂U ∂xl − 1 ρ ∂p ∂xl � − 2 3c2 ViVj ∂Vk ∂xk + 1 c2 � Vj ∂ ∂xi + Vi ∂ ∂xj ��5kT 2m + V 2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (24) Here we have introduced the fourth-order tensor ∆ijkl = δikδjl + δilδjk − 2 3δijδkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (25) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' LINEARIZED FIELD EQUATIONS The thermodynamic theory of a single relativistic fluid is described by the fields of particle four-flow N µ and energy-momentum tensor T µν where their hydrodynamic equations follow from the conserva- tion laws N µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='µ = ∂N µ ∂xµ + ΓµµσN σ = 0, T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='ν = ∂T µν ∂xν + ΓµνσT σν + ΓννσT µσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (26) 5 Above the semicolon refers to the covariant derivative and Γµνσ to the Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' From the knowledge of the expressions of the particle four-flow and energy momentum tensor com- ponents (19) – (24) and the conservation laws (26) one can obtain the field equations for the particle number density, momentum density and specific internal energy for a viscous and heat conducting fluid in the first post-Newtonian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Here we are interested in determining the linearized field equations and for that end we consider a background state of constant values for the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' absolute temperature and Newtonian gravitational potential denoted by ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' T0 and U0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' which are superposed by linear perturbed fields denoted by ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' V 1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Π1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' namely ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = ρ0 + ρ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' T (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = T0 + T1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = U0 + U1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (27) Vi(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = V 1 i (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = Φ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Πi(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t) = Π1 i (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (28) From the insertion of (19) into (26)1 follows the linearized field equation for the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' by tak- ing into account the expressions of the Christoffel symbols in the first post-Newtonian approximation – which can be found in [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' 8] – and of the representations (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' yielding ∂ρ1 ∂t + ρ0 ∂V 1 i ∂xi + 3ρ0 c2 ∂U1 ∂t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (29) The linearized field equations for the mass-energy and momentum densities are obtained from the time and spatial components of (26)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' by considering the expressions (19) – (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' the representations (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (28) and the Christoffel symbols in the first post-Newtonian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Hence it follows ∂ρ1 ∂t + ρ0 � 1 + kT0 mc2 � ∂V 1 i ∂xi + ρ0 c2 �3kT0 2m ∂T1 ∂t + 3∂U1 ∂t � − 5k2ρ0T0 2m2c2ν0 ∂2T1 ∂xi∂xi = 0, (30) ρ0 ∂V 1 i ∂t + k m � 1 − 1 c2 �5kT0 2m + 4U0 �� � T0 ∂ρ1 ∂xi + ρ0 ∂T1 ∂xi � − ρ0 � 1 − 4U0 c2 � ∂U1 ∂xi − 5k2ρ0T0 2m2c2ν0 ∂2T1 ∂t∂xi − kρ0T0 mν0 � 1 − 3U0 c2 � � ∂2V 1 i ∂xj∂xj + 1 3 ∂2V 1 j ∂xj∂xi � − ρ0 c2 � 2∂Φ1 ∂xi + ∂Π1 i ∂t � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (31) Since the constant values of the background state does not satisfy the Poisson equations (2) and (3) it is usual to take into account the ”Jeans swindle” (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' [18–20]) which requires that the Poisson equations are valid only for the perturbed fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Hence, by considering that ε = 3kT/2m = 3p/2ρ, the linearized Poisson equations become ∇2U1 = −4πGρ1, ∇2Φ1 = −4πGρ1 � U0 + 9k 4mT0 � − 4πGρ0 � U1 + 9k 4mT1 � , (32) ∇2Π1 i = −16πGρ0V 1 i + ∂2U1 ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (33) Let us find a solution of the coupled system of partial differential equations (29) – (33) in terms of a plane wave representation of the perturbed fields, namely ρ1(x, t) = ρe[i(κixi−ωt)], T1(x, t) = Te[i(κixi−ωt)], U1(x, t) = Ue[i(κixi−ωt)], (34) V 1 i (x, t) = Vie[i(κixi−ωt)], Φ1(x, t) = Φe[i(κixi−ωt)], Π1 i (x, t) = Πie[i(κixi−ωt)], (35) where κi denotes the wavenumber vector, ω the angular frequency and the overlined quantities the small amplitudes of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' We insert the plane wave representations (34) and (35) into the coupled system of partial differential 6 equations (29) – (33) and get a linearized system of algebraic equations for the amplitudes which reads ω∗ρ∗ − V∗ + 3U0 c2 U∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (36) ω∗ρ∗ − � 1 + 3c2 s 5c2 � V∗ + 3U0 c2 U∗ + 9c2 s 10c2 � ω∗ + iκ∗ ν∗ � T∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (37) � ω∗ + 4 5ν∗ � 1 − 3U0 c2 � iκ2 ∗ � V∗ − 3 5κ2 ∗ � 1 − c2 s c2 �3 2 + 4U0 c2s �� [ρ∗ + T∗] +κ2 ∗ U0 c2s � 1 − 4U0 c2 � U∗ − 3c2 s 2c2ν∗ iω∗κ2 ∗T∗ + c2 s c2 � 2κ2 ∗Φ∗ − ω∗Π∗ � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (38) κ2 ∗ U0 c2s U∗ = ρ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (39) κ2 ∗Φ∗ = �U0 c2s + 27 20 � ρ∗ + �U0 c2s U∗ + 27 20T∗ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (40) κ2 ∗Π∗ = 4V∗ − ω∗κ2 ∗ U0 c2s U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (41) Equations (38) and (41) result from the scalar product with κi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Furthermore, the above equations were written in terms of the dimensionless quantities κ∗ i = κi κJ , ω∗ = ω √4πGρ0 , ν∗ = ν0 √4πGρ0 , (42) ρ∗ = ρ ρ0 , T∗ = T T0 , V∗ = V iκi csκJ , U∗ = U U0 , Φ∗ = Φ c4s , Π∗ = Πiκi c3sκJ , (43) where κJ = √4πGρ0/cs denotes the Jeans wavelength, cs = � 5kT0/3m the sound speed and κ∗ = �κ∗ i κ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The system of algebraic equations for the amplitudes (36) – (41) admits a non-trivial solution if the determinant of the coefficients which correspond to the amplitudes vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Hence it follows the dispersion relation which connect the dimensionless angular frequency ω∗ with the dimensionless wavenumber κ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' namely ω3 ∗ + 9i 5ν∗ � κ2 ∗ + 4 3 � 1 − κ2 ∗ � 5 12 + U0 c2s � c2 s c2 �� ω2 ∗ + � 1 − κ2 ∗ − 4κ4 ∗ 5ν∗ + �33 10 + 2 κ2∗ +3κ2 ∗ 2 − 2U0 c2s (1 − 2κ2 ∗) − 12κ2 ∗ 5ν2∗ � 1 − U0κ2 ∗ c2s ��c2 s c2 � ω∗ + i ν∗ � κ2 ∗ � 1 − 3κ2 ∗ 5 � + � 2 + 27κ2 ∗ 10 � 1 + κ2 ∗ 3 � − 2κ2 ∗U0 c2s � 1 − 6κ2 ∗ 5 ��c2 s c2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (44) Here terms up to the order O(c−2) were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the case of a non relativistic and collisionless Boltzmann equation we have that cs/c → 0 and ν∗ → ∞ and we obtain from (44) Jeans solution [18] ω∗ = ± � λ2 J λ2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (45) Above we have introduced the wavelengths λ and λJ (Jeans wavelength) through the relationship κ∗ = κ/κJ = λJ/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' In the case of small wavelengths with respect to Jeans wavelength λJ/λ > 1 the dimensionless angular frequency is a real quantity and the perturbations propagate as harmonic waves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' On the other hand, for big wavelengths λJ/λ < 1 the angular frequency becomes a pure imaginary quantity and the perturbations will grow or decay in time, which will depend on the sign of the solution (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The perturbations which grow in time are referred as Jeans instability, which is associated with the gravitational collapse of self-gravitating gas clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The analysis of Jeans instability within the first and second post-Newtonian approximation by considering the Eulerian hydrodynamic equations were investigated in [21–23] and [24], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Here if we consider a collisionless Boltzmann equation where ν∗ → ∞ (44) reduces to ω3 ∗ + � 1 − κ2 ∗ + �33 10 + 2 κ2∗ + 3κ2 ∗ 2 − 2U0 c2s (1 − 2κ2 ∗) �c2 s c2 � ω∗ = 0, (46) 7 which is the dispersion relation in the first post-Newtonian approximation where dissipative effects are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' There is a difference of this expression with the one in [8], since here the constant value is 33/10 while there is 9/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The reason of this difference is that here we have considered the mass, mass-energy and momentum densities hydrodynamic equations while in the former work only the mass and momentum densities hydrodynamic equations were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' For big wavelengths with respect to Jeans wavelength λJ/λ < 1 three different values associated with the dimensionless angular frequencies can be obtained from (44) which correspond to the growth/decay of the perturbations: ω∗ = − i ν∗ λ2 J λ2 � 1 − 7c2 s 5c2 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' , (47) ω∗ = i � 1 − 1 2 λ2 J λ2 � 1 + 4 5ν∗ � + �43 20 − U0 c2s + λ2 λ2 J − 6 5ν∗ � c2 s c2 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' , (48) ω∗ = −i � 1 − 1 2 λ2 J λ2 � 1 − 4 5ν∗ � + �43 20 − U0 c2s + λ2 λ2 J + 6 5ν∗ � c2 s c2 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (49) On the other hand, if we expand the dimensionless wavenumber in power series of the reduced angular frequency κ∗ = a0 + a1ω∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' we get from the dispersion relation (44) the solution where the perturbations propagate as harmonic waves κ∗ = � 5 3 � 1 + �27 10 + U � c2 s c2 � + 2i 3ν∗ � 5 3 � 1 + 3ν2 ∗ 10 + �24 5 + 2U − ν2 ∗ �3U 10 + 36 25 �� c2 s c2 � ω∗+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (50) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' CONSTITUTIVE EQUATIONS As was previously said the thermodynamic theory of a single relativistic fluid is characterized by the fields of particle four-flow N µ and energy-momentum tensor T µν whose hydrodynamic equations are the conservation laws (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The representation of the particle four-flow and energy-momentum tensor in terms of non-relativistic quantities makes use of the four-velocity U µ –where U µUµ = c2 – and of the projector ∆µν = gµν − U µU ν/c2 – where gµν denotes the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The projector has the properties ∆µνUν = 0, ∆µν∆νσ = ∆µσ and in a local Minkowski rest frame where U µ = (c, 0) it reduces to ∆µν = diag(0, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Two representations for the particle four-flow and energy-momentum tensor in terms of non- relativistic quantities are the Eckart [17] and the Landau-Lifshitz [25] decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Here we shall use the Eckart decomposition where the particle four-flow and energy-momentum tensor are written as N µ = nU µ, (51) T µν = p⟨µν⟩ − (p + ̟) ∆µν + ǫ c2 U µU ν + 1 c2 � U µq(ν) + U νq(µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (52) Above n is the particle number density, p the hydrostatic pressure, ̟ the non-equilibrium pressure, p⟨µν⟩ the pressure deviator, q(µ) the heat flux and ǫ the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The energy density is a sum of two terms one related with the internal energy density ρε while the other with the mass density ρ, namely ǫ = ρc2(1 + ε/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The following projections of the particle four-flow and energy-momentum tensor define the non-relativistic quantities (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g [16]): n = 1 c2 N µUµ, ǫ = 1 c2 UµT µνUν, (p + ̟) = −1 3∆µνT µν (53) p⟨µν⟩ = � ∆µ σ∆ν τ − 1 3∆µν∆στ � T στ, q(µ) = ∆µ νUσT νσ, (54) In the first post-Newtonian approximation the components of the four-velocity read [2, 3, 8] U 0 = c � 1 + 1 c2 �V 2 2 + U �� , U i = ViU 0 c , (55) where V denotes the hydrodynamic three velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' 8 From the knowledge of the components of the metric tensor in the first post-Newtonian approxima- tion g00 = 1 − 2U c2 + 2 c4 � U 2 − 2Φ � , g0i = Πi c3 , gij = − � 1 + 2U c2 � δij, (56) and of the four-velocity components (55) we can determine the components of the projector, which read ∆00 = −V 2 c2 − 1 c4 � 6UV 2 + V 4 − 2ΠiVi � , ∆0i = −Vi c − 1 c3 � 2UVi + V 2Vi − Πi � , (57) ∆ij = − � 1 − 2U c2 � δij − ViVj c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (58) Now we introduce the non-relativistic pressure deviator pij = pij − pkkδij/3 whit δijpij = 0, (59) so that the components of the pressure deviator p⟨µν⟩ become [12] p⟨ij⟩ = pij + 1 2c2 (pikVkVj + pjkVkVi) , (60) p⟨00⟩ = pij ViVj c2 , p⟨0i⟩ = pij Vj c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (61) In terms of the non-relativistic heat flux vector qi the components of the heat flux q(µ) are q(i) = qi, q(0) = qi Vi c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (62) In the five field thermodynamic theory – where the basic fields are the mass density, momentum density and internal energy density – the pressure deviator, the dynamic pressure and the heat flux vector are given by constitutive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Here we can obtain the desired constitutive equations from the components of the energy-momentum tensor (19) – (24) combined with the decomposition expressions (53) and (54) and the components of the projection (57) and (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Hence it follows the constitutive equations for the non-relativistic heat flux vector and pressure deviator qi = − 5kp 2mν � 1 − c2 s c2 U c2s � ∂T ∂xi + p νc2 ∆ijkl ∂Vk ∂xl ��5kT 2m + 3U + V 2 2 � Vj − Πj � + p νc2 (V 2δij − ViVj) � Vk ∂Vk ∂xj − ∂T ∂xj � + p νc2 � V 2δij + ViVj 3 �� ∂U ∂xj − 1 ρ ∂p ∂xj � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (63) pij = − p ν � 1 + c2 s c2 �3 2 − U c2s �� ∆ijkl ∂Vk ∂xl + 2p 3νc2 ∂Vk ∂xk � ViVj − 1 3V 2δij � − p νc2 ∆ijkl �1 2 ∂V 2Vk ∂xl + Vk � ∂U ∂xl − 1 ρ ∂p ∂xl �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (64) The constitutive equation for the dynamic pressure ̟ does not show up in the first post-Newtonian approximation and it is known that in the kinetic theory of relativistic gases the coefficient of bulk viscosity – which relates the dynamic pressure with the velocity divergent – is of order O(c−4) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Let us fix our attention in the underlined linearized terms in (63) and (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Without the relativistic corrections they reduce to the non-relativistic constitutive equations of a viscous and heat conducting gas, namely qi = − 5kp 2mν ∂T ∂xi , pij = − p ν ∆ijkl ∂Vk ∂xl , (65) where the thermal conductivity λ and the shear viscosity µ coefficients are those of the non-relativistic BGK model λ = 5kp 2mν , µ = p ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (66) 9 With the first post-Newtonian correction these coefficients read λ = 5kp 2mν � 1 − c2 s c2 U c2s � , µ = p ν � 1 + c2 s c2 �3 2 − U c2s �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' (67) We note that the coefficients of shear viscosity and thermal conductivity do depend on the Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' On the basis of a non-relativistic kinetic theory the influence the gravity on the thermal coefficient was first reported in [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' Within the framework of a relativistic kinetic theory the transport coefficients of shear viscosity, thermal conductivity and bulk viscosity were obtained by considering a Schwarzschild metric in [28] and the diffusion coefficient in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' CONCLUSIONS In this work we have examined a relaxation-time model for the post-Newtonian Boltzmann equation and determined the non-equilibrium distribution function by using the Chapman-Enskog method and the equilibrium post-Newtonian Maxwell-J¨uttner distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' The components of the energy- momentum tensor were calculated by using the equilibrium and non-equilibrium distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' From the conservation laws of the particle four-flow and energy-momentum tensor the linearized field equations for the mass, momentum and internal energy densities were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' A plane wave solution of these linearized field equations coupled with the three post-Newtonian Poisson equations was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'} +page_content=' By using the Eckart decomposition of the energy-momentum tensor the constitutive equations for the viscous stress and heat flux vector were obtained and it was shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.' 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A northern sky survey in intensity +and polarization at 10–20 GHz with the Multi-Frequency Instrument +J. A. Rubiño-Martín,1,2★ F. Guidi,1,2,3 R. T. Génova-Santos,1,2 S. E. Harper,4 D. Herranz,5 +R. J. Hoyland,1,2 A. N. Lasenby,6,7 F. Poidevin,1,2 R. Rebolo,1,2,8 B. Ruiz-Granados,1,2,9 +F. Vansyngel,1,2 P. Vielva,5 R. A. Watson,4 E. Artal,10 M. Ashdown,6,7 R. B. Barreiro,5 +J. D. Bilbao-Ahedo,5,11 F. J. Casas,5 B. Casaponsa,5 R. Cepeda-Arroita,4 E. de la Hoz,5,11 +C. Dickinson,4 R. Fernández-Cobos,5,12 M. Fernández-Torreiro,1,2 R. González-González,1,2 +C. Hernández-Monteagudo,1,2 M. López-Caniego,13,14 C. López-Caraballo,1,2 +E. Martínez-González,5 M. W. Peel,1,2 A. E. Peláez-Santos,1,2 Y. Perrott,6,15 L. Piccirillo,4 +N. Razavi-Ghods,6 P. Scott,6 D. Titterington,6 D. Tramonte,1,2,16,17 R. Vignaga.1,2 +1Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain +2Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain +3Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France. +4Jodrell Bank Centre for Astrophysics, Alan Turing Building, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, +Oxford Road, Manchester M13 9PL, Manchester, UK +5Instituto de Física de Cantabria (IFCA), CSIC-Univ. de Cantabria, Avda. los Castros, s/n, E-39005 Santander, Spain +6Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK +7Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +8Consejo Superior de Investigaciones Científicas, E-28006 Madrid, Spain +9Departamento de Física. Facultad de Ciencias. Universidad de Córdoba. Campus de Rabanales, Edif. C2. Planta Baja. E-14071 Córdoba, Spain. +10Departamento de Ingenieria de COMunicaciones (DICOM), Laboratorios de I+D de Telecomunicaciones, Plaza de la Ciencia s/n, E-39005 Santander, Spain +11Departamento de Física Moderna, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain +12Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. los Castros, s/n, E-39005 Santander, Spain +13Aurora Technology for the European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 +Villanueva de la Cañada, Madrid, Spain +14Universidad Europea de Madrid, 28670, Madrid, Spain +15School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand +16Purple Mountain Observatory, CAS, No.10 Yuanhua Road, Qixia District, Nanjing 210034, China +17NAOC-UKZN Computational Astrophysics Center (NUCAC), University of Kwazulu-Natal, Durban 4000, South Africa. +Accepted 2022 November 11. Received 2022 November 1; in original form 2022 July 29 +ABSTRACT +We present QUIJOTE intensity and polarization maps in four frequency bands centred around +11, 13, 17 and 19 GHz, and covering approximately 29 000 deg2, including most of the Northern +sky region. These maps result from 9 000 h of observations taken between May 2013 and June +2018 with the first QUIJOTE instrument (MFI), and have angular resolutions of around 1◦, +and sensitivities in polarization within the range 35–40 𝜇K per 1-degree beam, being a factor +∼ 2–4 worse in intensity. We discuss the data processing pipeline employed, and the basic +characteristics of the maps in terms of real space statistics and angular power spectra. A number +of validation tests have been applied to characterise the accuracy of the calibration and the +residual level of systematic effects, finding a conservative overall calibration uncertainty of +5 %. We also discuss flux densities for four bright celestial sources (Tau A, Cas A, Cyg A and +3C274) which are often used as calibrators at microwave frequencies. The polarization signal +in our maps is dominated by synchrotron emission. The distribution of spectral index values +between the 11 GHz and WMAP 23 GHz map peaks at 𝛽 = −3.09 with a standard deviation of +0.14. The measured BB/EE ratio at scales of ℓ = 80 is 0.26 ± 0.07 for a Galactic cut |𝑏| > 10◦. +We find a positive TE correlation for 11 GHz at large angular scales (ℓ ≲ 50), while the EB and +TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150. The maps discussed +in this paper are publicly available. +Key words: cosmology: observations – cosmic microwave background +★ E-mail: jalberto@iac.es +© 2022 The Authors +arXiv:2301.05113v1 [astro-ph.GA] 12 Jan 2023 + +2 +Rubiño-Martín et al. +1 +INTRODUCTION +Measurements of the Cosmic Microwave Background (CMB) +anisotropies provide one of the most powerful tools in modern cos- +mology, playing a fundamental role in our current understanding of +the physics of the early Universe and structure formation (Bennett +et al. 2013; Planck Collaboration et al. 2020a). Moreover, CMB +polarization observations open a window to probe the amplitude +of primordial gravitational waves generated during the inflationary +epoch (Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997). Fol- +lowing this scientific motivation, observations of B-modes at large +angular scales have progressed substantially over the last few years. +Current best upper limits on the tensor-to-scalar ratio come from +the BICEP/Keck 2018 CMB polarization data (Ade et al. 2021), +and give 𝑟 < 0.036 at 95% confidence level, which improves to +𝑟 < 0.032 when adding the latest Planck PR4 data (Tristram et al. +2022). Upcoming ground-based experiments like Simons Observa- +tory (Ade et al. 2019) or CMB-S4 (Abazajian et al. 2022), and space +missions like LiteBIRD (LiteBIRD Collaboration et al. 2022) will +improve these constraints in the coming years. +Due to the low amplitude of this primordial B-mode signal, +the control and removal of diffuse Galactic foreground contamina- +tion in polarization is becoming a key challenge for current and +future CMB experiments. Basically there are two main Galactic +foregrounds that are known to emit linearly polarized radiation: +the synchrotron emission resulting from cosmic ray electrons ac- +celerated around the Galactic magnetic field lines, and the thermal +radiation from interstellar dust grains also aligned with the mag- +netic field (Bennett et al. 2013; Planck Collaboration et al. 2016g, +2020d). Anomalous microwave emission (AME) has been also de- +tected in intensity, but no polarization has been measured up to date +(Rubiño-Martín et al. 2012a; Dickinson et al. 2018). Although there +are theoretical motivations to expect negligible polarization levels if +AME is produced by spinning dust grains (Draine & Hensley 2016), +improved low frequency observations will be needed to consolidate +our understanding of this physical process. +The Planck satellite (Planck Collaboration et al. 2020a) pro- +duced seven full sky polarization maps covering the frequency range +between 30 and 353 GHz. The Wilkinson Microwave Anisotropy +Probe (WMAP) satellite (Bennett et al. 2013) scanned the full sky +in polarization in five bands between 23 and 94 GHz. The analysis of +these data shows that, for a B-mode signal with amplitude 𝑟 = 10−3 +(which is the target of the LiteBIRD space mission), there is no +frequency domain or sky region where the sum of the synchrotron +and thermal dust foregrounds is subdominant with respect to the +expected CMB B-mode signal (Planck Collaboration et al. 2016a; +Krachmalnicoff et al. 2016). Moreover, further analyses of these +and other datasets show increasing evidence of complexity in the +spectral and spatial behaviour of the Galactic dust and synchrotron +emissions (Choi & Page 2015; Planck Collaboration et al. 2017a; +Krachmalnicoff et al. 2018; Fuskeland et al. 2021; Weiland et al. +2022; de Belsunce et al. 2022). +The situation is particularly complex for the polarized syn- +chrotron emission. The sensitivity of the low frequency channels +from Planck and WMAP does not allow the detection of polarized +synchrotron signal at intermediate and high Galactic latitudes, and +therefore we are lacking a detailed spectral modelling of this emis- +sion precisely in the regions of cosmological interest. In this context, +there is a need for complementing the existing satellite observations +with measurements at lower frequencies in order to improve our de- +scription of the foregrounds at the required level for B-mode studies. +There are only a limited number of radio surveys that preserve the +large-scale structure of Galactic emission, and most of them pro- +vide only intensity measurements (Haslam et al. 1982; Berkhuijsen +1972; Reich et al. 2001; Jonas et al. 1998), but this situation is +now changing. The S-band Polarization All-Sky Survey (S-PASS; +Carretti et al. 2019) recently provided the first map of the polar- +ized radio emission over the southern sky at declinations below −1◦ +taken with the Parkes radio telescope at 2.3 GHz. The C-Band All +Sky Survey (C-BASS; Jones et al. 2018) will cover the full sky at +5 GHz, and the maps of the northern sky will be soon available. +With the aim of providing spectral coverage complementary +to WMAP and Planck at intermediate frequencies, the Q-U-I JOint +Tenerife Experiment (QUIJOTE, Rubiño-Martín et al. 2010) is a sci- +entific collaboration between the Instituto de Astrofisica de Canarias +(IAC), the Instituto de Fisica de Cantabria (IFCA), the Universities +of Cantabria, Manchester and Cambridge, and the IDOM company. +It has the goal of characterising the polarization of the CMB and +other Galactic and extragalactic physical processes in the frequency +range 10–40 GHz and at large angular scales ( >∼ 1◦). QUIJOTE has +been designed to have the required sensitivity to detect a primordial +gravitational-wave component if the tensor-to-scalar ratio is larger +than 𝑟 = 0.05. The experiment is located at the Teide Observatory +(altitude of 2,400 m a.s.l) in Tenerife (Canary Islands), and con- +sists of two telescopes equipped with three instruments: the Multi- +Frequency Instrument (hereafter, MFI), operating at 10–20GHz, the +Thirty-GHz Instrument (TGI) and the Forty-GHz Instrument (FGI). +The two QUIJOTE telescopes, QT-1 (Gomez et al. 2010) and QT-2 +(Sanquirce et al. 2014; Sanquirce-García et al. 2016), are based on +an offset crossed-Dragone design with projected apertures of 2.25 +and 1.89 m for the primary and secondary mirrors respectively, +and provide optimal polarization properties (polarization leakage +≤ −25dB), low sidelobes (≤ −40 dB) and highly symmetric beams +(ellipticity ≤ 2 %). +MFI is a multi-channel instrument that has been operating +between November 2012 and October 2018 mounted on the first +QUIJOTE telescope, QT-1. MFI consists of four polarimeters (also +called here "horns"). Horns 1 and 3 operate in the band 10–14 GHz, +while horns 2 and 4 operate at 16–20 GHz. Using frequency filters +in the back-end module (hereafter BEM) of the instrument, each +horn provides outputs in two frequency sub-bands, each one with +an approximate bandwidth of Δ𝜈 = 2 GHz. There are a total of 8 +outputs for each polarimeter, and these are then fed into the Data +Acquisition Electronics (DAE). In total, the MFI provides four fre- +quency bands centred around 11, 13, 17 and 19 GHz, with each band +covered by two independent horns. The approximate angular resolu- +tion, given in terms of the full width at half-maximum, is 52 arcmin +for the low-frequency bands (11 and 13 GHz), and 38 arcmin for +the 17 and 19 GHz channels. During the lifetime of the instrument, +we had basically two instrumental configurations for the MFI. The +main difference of the second configuration with respect to the first +one is the integration of 90◦ hybrid couplers in each polarimeter, +giving correlated outputs in all four detectors. A more detailed de- +scription of the instrument can be found in Hoyland et al. (2012); +Pérez-de-Taoro et al. (2016), and will be included in a future paper +(Hoyland et al., in prep). A complete description of the MFI instru- +ment characteristics, as well as the MFI data processing pipeline, is +included in an accompanying paper (Génova-Santos et al. 2023). +As described in Rubiño-Martín et al. (2010), most of the +QUIJOTE-MFI observing time was dedicated to two main surveys: +a shallow Galactic survey (hereafter the "wide survey") covering +all the visible sky from Tenerife at elevations larger than 30◦, and +a deep cosmological survey covering approximately 3 000 deg2 in +three separated sky patches in the northern sky. In addition to those +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +3 +two main surveys, a fraction of the MFI observing time was dedi- +cated to raster scan observations in some selected Galactic regions. +Data from some of those MFI raster scan observations were already +presented in three QUIJOTE collaboration publications (Génova- +Santos et al. 2015, 2017; Poidevin et al. 2019), where we charac- +terised the presence of AME towards several Galactic molecular +complexes, as the Perseus region, W43, W47 or Taurus, and to- +wards a supernova remnant, W44. In particular, the study of W43 +provides the strongest upper limits to date on the polarization frac- +tion of the AME (Génova-Santos et al. 2017). Additional raster scan +observations were carried out in W51, IC443, rho-Ophiucus, and +M31, among others. +A preliminary version of the MFI wide survey maps, in com- +bination with C-BASS North data, were used in the study of the +𝜆-Orionis region (Cepeda-Arroita et al. 2021). This paper presents +the final maps of the QUIJOTE-MFI wide survey. Section 2 de- +scribes the observations and the data processing pipeline. The final +maps are presented in Section 3. The validation and characterisation +of these maps is presented in Section 4. An assessment of the overall +calibration uncertainty of the maps is discussed in Section 5, while +Section 6 describes the generation of specific noise simulations for +the QUIJOTE MFI wide survey. Sections 7, 8 and 9 discuss some +of the basic properties of the maps both in real and harmonic space, +including the photometry results of some bright radio sources. Fi- +nally, Section 10 describes the data products and associated scien- +tific papers accompanying this paper. All of them are devoted to +the understanding of the low frequency Galactic foregrounds in in- +tensity and polarization, either in the full QUIJOTE MFI footprint +or in localised regions, and using various analysis techniques. The +conclusions of this work are presented in Section 11. +2 +THE QUIJOTE-MFI WIDE SURVEY DATA +The QUIJOTE wide survey is a shallow survey which covers all +the visible sky from the Teide Observatory (latitude +28.3◦) with +elevations greater than 30◦ (more than 29 000 deg2). This was one +of the main scientific objectives of QUIJOTE (Rubiño-Martín et al. +2012b), and in particular, of the MFI instrument. This paper presents +the QUIJOTE MFI wide survey maps, which were obtained with +approximately 9 000 h of observing time. The four final maps at +nominal frequencies 11, 13, 17 and 19 GHz, smoothed to 1 degree +resolution, are shown in Figs. 1, 2, 3 and 4, respectively. All maps +were generated using the HEALPix1 pixelization scheme (Górski +et al. 2005) with 𝑁side = 512. In HEALPix the sphere is divided into +12𝑁side2 pixels of equal area. In particular, 𝑁side = 512 corresponds +to a pixel size of approximately 6.9 arcmin on the sky. Figure 5 also +shows the polarized intensity (𝑃 = +√︁ +𝑄2 + 𝑈2), the polarization +angle direction2 (𝛾 = 0.5 arctan(−𝑈/𝑄)), and the direction of mag- +netic field lines for the 11 GHz map. In the following subsections +we describe the observations, the data processing pipeline, the map- +making and the specific post-processing and recalibration applied +to these maps. +1 https://healpix.sourceforge.io +2 QUIJOTE polarization maps use the COSMO convention from HEALPix, +so we use a minus sign in the definition of 𝛾 to recover the IAU convention +for the angle. +2.1 +Observations +The maps described in this paper are based on MFI observations +carried out between May 2013 and June 2018 using the so-called +"nominal mode", which consists of continuous (360◦) azimuth scans +at a constant telescope elevation. The default azimuth scan speed +was 𝑣AZ = 6 deg s−1 from the beginning of the survey until January +9th 2014, but this was increased to 𝑣AZ = 12 deg s−1 after this date, +in order to reduce the 1/ 𝑓 noise contribution in the intensity maps. In +this observing mode, every day each MFI horn covers a continuous +band of 360◦ in right ascension, and a certain declination range +specified by the elevation of the telescope. As in all QUIJOTE- +MFI observations, and in order to minimize systematic effects in +the polarization parameters, observations are carried out in four +discrete positions of the polar modulators 𝜃pm =(0◦, 22.5◦, 45◦ and +67.5◦). In the wide survey, each observation at a given elevation +and modulator angle position has a typical duration of 24 h. +The combination of multiple elevations allows us to obtain a +more homogeneous sampling of the sky. Table 1 contains the final +set of telescope elevations considered here to produce the maps, to- +gether with the total number of hours observed and used in each case. +In total, there are approximately 9 200 h of observations, equivalent +to 383 observing days. Almost all of this observing time was suit- +able for use in the preparation of the intensity maps. However, the +final polarization maps only use of the order of 5 700 h, as explained +below. +Observations are also separated in periods of several months. +The definition of each period is usually associated with changes +either in the MFI instrument configuration, telescope configuration, +or simply to new observing cycles after instrument maintenance. +A complete description of those periods, as well as the associated +instrument changes, can be found in Génova-Santos et al. (2023). +We note that for the MFI wide survey, we conducted observations +only during periods 1, 2, 5 and 6. The global dates and effective +epoch (year) for each of those periods are listed in Table 2. +As noted in this table, an extended shielding was installed in the +first QUIJOTE telescope (QT-1) at the beginning of period 2. The +main reason for this was to minimize the impact of far sidelobes due +to the emission of geo-stationary satellites, which were particularly +important for horn 1 (Génova-Santos et al. 2023). In addition, during +the operations horn 1 was either not operative (periods 5 and 6) or +had problems with the positioning of the polar modulator (period +2). Because of these reasons, although wide-survey maps of horn +1 have been produced for internal consistency tests, they have not +been used for this paper because they are significantly affected by +systematic effects. +2.2 +Data processing pipeline +A complete description of the MFI data processing pipeline can be +found in the MFI pipeline paper (Génova-Santos et al. 2023). Here, +we summarize the basic characteristics of the MFI data, and we +discuss those aspects which are specific of the MFI wide survey. +Each MFI polarimeter is divided into a lower and upper band +of approximately 2 GHz bandwidth which is defined by the band- +pass filters. Each sub-band has four outputs, which are labelled as +(𝑉x+y,𝑉x−y,𝑉x,𝑉y). The first two outputs are called "correlated" +channels because in the first (original) configuration of the instru- +ment they passed through a 180◦-hybrid, and therefore they have +correlated (common) 1/ 𝑓 noise properties. The second pair is called +"uncorrelated" channels, and in the original configuration provided +two outputs with independent noise. The first instrument configu- +MNRAS 000, 1–58 (2022) + +4 +Rubiño-Martín et al. +Figure 1. QUIJOTE MFI maps at 11 GHz in Galactic coordinates, smoothed to 1 degree resolution and using 𝑁side = 512. Top: intensity 𝐼. Middle: polarization +𝑄 component. Bottom: polarization 𝑈 component. +MNRAS 000, 1–58 (2022) + +QUJOTE1H311GHz(1deg) +5 +mk +20QUJOTEQH311GHz(1deg) +mKQUOTEUH311GHz（1deg） +mKQUIJOTE MFI wide survey +5 +Figure 2. QUIJOTE MFI maps at 13 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈 +component. +MNRAS 000, 1–58 (2022) + +QUJOTE1H313GHz(1deg) +5 +mk +20QUJOTEQH313GHz(1deg) +1 +mKQUJOTEUH313GHz(1deg) +1 +mK6 +Rubiño-Martín et al. +Figure 3. QUIJOTE MFI maps at 17 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈 +component. +MNRAS 000, 1–58 (2022) + +QUIJOTE117GHz combined H2+H4(1deg) +5 +mk +20QUijOTEQ17GHzcombinedH2+H4(1deg) +mkQUjOTEU17GHzcombinedH2+H4(1deg) +mkQUIJOTE MFI wide survey +7 +Figure 4. QUIJOTE MFI maps at 19 GHz smoothed to 1 degree resolution. Top: intensity 𝐼. Middle: polarization 𝑄 component. Bottom: polarization 𝑈 +component. +MNRAS 000, 1–58 (2022) + +QUJOTE119GHz combinedH2+H4(1deg) +5 +mk +20QUljOTEQ19GHzcombinedH2+H4(1deg) +mkQUljOTEU19GHzcombinedH2+H4(1deg) +mk8 +Rubiño-Martín et al. +Figure 5. QUIJOTE MFI maps at 11 GHz smoothed to 1 degree resolution. Top: polarized intensity 𝑃 = +√︁ +𝑄2 + 𝑈2. Middle: polarization angle. Bottom: +Polarization angle at 11 GHz, rotated by 90◦ to indicate the direction of the Galactic magnetic field projected on the plane of the sky. The colours represent the +polarized intensity signal. The "drapery" pattern was obtained with the healpy routine line_integral_convolution, and it is smoothed to 2◦ for display purposes. +MNRAS 000, 1–58 (2022) + +QUUJOTEP11GHz(1deg) +0 +mk +1.3QUJOTEang11GHz(1deg) +-90 +deg +90MFI 11GHz - LICQUIJOTE MFI wide survey +9 +Table 1. List of telescope elevations used for the wide survey observations with the QUIJOTE MFI instrument. The second column indicates the total observing +time (𝑇observed) in hours dedicated to each elevation. Columns 3 to 6 show the total observing time for the actual subset of observations used for the final +intensity (𝑇used,I) and polarization (𝑇used,P) maps. In the later case, different subsets of data are used for each particular horn. Observations are separated in +periods (column 7), which correspond to specific epochs (column 8) and instrumental configurations (see text for details). +Elevation (◦) +𝑇observed (h) +𝑇used,I (h) +𝑇used,P,H2 (h) +𝑇used,P,H3 (h) +𝑇used,P,H4 (h) +Period +Range of Dates +30 +121.9 +0.0 +0.0 +0.0 +0.0 +1 +06/2013–07/2013 +60 +986.5 +986.5 +0.0 +0.0 +0.0 +1 +05/2013–03/2014 +65 +665.2 +665.2 +0.0 +0.0 +0.0 +1 +05/2013–03/2014 +70 +394.9 +0.0 +0.0 +0.0 +0.0 +1 +06/2013–03/2014 +30 +829.4 +829.4 +829.4 +829.4 +0.0 +2 +08/2014–03/2015 +40 +489.3 +489.3 +489.3 +489.3 +0.0 +2 +08/2014–01/2015 +50 +564.7 +564.7 +564.7 +564.7 +0.0 +2 +08/2014–10/2015 +60 +91.7 +91.7 +91.7 +91.7 +0.0 +2 +06/2014–09/2014 +65 +128.9 +128.9 +128.9 +128.9 +0.0 +2 +08/2014–10/2014 +30 +200.1 +0.0 +0.0 +0.0 +0.0 +5 +08/2016–10/2016 +40 +324.6 +324.6 +0.0 +324.6 +324.6 +5 +08/2016–10/2016 +50 +488.6 +488.6 +0.0 +488.6 +488.6 +5 +08/2016–10/2016 +60 +198.4 +198.4 +0.0 +198.4 +198.4 +5 +08/2016–09/2016 +35 +1998.6 +1998.6 +1998.6 +1998.6 +1998.6 +6 +12/2017–06/2018 +50 +326.7 +326.7 +326.7 +326.7 +326.7 +6 +03/2017–04/2017 +60 +552.5 +552.5 +552.5 +552.5 +552.5 +6 +12/2016–02/2017 +65 +430.7 +430.7 +430.7 +430.7 +430.7 +6 +03/2017–04/2017 +70 +400.8 +400.8 +400.8 +400.8 +400.8 +6 +02/2017–04/2017 +TOTAL: +9193.6 +8476.6 +5813.3 +6824.9 +4720.9 +Table 2. Definition of the four observing periods during which we carried out wide survey observations with the QUIJOTE MFI instrument. Last column +indicates the instrument configuration and main changes. Configuration 1 corresponds to the original MFI design (Hoyland et al. 2012), while configuration 2 +corresponds to the installation of 90◦-hybrids (Pérez-de-Taoro et al. 2016). See text for details. +Period +From +To +Effective year +Comments +(dd/mm/yyyy) +(dd/mm/yyyy) +1 +12/11/2012 +10/04/2014 +2013.7 +Configuration 1 for all horns. No extended shielding. +2 +11/04/2014 +30/11/2015 +2014.9 +Horn 1 in configuration 2. Extended shielding installed. +5 +01/05/2016 +14/10/2016 +2016.7 +All horns in configuration 2. Horn 1 not operative. +6 +15/10/2016 +01/11/2018 +2017.8 +All horns in configuration 2. Horn 1 not operative. +ration (Hoyland et al. 2012) was used during periods 1 and 2 (see +Table 2), but a new configuration was later implemented using 90◦- +hybrids (Pérez-de-Taoro et al. 2016). In this second configuration, +all MFI channels are formally correlated, but for historical reasons +we maintain the notation of correlated and uncorrelated channels. +The sum of pairs of channels provides two independent mea- +surements of the intensity. For example, for the first MFI configu- +ration, we have +𝑉x + 𝑟u𝑉y = 𝑠x𝑔2𝐼 +(1) +𝑉x+y + 𝑟c𝑉x−y = 𝑠x+y𝑔2𝐼, +(2) +while the difference of the pairs of channels provides two measure- +ments of the linear polarization +𝑉x − 𝑟u𝑉y = 𝑠x𝑔2� +𝑄 cos(4𝜃pm + 2𝛾p) − 𝑈 sin(4𝜃pm + 2𝛾p) +� +(3) +𝑉x+y − 𝑟c𝑉x−y = 𝑠x+y𝑔2� +𝑄 sin(4𝜃pm + 2𝛾p) + 𝑈 cos(4𝜃pm + 2𝛾p) +� +, +(4) +where 𝑉𝑖 +represents the output voltage for channels 𝑖 +∈ +{x, y, x + y, x − y}, 𝑠x and 𝑠x+y are the responsivities of those +branches in the MFI instrument, 𝑔 represents the voltage gain of +the two MFI Low Noise Amplifiers (here taken to be the same in +the two LNAs for simplicity), 𝑟c and 𝑟u are the so-called r-factors +which measure the possible gain and responsivity imbalance in the +pair of channels, 𝜃pm is the position angle of the polar modulator, +and 𝛾p is the parallactic angle (see details in Génova-Santos et al. +2023). When the two channels in the pair have correlated noise, +then the difference cancels significantly the 1/ 𝑓 component. In the +MFI pipeline, maps for correlated and uncorrelated channels are +produced separately, and combined afterwards. Due to their noise +properties, in polarization we use only those pair of channels with +common 1/ 𝑓 properties, i.e. the "correlated" channels during pe- +riods 1 and 2, and both of them ("correlated" and "uncorrelated" +channels) for periods 5 and 6. +The MFI data sampling rate is 1 ms. For the wide survey, all +time streams (hereafter Time-Ordered Data or TODs) are binned in +40 ms samples. Note that this is different from the binning scheme +of 60 ms used for raster scan observations in the past (e.g. Génova- +Santos et al. 2017), due to the higher azimuth scan speed. The bin- +ning process allows us to assign a variance 𝜎2 +𝑖 to each binned sam- +MNRAS 000, 1–58 (2022) + +10 +Rubiño-Martín et al. +Table 3. QUIJOTE-MFI basic peformance parameters. Values for 11 and 13 GHz correspond to horn 3 of MFI. Values for 17 and 19 GHz have been obtained +as the weighted average of horns 2 and 4, using the relative weights described in Table 9. +Parameter +11 GHz +13 GHz +17 GHz +19 GHz +MFI horns contributing to these bands +3 +3 +2,4 +2,4 +Centre frequency (nominal), 𝜈0 (GHz) +11.1 +12.9 +16.8 +18.8 +Effective frequency for 𝛼 = −1, 𝜈𝑒 (𝛼 = −1) (GHz) +10.98 +12.89 +16.85 +18.85 +Bandwidth (GHz) +2.17 +2.20 +2.24 +2.34 +Beam FWHM (arcmin) +55.38 +55.84 +38.95 +40.32 +Main beam solid angle, Ωmb (10−4sr) +2.748 +2.781 +1.362 +1.428 +Beam ellipticity𝑎, 𝑒 +0.013 +0.040 +0.034 +0.035 +Antenna sensitivity, Γ (𝜇KCMB/Jy) +961.9 +703.8 +847.0 +645.2 +White-noise level in timelines (𝜇KCMBs1/2) +858 +697 +773 +866 +Knee frequency 𝑓k in polarization (mHz) +254 +198 +223 +556 +1/ 𝑓 slope in polarization +1.95 +1.86 +1.73 +1.34 +Overall calibration uncertainty I (%) +5 +5 +5 +5 +Overall calibration uncertainty Q,U (%) +5 +5 +6 +6 +𝑎 The ellipticity is defined here as 𝑒 = 1 − FWHMmin/FWHMmax. +Table 4. Colour correction coefficients, 𝐶(𝛼, 𝜈0) = 𝑐0 + 𝑐1𝛼 + 𝑐2𝛼2. +The colour corrected temperature is obtained as 𝐶(𝛼, 𝜈0)𝑇 , being 𝑇 the +uncorrected one. +Band +𝜈0 +𝑐0 +𝑐1 +𝑐2 +11 +11.1 +0.981 +0.0125 +-0.0015 +13 +12.9 +1.001 +0.0018 +-0.0012 +17 +16.8 +1.007 +-0.0022 +-0.0007 +19 +18.8 +1.007 +-0.0020 +-0.0008 +ple 𝑖, which we used to define the associated weights (𝑤𝑖 = 1/𝜎2 +𝑖 ). +When propagated through the entire pipeline, the resulting weight +maps are used for the combination of maps from correlated and +uncorrelated channels, and will be used also in the noise character- +ization. +Table 3 contains the summary of basic parameters (central fre- +quencies, beams, solid angles) for all MFI horns, extracted from +Génova-Santos et al. (2023). We also include the calibration uncer- +tainties discussed in Sect. 5, and representative noise characteristics +(knee frequencies and 1/ 𝑓 slopes) that we have obtained from this +data. Table 4 also presents the colour corrections for these maps, de- +rived from the associated bandpasses as explained in Génova-Santos +et al. (2023). Colour corrections are presented here in terms of sec- +ond order polynomials as a function of the spectral index 𝛼. For a +sky emission having a flux density law 𝑆𝜈 ∝ 𝜈𝛼, the coefficients +𝐶(𝛼, 𝜈0) provide the multiplicative correction factor to the mea- +sured flux density for the MFI frequency map at nominal frequency +𝜈0. These corrections are identical for intensity and polarization. +Throughout the paper, we use the following notation to refer +to specific MFI maps per horn and frequency. We will use three +numbers, the first one refering to the horn number (i.e. 2, 3 or 4), +and the other two indicating the nominal frequency (i.e. 11, 13, 17 +or 19). For example, the 19 GHz map for horn 4 will be cited either +as 𝑚4,19, 𝑚419, or directly, 419 map. We recall that each map will +be made, in principle, from the contribution of both the correlated +and uncorrelated channels. In some case, we use the same notation +to refer to channels. For example, the correlated channels of 419 are +obtained from the 𝑉x+y and 𝑉x−y outputs of horn 4 at 19 GHz. +In the following, we discuss specific additions to the MFI +pipeline in the case of the wide survey. In particular, we discuss +the gain model for wide survey data and the specific data flagging +applied in ”nominal mode". After this, we present our approach +to correct for Radio Frequency Interference (RFI) signals and at- +mospheric contamination in the MFI wide survey data. For these +corrections, the general philosophy adopted in our pipeline follows +a two step approach. We first implement specific methods to de- +tect and mitigate the effect of RFI and atmospheric signals both +at the TOD (see Sect. 2.2.3 and 2.2.4) and at the map-level in the +post-processing stage (Sect. 2.4). Then, a detailed assessment is +made later of residual signals in the maps by a variety of techniques +(Sect. 4). In practice, the values of uncertainties in calibration and +other error budgets are increased appropriately if there is clear evi- +dence of residual effects still being present in the maps (Sect. 5). +2.2.1 +Gain model +Gain calibration and the associated relative gain factors (𝑟c and 𝑟u) +between pairs of channels are based on Cas A and Tau A observa- +tions taken during each period. Relative gain variations with respect +to the mean gain value 𝐺0 during the full period are traced using +the signal of a thermally stabilized calibration diode, located at the +centre of the secondary mirror. Every 30 s, the diode injects a signal +during 1 s, which is used to measure the relative gain of each chan- +nel, 𝛿𝐺(𝑡) ≡ 𝐺(𝑡) −𝐺0 (see Génova-Santos et al. 2023, for details). +Nominal mode observations used for the wide survey usually have +a duration of one day for each polarimeter position. Specifically +for this nominal mode data, a smooth (interpolated) gain model is +obtained by applying a top-hat smoothing kernel on the individual +gain measurements. The width of this kernel is 30 minutes for low +frequency channels, and 120 minutes for high frequency ones, due +to the different signal-to-noise ratio of the diode signal in the differ- +ent channels. We have checked that the typical MFI gain variations +occur on timescales longer than those. These interpolated models +are used to correct the instrument gain as +𝐺(𝑡) = 𝐺0 +� +1 + 𝛿𝐺(𝑡) +𝐺0 +� +. +(5) +Once these interpolated gain models are generated for the entire +survey, they are inspected in order to find residual features (peaks +or jumps) in the models. These features are introduced in flagging +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +11 +tables which are later applied during the generation of the calibrated +TOD. +2.2.2 +Data flagging +Génova-Santos et al. (2023) describes the basic data flagging that +is applied by default to all MFI observations, including flags due +to voltage ranges, house-keeping parameters, emission of the Sun +and Moon (using a 10◦ exclusion radius), and also the emission of +geo-stationary satellites. In particular, this last flagging produces +the empty strip around declination zero degrees that is seen in the +11 and 13 GHz maps (Figures 1 and 2), and also the noise increase +in the same region in the 17 and 19 GHz maps (Figures 3 and 4), +due to the lower number of independent crossings in the area. +For the wide survey, a specific flagging based on the root mean +square (rms) of the data in each scan has been implemented as +follows. A first version of the wide survey maps is produced with +the default pipeline. From here, and separately for each period, we +compute the rms of the data minus the reprojected version of that +map onto the TOD, in scales of 30 s. This time value corresponds to +the length of one azimuth scan at the default scanning speed, and to +the length of half azimuth scan for the scanning speed used in part +of period 1. Histograms with the distribution of these rms values +are built for each channel and period, and are used to flag those +scans with extreme rms values (either above 1.7 times the median +rms value in the entire period, or below 0.5 times that median rms). +The fraction of excluded data using this procedure depends on the +channel, but typically is of the order of 10–20 per cent. Once this +flagging is applied, no obvious residual spikes or rings are visible in +the reconstructed maps. Finally, for the final wide survey maps we +also exclude Jupiter, Venus and Mars, using a 2◦ exclusion radius +directly in the TOD. Appendix A contains detailed tables with the +percentage of used (and flagged) data for each MFI channel in every +observing period. Those fractions of used data apply to the total +number of used hours in each case, which were listed in Table 1. On +average we are using 61 % of the data after applying all the different +flags. Out of the flagged 39 %, most of it (approximately three +quarters) is excluded in the specific post-processing stage described +in this subsection. The percentage of used data was slightly lower +in period 2 (52 %), and higher in period 6 (68 %). +2.2.3 +RFI correction +Specifically for the wide survey data, residual random spikes as well +as possible RFI signals from satellites not identified in our standard +pipeline are flagged using a dedicated matched-filter code that is +applied to the one-dimensional TOD. The only assumption is that +the object to be detected is unresolved, and thus should match the +beam profile. The code3 excludes the location of the known bright +radio-sources, which are also easily detected in the TOD. +Residual RFI signals appear at fixed azimuth (AZ) locations. In +the case of QUIJOTE MFI, most of these signals are due to the radio +emission of geo-stationary satellites entering through the beam far +sidelobes. These signals were particularly visible in period 1 and at +low frequencies (horns 1 and 3), until the installation of the extended +shielding of the first QUIJOTE telescope was completed. All other +periods are much less affected, due to the significant suppression +of the far sidelobes. Because of this reason, period 1 was used for +the intensity maps only, and not for polarization. In order to remove +3 https://gitlab.com/HerranzD/quijote-satdet +these RFI signals, we generate spatial templates in the azimuth +direction, by obtaining stacks of the TOD signal as a function of +AZ, 𝑓 (AZ). These templates are computed for each period and each +elevation separately, and thus rely on the assumption that the RFI +signal is stable in time during the whole period. The templates are +generated both for the sum and difference of MFI channels, and +thus, they are applied to the intensity and the polarization TOD. +Finally, a smoothed version of these templates (in scales of 10◦) +is subtracted from the TOD. Figure 6 shows two examples of the +global RFI patterns removed using this procedure. These figures are +obtained as the difference between the end-to-end MFI maps with +and without applying the RFI correction at the TOD level. We also +note that once the final maps are produced, any residual RFI signals +are effectively corrected in the post-postprocessing stage, using a +function of the declination as described in Sect. 2.4. +Some remaining RFI features and glitches are removed after +a careful inspection of the final maps. For this purpose, separate +maps for each elevation and period are produced. Once a particular +RFI feature is identified in these maps, the corresponding location is +introduced in specific flagging tables for each period and elevation, +which are later applied to the calibrated TOD. +2.2.4 +Atmospheric correction +Although the observations are done at (nominal) constant elevation, +there are still some residual variations due to changes in the atmo- +spheric contribution along different directions. These variations are +seen in the data as correlated patterns repeating in azimuth on very +large angular scales, and with the amplitude increasing strongly with +frequency, as expected for MFI frequencies due to the proximity of +the 22 GHz atmospheric water line (see e.g. Paine 2019). It also +evolves and changes on the scale of several hours, which is expected +due to varying integrated water vapour content along lines of sight +as weather systems blow over the site. It is possible to try to remove +these effects especially at the more troublesome higher frequencies +by a Principal Component Analysis (PCA) decomposition to look +for these correlated signals. +To model this atmospheric component in the MFI intensity +data, only broad scale features are removed by using baselines up +to only 5 harmonics over the azimuth scans. A mask is used to +avoid bright emission from the Galactic plane and strong point +sources. The baseline atmospheric patterns are generated over an +hour, as a compromise between good signal to noise and the time +evolution of the atmosphere. The PCA decomposition method used +is implemented in Python, using the sklearn module (Pedregosa +et al. 2011) on all the channels. +The first most significant component found is one that in- +creases strongly with frequency, with the spectrum expected for +water vapour. A histogram of the ratios between 17 and 19 GHz, the +two most strongly affected frequencies, shows a clear broad peak +at 0.42 near the values expected from atmospheric models for the +Teide Observatory of 0.49 (see e.g. Paine 2019, and typical PWV +conditions of 3–4 mm), although this sits on a smaller but much +broader distribution. Points outside the range 0.3 to 0.6 appear to +be for dryer conditions, with the implication that the water vapour +signal is too weak to be reliably recovered. It was decided to use +this range ratio of 17 to 19 GHz signal as an indicator of a usable +atmospheric signal that can be removed. The removal is done by +subtracting the PCA template with the coefficient found for each +frequency channel at the TOD level. +Maps of this atmospheric emission can be produced running +the full pipeline with and without this atmospheric correction, and +MNRAS 000, 1–58 (2022) + +12 +Rubiño-Martín et al. +Figure 6. RFI patterns removed from the maps of the QUIJOTE MFI wide survey. Top row corresponds to the RFI emission at 11 GHz (horn 3, labelled as +"311"), while the bottom row corresponds to 17 GHz (horn 4, labelled as "417"). From left to right, we show the residuals for intensity, Stokes Q and Stokes U +parameters. The colour scale is the same in all six panels, corresponding to the temperature range ±0.2 mK. For visualization purposes, all maps are smoothed +to one degree resolution. +then taking the difference of the two resulting maps. The atmo- +spheric emission maps for horns 3 and 4 are shown in Figure 7. +The map for horn 2 is similar to the one for horn 4, so it is omitted +for clarity. As expected, this atmospheric contribution is more rel- +evant at higher MFI frequencies, and affects large angular scales. +As shown below (see Sect. 2.5), when doing a spherical harmonic +expansion of the maps, this correction is only relevant in the in- +tensity maps at multipoles ℓ ≲ 15 for 11 GHz, and ℓ ≲ 25 for +19 GHz. No atmospheric correction is needed in polarization for +the MFI wide survey maps. When a similar procedure is applied +to the polarization data, the results are consistent with essentially +unpolarized atmospheric emission. +2.3 +Map-making +The QUIJOTE MFI wide survey maps are produced using the PI- +CASSO code (Guidi et al. 2021), a destriping algorithm based on +the MADAM approach (Keihänen et al. 2005, 2010) but specifi- +cally implemented and optimised for QUIJOTE MFI. The destriping +technique corrects for a correlated noise component by modelling +the 1/ 𝑓 drifts in the TOD with a set of consecutive offsets with a +given time length 𝑡b, the so-called baselines. The PICASSO code +has been tested extensively using realistic simulations matching the +actual observations of the MFI wide survey and with realistic noise +properties (Guidi et al. 2021). In these conditions, the reconstructed +maps preserve all angular scales with high fidelity, and in particular, +we expect a signal error better than 0.001 per cent at 20 < ℓ < 200. +Those realistic simulations were also used to set the reference +parameters adopted for the production of the final MFI wide survey +maps. In particular, we use a baseline length of 𝑡b = 2.5 s for the +entire survey. Maps are generated using the HEALPix pixelization +scheme with 𝑁side = 512. The specific priors for the 1/ 𝑓 noise +properties (knee frequency 𝑓𝑘, slope 𝛾, and cutoff frequency 𝑓cut) +are shown in Table 5, both for the intensity and polarization maps. +In the later case, the parameters are different depending on the +noise levels of the corresponding pair of channels (i.e. if they are +correlated or uncorrelated channels). As discussed in Guidi et al. +Table 5. Map-making parameters and related information. We consider three +different cases of use with the PICASSO code: intensity maps, polarization +maps with correlated channels, and polarization maps with uncorrelated +channels. For each case, we quote the prior values for the knee frequency +𝑓k, the slope of the 1/ 𝑓 noise component 𝛾, and the low cut-off frequency +𝑓cut, as well as the 𝑁side value of the HEALPix map and the baseline length +𝑡b (in seconds). See text for details. +Case +𝑓k +𝛾 +𝑓cut +𝑁side +𝑡b +[Hz] +[Hz] +[s] +I +40.0 +1.5 +0.033 +512 +2.5 +Q,U corr +0.3 +1.8 +0.033 +512 +2.5 +Q,U uncorr +40.0 +1.5 +0.033 +512 +2.5 +(2021), those priors are assumed to be stationary parameters for the +whole survey. +2.4 +Post-processing of MFI wide survey maps +2.4.1 +Combination of maps +For each horn and frequency sub-band, maps for the correlated +and uncorrelated pairs are produced running the PICASSO code +separately for each one of them. These maps are combined at this +post-processing stage, using the weight maps which are also pro- +duced by the map-making code as the propagation of the individual +weights for each sample in the binned TOD. The combination of +correlated (𝑥c) and uncorrelated (𝑥u) maps is done with the usual +formula for the weighted arithmetic mean: +𝑚 = 𝑤c𝑥c + 𝑤u𝑥u +𝑤c + 𝑤u +. +(6) +Given that both correlated and uncorrelated channels share the same +amplifier, we expect a high level of correlation between the two +intensity measurements. As shown below in Sect. 4.3.4, this cor- +relation is indeed of the order of 90–95 per cent for the intensity +channels, and consistent with zero for the polarization ones. Al- +though in principle it is possible to construct a minimum variance +MNRAS 000, 1–58 (2022) + +RFlmap (311,I) +-0.2 +mK +0.2RFl map (311,Q) +-0.2 +mk +0.2RF map (311,U) +-0.2 +mk +0.2RFl map (417,I) +-0.2 +mK +0.2RFl map (417,Q) +-0.2 +mk +0.2RFl map (417,U) +-0.2 +mk +0.2QUIJOTE MFI wide survey +13 +Figure 7. Atmospheric pattern removed from the intensity maps of the +QUIJOTE MFI wide survey. From top to bottom, we have the atmospheric +emission at 11 GHz (horn 3), 13 GHz (horn 3), 17 GHz (horn 4) and 19 GHz +(horn 4). The colour scale is the same in the four panels (±3 mK), in order to +visualise the increasing contribution of the atmospheric emission at higher +MFI frequencies. For visualization purposes, all maps are smoothed to one +degree resolution. +estimator accounting for these correlations in the intensity pairs, we +still use equation 6 for the combination of the intensity (correlated +and uncorrelated) maps, in order to have a more robust estimate of +the combination (see e.g. Schmelling 1995). +From equation 6, we can derive the expression for the weight +map of the linear combination as +𝑤 = +(𝑤c + 𝑤u)2 +𝑤c + 𝑤u + 2𝜌√𝑤c𝑤u +, +(7) +where 𝜌 stands for the correlation fraction between correlated and +uncorrelated channels. +The map-making code also produces an estimate of the co- +variance matrix in polarization, 𝑐𝑜𝑣(𝑄,𝑈), as well as the condition +number (𝑟cond) map (see equations 44 and 45 in Guidi et al. 2021). +Before forming the combination of the polarization maps in the +wide survey, those pixels with 𝑟cond > 3 are excluded. In practice, +this only affects a small number of pixels close to the boundary of +the satellite strip, as well as to the north celestial pole. In particular, +for the 419 map (i.e. horn 4 at 19 GHz) the number of affected pixels +is slightly larger in those areas. Appendix C contains the 𝑟cond maps +for all the MFI wide survey maps, together with other relevant maps, +as discussed in Sect. 3. Once the combination of the correlated and +uncorrelated maps is carried out in polarization, the corresponding +weight maps (𝑤𝑄, 𝑤𝑈) and covariance matrices 𝑐𝑜𝑣(𝑄,𝑈) are also +derived. Appendix C also presents images of the 𝑐𝑜𝑣(𝑄,𝑈) maps +for all horns and frequencies. These maps show that, as expected, +the normalized covariance 𝑐𝑜𝑣(𝑄,𝑈)/(𝜎𝑄𝜎𝑈) is very small (well +below 0.01 %), so effectively each pair of 𝑄 and 𝑈 maps are almost +independent. +2.4.2 +Residual interference: the FDEC filtering +After the map-making step, the resulting maps still present some +residual RFI and large-scale patterns, which are corrected during +this post-processing stage. As described in Sect. 2.2.3, residual RFI +signals appear at fixed azimuth locations, so during the map-making +process these features are projected onto the maps in stripes of con- +stant declination. This residual RFI is removed using a function of +the declination, 𝑓 (𝛿) (hereafter FDEC4), which is extracted directly +from the maps as the median of all pixels with the same declination. +This template function is built using a |𝑏| < 10◦ mask to exclude the +Galactic emission, and specific masks in intensity and polarization +for each frequency channel excluding the 10 per cent of the brightest +pixels. The procedure is applied both in intensity and polarization. +In polarization, the maps are first rotated to local (equatorial) coor- +dinates in order to extract the correction function. In this way, the +RFI contamination from static sources in local coordinates appears +as a constant signal in a given declination band. +Figure 8 shows the correction functions for intensity and po- +larization for all MFI maps based on correlated channels. Similar +curves are obtained for uncorrelated channels. Note that in this fig- +ure, the panel for Stokes Q parameter corresponds to equatorial +coordinates. As expected for RFI signals, these correction functions +are larger in the vicinity of the geo-stationary strip (around declina- +tion zero) and at low declinations (corresponding to low elevation +values of the telescope, where the RFI is expected to be larger). We +also note that they are also larger in intensity than in polarization. +Once these correction functions 𝑓 (𝛿) are derived, they are re- +projected onto a map in order to produce a RFI template. These +4 https://github.com/jarubinomartin/sancho.git +MNRAS 000, 1–58 (2022) + +Atmosphere (311,D) +mk +-3 +3Atmosphere (313,1) +mk +-3 +3Atmosphere (417,) +mK +-3 +3Atmosphere (419,D) +-3 +mk +314 +Rubiño-Martín et al. +Figure 8. Examples of 𝑓 ( 𝛿) correction functions (FDEC) to remove resid- +ual RFI in the MFI maps. Top: Stokes I FDEC for correlated channels. +Bottom: Stokes Q parameter in equatorial (RADEC) coordinates for corre- +lated channels. +templates are subtracted from the data before carrying out the com- +bination of correlated and uncorrelated maps. Figure 9 illustrates +the final FDEC correction applied to the maps of horn 3 at 11 GHz, +after combining the correlated and uncorrelated maps in intensity. +2.4.3 +Monopole and dipole removal +Finally, a monopole and a dipole component are subtracted from the +correlated and uncorrelated maps before their combination, using +the remove_dipole routine of HEALPix with a Galactic mask +excluding the region |𝑏| < 10◦. The removed dipole is consistent +with the expected CMB dipole, as discussed in Sect. 5.3.2. +2.5 +Effective transfer function +The PICASSO map-making code essentially preserves all angular +scales in the MFI wide survey maps. The expected signal error is +better than 0.001 per cent in the multipole range 20 < ℓ < 200 both +for intensity and polarization maps, and stays well within one per +cent down to ℓ = 10 (Guidi et al. 2021, and see also Fig. 10). How- +ever, some of the specific procedures applied in the MFI pipeline to +correct for RFI signals and atmospheric contributions might have +an impact on the effective transfer function of the wide survey. In +particular, we should consider the impact of the RFI (Sect. 2.2.3) +Figure 9. Example of the effective correction map based on a function +declination (FDEC) for the 311 map (horn 3 at 11 GHz). Top: Stokes I, with +a colour scale in the range ±2 mK. Middle and bottom: Stokes Q and U +parameters, with a colour scale in the range ±1 mK. +and atmospheric (Sect. 2.2.4) corrections at the TOD level, and the +RFI correction at the post-processing stage using a function of the +declination FDEC (see previous subsection). In terms of their am- +plitudes at the map level, the largest correction corresponds to the +third case (subtracting a function of declination), so we discuss the +transfer function of this case in detail. +It is important to note that, by construction, after applying this +FDEC correction, the zero mode at constant declination will be +missing from the maps. To characterize its impact on the effective +transfer function of the wide survey, we follow the methodology +described in Sect. 6.3 of Guidi et al. (2021). Here, we use simulations +in the ideal case including CMB and foregrounds, but without a +noise component. The transfer function is then computed in terms of +the power spectra of the map with residuals 𝐶res +ℓ +(i.e. reconstructed +map minus input sky) and that of the reconstructed map 𝐶map +ℓ +, both +MNRAS 000, 1–58 (2022) + +Stokes I, corr +217c +219c +311c +313c +417c +419c +()[mk] +-2 +-20 +0 +20 +40 +60 +80 +Declination [deg]Stokes +corr +10 +217c +219c +311c +313c +0.5 +417c +419c +[mk] +0.0 +-0.5 +-20 +0 +20 +40 +60 +80 +Declination [deg]FDEC (311, I) +-2.0 +2.0 mKFDEC (311, Q) +-1.0 +1.0 mKFDEC (311, U) +-1.0 +1.0 mKQUIJOTE MFI wide survey +15 +Figure 10. Transfer function (TF) of the QUIJOTE MFI wide survey map +at 11 GHz, after accounting for the post-processing stage of a subtraction of +a function of the declination (FDEC). The TF for TT is marked with circles +connected by red solid lines; the EE case with triangles and red dashed lines, +and the BB with diamonds and red dotted lines. As a reference, in green we +also include the TF of the PICASSO map-making code (Guidi et al. 2021). +computed within the same mask, using this expression: +𝑓ℓ = +1 +1 − 𝐶res +ℓ /𝐶map +ℓ +. +(8) +Figure 10 presents the result obtained for the 311 case. As expected, +we find that the FDEC correction is affecting low multipoles (ℓ ≲ +15). The reconstruction of the sky signal is better than one per cent +down to ℓ ≈ 10 in intensity. In polarization, the correction stays +within one per cent down to ℓ ≈ 30, being at ℓ = 10 of the order of +20 % for BB, and 5 % for EE. Because of this reason, and although +we are able to reconstruct the sky signal to lower multipoles, as a +conservative approach the power spectra analyses in this paper will +be restricted to ℓ ≥ 30, so no transfer function correction will be +needed. Appendix B contains a more detailed discussion on how +a given map is affected by the FDEC filtering. The impact of the +RFI and atmospheric corrections at the TOD level is discussed in +detail in Sect. 4.4, although we anticipate that their impact is lower +than the 𝑓 (𝛿) discussed here (except maybe for 19 GHz, where +the atmospheric contribution becomes comparable to the FDEC +correction). +2.6 +Recalibration of the wide survey maps using Tau A +Once the MFI wide survey maps are produced using the pipeline +described above, we re-evaluate three aspects of the calibration +using Tau A: i) the global calibration scale in intensity, ii) the +polarization angle calibration, and iii) the polarization efficiency. +We discuss them in detail here. +2.6.1 +Global recalibration in intensity +Tau A and Cas A are the two main primary calibrators of QUI- +JOTE MFI (Génova-Santos et al. 2023). Daily observations of these +sources in raster scan mode are used to obtain the overall gain scale +in intensity for each MFI channel in every observing period. How- +ever, as daily calibrator observations might suffer from 1/ 𝑓 noise +and other uncertainties, we recalibrate the MFI wide survey maps +in the post-processing stage. For this recalibration, we use Tau A as +the reference source, because it is located on a cleaner background +than Cas A. +For this, we first generate wide survey maps for each individual +period (four maps in total for each horn and frequency). These four +maps per period are degraded to one degree angular resolution, +and then we apply beam fitting photometry (hereafter BF1d) on +Tau A. The derived flux densities are compared, accounting for +colour corrections, with a spectral emission model that we have +specifically obtained for Tau A, using WMAP and Planck data +together with some ancillary measurements, and applying the same +BF1d methodology. The new model will be presented and discussed +in detail in a separate paper (Génova-Santos & Rubiño-Martín, in +preparation), and builds on that presented in Weiland et al. (2011), +but including several improvements: i) improved treatment of the +colour-corrections and beam effects on WMAP data, ii) inclusion +of Planck data, iii) improved variability model. The adopted Tau A +model for the recalibration of MFI maps has the shape +𝑆𝜈(Tau A) = 358.3 +� +𝜈 +22.8 GHz +�−0.297 +Jy. +(9) +This model is evaluated at epoch 2016.3, which corresponds to +the effective central epoch of the wide survey, and we use a secular +decrease of −0.218 % yr−1 (Weiland et al. 2011). From this compar- +ison, we derive global recalibration factors for each MFI frequency +map and for each individual period, accounting for the secular de- +crease of Tau A and the effective epochs in each period (see values +in Column 4 of Table 2). The mean value of these recalibration fac- +tors results in an overall 4 per cent recalibration of the wide survey +maps. The accuracy of the MFI wide survey intensity calibration is +discussed in Sect. 5.2. +2.6.2 +Polar angle recalibration +The reference angle for each MFI polarimeter (i.e. the reference for +𝜃pm in equations 3 and 4) changes across the spectral band, and +thus from band to band. For this reason, the reference angle for +each frequency map is calibrated separately, despite of the fact that +the two frequency bands of the same horn share the same polar +modulator. This procedure is based on daily Tau A observations, +and it is described in Génova-Santos et al. (2023). In particular, the +adopted model for the Tau A angle in Galactic coordinates is given +by +𝛾Tau A = 𝛾0 + 𝑅𝑀𝜆2, +(10) +where 𝑅𝑀 = −1406 ± 12 deg m−2 and 𝛾0 = −88.31◦ ± 0.25◦. Our +daily calibration provides a reference polar angle for Tau A with a +statistical error of approximately 1◦ within a period. But similarly +to the intensity calibration, daily observations of Tau A might suffer +from 1/ 𝑓 noise or other effects, so the polar angles of the final wide +survey maps are recalibrated in each period with Tau A again. As +for the global recalibration in intensity, we also use BF1d in Tau A +to extract the fluxes in Stokes Q and U parameters in the maps per +period. From there, recalibration offsets in the reference angles are +computed for each channel and each period, and applied in order to +generate the final maps. The accuracy of the angle calibration in the +MFI wide survey is discussed in Sect. 5.5. +MNRAS 000, 1–58 (2022) + +Transfer function. 11 GHz +1.3 +EE + = 1/(1 - C, res/Ce,map) +BB +1.2 +FDEC TT +FDEC EE +FDEC BB +1.1 +1.0 +101 +10216 +Rubiño-Martín et al. +Table 6. Polarization efficiency for horns 2, 3 and 4 in period 6. Error bars +for all measurements are 2 per cent. See text for details. +Channel +𝜌corr +𝜌uncorr +217 +0.84 +0.98 +219 +0.86 +0.96 +311 +0.89 +0.98 +313 +0.83 +0.97 +417 +1.00 +0.93 +419 +0.99 +0.91 +Table 7. Change in the polarization efficiency for horns 2, 3 and 4 in period +6 due to errors in the 𝑟-factor. See text for details. +Channel +Horn 2 +Horn 3 +Horn 4 +Low freq, corr +−0.075 +0.021 +−0.006 +High freq, corr +−0.113 +0.029 +0.016 +Low freq, uncorr +0.028 +0.004 +0.005 +High freq, uncorr +−0.020 +−0.002 +0.011 +2.6.3 +Polar efficiency +Detailed measurements of the polar efficiency of the MFI polarime- +ters in horns 2, 3 and 4 were obtained in period 6, once the MFI +observations concluded. The description of the instrumental setup +and the final measurements are presented in Génova-Santos et al. +(2023), and summarized in Table 6. +In order to transfer this polar efficiency information to the other +observing periods where we do not have laboratory measurements, +we use again BF1d photometry on Tau A, using the MFI wide +survey maps per period. The polar efficiency in each period 𝑝 is +transferred from period 6 according to the relative value of the Tau +A polarized intensity 𝑃TauA(𝑝) in that period and in period 6, i.e. +using the ratio 𝑃TauA(𝑝)/𝑃TauA(6). On average, this photometry +method introduces errors of approximately 1 % for horn 3, and 2 % +for horns 2 and 4. +Finally, we also account for possible errors in the determination +of the 𝑟 factors in equations 3 and 4, using wide survey data as +follows. As shown in Appendix D, an error 𝜖 in the determination +of the 𝑟 factors translates into a modification of the polar efficiency, +and the appearance of a small leakage term in the TOD polarization +timeline which is proportional to the intensity map. We use the +PICASSO map-making code to fit for an intensity-to-polarization +leakage global component in period 6 data, in a two step process. +First, we solve for the intensity map 𝐼 for each case (i.e. horn, +frequency and channel), and then we use it to fit for an additional +term 𝛼𝐼 when solving for the polarization map in equations 3 and +4. These values are used to correct for the polar efficiency of each +channel in period 6, using the equations derived in Appendix D. +Table 7 shows the effective correction terms 𝛼 ≡ 𝜖/(2𝑟). We can +see that in the case of horn 3, this correction introduces a change +of 2–3 per cent in correlated channels, and below 1 per cent for +uncorrelated channels. Horn 4 is almost unaffected, while the largest +correction factor appears for the correlated channels in horn 2. The +accuracy of the polar efficiency calibration in the MFI wide survey +is discussed again in Sect. 5.1. +3 +MFI WIDE SURVEY MAPS: INTENSITY AND +POLARIZATION +Following the methodology described in the previous section, we +produced intensity and polarization maps for each MFI horn and fre- +quency. Images of these individual maps (per horn and frequency) +are shown in Appendix C, at their original resolution (i.e. the angular +resolution listed in Table 3). The resulting maps cover a sky fraction +of 𝑓sky = 0.75, 0.71 and 0.73 (equivalent to sky areas of 30 900, +29 300 and 30 100 deg2) for horns 2, 3 and 4, respectively. All MFI +maps are produced in CMB thermodynamic units (mKCMB). For +simplicity, throughout this paper we drop the subindex CMB and +use the notation mK. Nevertheless, we recall that the correction +to Rayleigh-Jeans units is very small at MFI frequencies (at most +1 per cent at 19 GHz). Smoothed maps at 1◦ resolution are gen- +erated by convolving those original maps with the corresponding +transfer function 𝑇ℓ ≡ 𝑊1 deg +ℓ +/𝑊MFI +ℓ +, which converts the spherical +harmonic window function for each horn (𝑊MFI +ℓ +) into a gaussian +beam with FWHM= 1◦ (𝑊1 deg +ℓ +). All maps are displayed in Galactic +coordinates. We recall that QUIJOTE-MFI Stokes Q and U param- +eter maps and data follow the COSMO convention for polarization +angles from HEALPix. Grey regions correspond to the sky areas +not observed by QUIJOTE MFI: the southern sky (approximately +below 𝛿 = −34◦); a small area around the North Celestial Pole +(NCP) for some of the horns (depending on their location in the +MFI focal plane); and the band of geostationary satellites close to +declination zero degrees, which mainly emit at 11 and 13 GHz. +Appendix C also contains the associated number of hits (𝑁hit) +and weight maps. Both set of maps are outputs of the PICASSO +map-making code. The hit maps (𝑁hit) correspond to the total num- +ber of 40 ms samples in each HEALPix pixel of 𝑁side = 512 reso- +lution. The weight maps correspond to the propagation through the +map-making process of the errors (weights) associated with each +individual 40 ms sample. Both sets of maps clearly show the im- +print of the scanning strategy of the QUIJOTE MFI wide survey. +The ring structures around the North Celestial Pole correspond to +the boundaries of the different elevations considered in the survey. +Due to projection effects, the number of hits is significantly larger +in those borders (and thus, the noise levels are smaller). In the low +declination band of the maps (below the masked area due to geosta- +tionary satellites), the number of hits is significantly lower due to +the combined effect of a lower number of observations at these low +elevations (mainly 30◦, 35◦ and 40◦), and projection effects. We +recall that the number of hits in the intensity maps is larger than in +polarization due to the fact that some intensity data are not used in +polarization (period 1 data are not used for any polarization maps; +data from period 2 are not used in polarization for horn 4; and data +from period 5 are not used in polarization for horn 2; see summary +information in Table 8). +The final QUIJOTE MFI wide survey maps at 11 and 13 GHz +presented in Fig. 1 and 2 are directly the maps from horn 3, smoothed +to 1◦ resolution. The final maps at 17 and 19 GHz in Fig. 3 and 4 +have been produced as a linear combination of those for horns 2 and +4. For simplicity in the computation of effective beams, frequencies +and colour corrections, we adopted constant weights for this com- +bination. We have checked that the resulting maps have comparable +noise levels to the maps obtained using spatially-varying weights +based on the actual weight maps for each individual map in the +combination. Thus, the combined maps at 17 GHz can be obtained +as +𝑚17 = 𝑤2,17𝑚2,17 + 𝑤4,17𝑚4,17 +(11) +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +17 +Table 8. List of periods contributing to each final MFI map per horn. +Column 1 indicates the map per horn with the usual notation: the first +number indicates the horn/pixel (column 2), and second and third numbers +indicate the nominal frequency (column 3). Column 4 shows the list of +periods contributing to the map based on the correlated channels 𝑉x+y and +𝑉x−y. Column 5 shows the list of periods used for the map based on the +uncorrelated channels 𝑉x and 𝑉y. The final map is the combination of both +correlated and uncorrelated maps. +Map +Horn/Pixel +Nominal Freq. (GHz) +Corr +Uncorr +Intensity +311 +3 +11 +1,2,5,6 +1,2,5,6 +313 +3 +13 +1,2,5,6 +1,2,5,6 +217 +2 +17 +1,2,5,6 +1,2,5,6 +219 +2 +19 +1,2,5,6 +1,2,5,6 +417 +4 +17 +1,2,5,6 +1,2,5,6 +419 +4 +19 +1,2,5,6 +1,2,5,6 +Polarization +311 +3 +11 +2,5,6 +5,6 +313 +3 +13 +2,5,6 +5,6 +217 +2 +17 +2,6 +6 +219 +2 +19 +2,6 +6 +417 +4 +17 +5,6 +5,6 +419 +4 +19 +5,6 +5,6 +Table 9. Constant weight factors used to produce the combined 17 and +19 GHz MFI wide survey maps. We include only the weight factors for +horn 4, as those for horn 2 can be obtained as 𝑤2,17 = 1 − 𝑤4,17 and +𝑤2,19 = 1 − 𝑤4,19. +I +Q +U +𝑤4,17 +0.362 +0.732 +0.732 +𝑤4,19 +0.419 +0.788 +0.788 +for 𝑚 = 𝐼, 𝑄,𝑈, and similarly for 19 GHz, we have +𝑚19 = 𝑤2,19𝑚2,19 + 𝑤4,19𝑚4,19. +(12) +Table 9 contains the final weights used for this linear combination. +These values have been derived from the white noise level of the +individual frequency maps for each horn, using optimal (inverse +variance) weights. We note that horn 2 dominates the linear com- +bination in intensity, while horn 4 contributes with a higher weight +to the polarization maps. The actual values of noise levels for these +maps are discussed in Sect. 4.3. +The final maps in polarization (Figs. 1–4) are dominated by the +Galactic synchrotron emission (the spectral index of the observed +signal is discussed below in Sect. 7 and 8). Large scale features such +as the Fan region or the North Polar Spur are clearly seen in the four +frequency maps. The MFI instrument is not optimized to measure +the intensity signal, and thus the intensity maps present worse noise +properties. In particular, the two highest frequency channels show +clear large scale 1/ 𝑓 residuals, particularly at negative declinations, +due to the fact that they are observed only with the lower elevations +(higher air masses). +3.1 +Analysis masks +Figure 11 shows the footprint of the different analysis masks which +are specific for the QUIJOTE wide survey. There are three distinct +regions that are considered when building these masks: +• Satellite band ("sat"). The masked region around declination +zero is used to block the RFI contamination of geostationary satel- +lites mainly affecting 11 and 13 GHz maps. In the MFI pipeline, +the emission from each geostationary satellite is flagged at the TOD +level using a mask of 5◦ radius around each satellite. Other satellites +or RFI signals are flagged as described in Section 2. After this pro- +cess, the resulting masked area (with zero number of hits) is located +approximately between declinations −10◦ to −2◦ (note that geo- +stationary satellites are seen at slightly negative declinations from +the Teide Observatory). The proposed mask to remove the satellite +band (−12◦ < 𝛿 < 6◦) is a conservative choice based on a close +inspection of the final maps, extending the unobserved area by two +degrees in the negative declination direction, and by eight degrees +in the positive direction. This choice accounts for low-level RFI +residuals in the intensity maps (some of the residual RFI signals +corrected during the post-processing stage are located in that area), +while keeping a relatively high number of hits per pixel. +• North Celestial Pole ("NCP") region. Given the latitude of the +Teide Observatory (28.30◦N) and the minimum elevation observed +with QUIJOTE MFI (EL= 30◦), some of the maps present a small +area of unobserved pixels around the NCP, depending on the loca- +tion of the MFI horns in the focal plane. The maximum observed +declination is approximately 86◦ for horn 3, and 87.5◦ for horn 2. +Horn 4 covers up to 90◦ in declination. In any case, the pixels sur- +rounding this NCP area are only accesible with the lowest elevation +bands, which usually present the largest levels of atmospheric con- +tamination in the intensity maps, particularly at 19 GHz. For this +reason, for some of the analysis we mask the region above 𝛿 = 70◦, +in order to keep a sky area that is observed practically by all the +elevations considered in the survey. +• Low (negative) declinations ("lowdec"). Similarly to the NCP +area, this region is only observed when using low elevations (below +40◦), and thus the corresponding intensity maps, specially at the +two highest frequencies, are more affected by 1/ 𝑓 residuals from +atmospheric emission (see Fig. 3 and 4, and also the individual maps +for horns 2 and 4 in Appendix C). The proposed mask to exclude +this area covers all declinations below 𝛿 = −12◦. +All different combinations of those three masked regions produce +the reference set of specific masks for the MFI wide survey used +in this and all accompanying papers. In particular, unless other- +wise stated, the default analysis mask used in most of the scientific +analyses in this paper, and in particular, in all power spectrum +computations, corresponds to the superposition of the three re- +gions (sat+NCP+lowdec). This mask preserves a sky fraction of +𝑓sky = 0.418, equivalent to approximately 17 200 deg2. +4 +DATA VALIDATION +In order to characterize the properties of the wide survey maps, +we carry out a number of tests and studies in this section. Most +of them rely on different types of null tests, which can be used to +detect possible remaining systematic effects in the data, including +residual RFI signals, calibration issues, changes in the operational +or instrumental conditions, or even unknown effects. +MNRAS 000, 1–58 (2022) + +18 +Rubiño-Martín et al. +Figure 11. Footprint of the wide survey in Galactic coordinates, and pro- +posed analysis masks. The background image corresponds to the 9-yr +WMAP-K band polarized intensity map (Bennett et al. 2013). Light colours +indicate the observed MFI wide survey regions. The band excluded due to +satellite contamination corresponds to −12◦ < 𝛿 < 6◦. The default mask +adopted for the analyses in this paper preserves the band 6◦ < 𝛿 < 70◦, +which is marked as the brightest region in the image. This mask is labelled +as sat+NCP+lowdec (see text for details). +4.1 +Null tests +A “null test” is defined as the difference between the maps produced +from two independent sub-sets of files from the full data base, which +are expected to give the same signal under the assumption of a +perfect calibration and no systematic effects. Null tests have been +shown to be a powerful mean to assess the contribution of residual +systematic effects in CMB analyses (e.g. Planck Collaboration et al. +2014c, 2016d). For the characterization of the QUIJOTE MFI wide +survey data, we produced the following set of null tests: +(i) Half mission. The full database is divided in two halves. The +separation is done according to the calendar date inside each period +and each elevation, producing maps labelled as “half1” and “half2”. +In this way, both null test maps contain data from all periods, and +have a similar sky coverage. This is the reference null test used to +characterize the overall noise properties. +(ii) Rings. The MFI wide survey maps are produced using the so- +called nominal observing mode, in which the QUIJOTE telescope +scans the sky using a circular scanning strategy with a continuous +movement in azimuth direction while maintaining a constant ele- +vation. Each azimuth scan is called a "ring". For this null test, the +full database is divided in odd (“rings1”) and even (“rings2”) rings. +With the nominal azimuth scan speed of 12 deg s−1, each ring is +completed in 30 s, so this null test can be used to test for instrumen- +tal variations in these short time scales. As the instrument gain is +stable in time scales much longer than one minute, this null test is +not expected to reflect gain variations, and will essentially contain +white noise plus a 1/ 𝑓 -noise component in scales of 30 s. +(iii) Daynight. In order to evaluate possible residual system- +atic effects due to day-night variations of the system gain or cal- +ibration factors, this null test is produced by dividing the full +database into day observations (“daynight1”) and night observa- +tions (“daynight2”). For simplicity, we define here “day” as all +observations from 8 AM to 8 PM (UT). +(iv) PWV. Using the information from GPS measurements at the +Teide Observatory of the precipitable water vapour (PWV) content +of the atmosphere during each individual observation5, we divide +5 The GNSS antenna that provides these PWV measurements is located at +the full data base in two sets of low (“pwv1”) and high (“pwv2”) pwv +values. As in the case of the half mission null test, the separation is +done inside each period and elevation, to guarantee that both splits +contain a similar sky coverage. As a reference, the resulting median +pwv in these two data splits is 2 mm and 5.2 mm, for "pwv1" and +"pwv2", respectively. +(v) Halfrings. This null test separates the data by dividing each +ring in two halves. Data taken with telescope azimuth values 0◦ ≤ +𝐴𝑍 ≤ 180◦ correspond to "halfring1", while data with 𝐴𝑍 > 180◦ +are part of "halfring2". Although these maps are expected to be +noisier than the other null tests due to 1/ 𝑓 contributions (note that +in this case we are basically decreasing by a factor of two the number +of independent crossings in each pixel when solving the conjugate +gradient inside the map-making algorithm), they are still extremely +useful to detect residual RFI signals arising from local structures, +which usually appear at fixed AZ values. Moreover, these maps can +be also used to test residual pointing errors. +(vi) 𝑇BEM. As explained in Génova-Santos et al. (2023), the +overall gain of the instrument is strongly correlated with the physical +temperature in the electronic boxes containing the Back-End Module +(BEM) of the MFI. As a further test to explore possible residual +variations after our gain model correction, we use the values of one +of the temperature sensors 𝑇BEM, which is monitored every second +as part of the house-keeping data, to separate the data in two halves, +according to low ("tbem1") and high ("tbem2") values of the BEM +temperature. As a reference, the median temperature for these two +data splits is 8.1◦C and 16.1◦C, respectively. As for the half mission +and PWV null tests, we do the division in two halves for each period +and elevation configuration separately, and then we combine the +sub-lists. For simplicity, we refer to this case as "tbem null test" in +the text. +Two separated lists of calibrated TOD files are produced for +each one of those six null tests cases, and the corresponding maps +ℎ1 and ℎ2 are produced with fully independent runs of the map- +making code. The post-processing of each null test is identical to +the procedure applied to the full maps. From this point, a "null-test +difference map" can be produced for each case, as +𝑛 = ℎ1 − ℎ2 +𝑤 +, +(13) +where the normalizing weight is computed as +𝑤 = +√︃ +(𝑤1 + 𝑤2)(𝑤−1 +1 + 𝑤−1 +2 ). +(14) +Here 𝑤1 and 𝑤2 are the individual weight maps of the null tests +ℎ1 and ℎ2, respectively. They are computed as 𝑤𝑖 = 1/𝜎2 +𝑖 , with +𝑖 = 1, 2. Defined in this way, equation 13 provides a map with +similar noise levels as the residual noise for the weighted-sum of +the two halves (see e.g. Planck Collaboration et al. 2014b, 2016f). +4.1.1 +Null tests with a common baseline solution +For those six cases listed above we have also produced a different +set of null test maps, named as "null test with common baselines", +as follows. First, we run the map-making code for the complete +database, and record the baseline solutions. Then, each pair of null +test maps is generated using that recorded solution, instead of solv- +ing for the baselines with half of the data only, as it was the case +before. By construction, this procedure cancels out an important +the Izaña Atmospheric Observatory (IZO) just 1.4 km away from QUIJOTE, +and virtually at the same altitude (≈ 10 m below). +MNRAS 000, 1–58 (2022) + +QUljOTE MFI masks +6 =.70 +6 +12QUIJOTE MFI wide survey +19 +Figure 12. Half-mission null test difference maps for horn 3 11 GHz. Top row shows the Stokes I (left), Q (centre) and U (right) difference maps for the case +of "independent baselines". Bottom row corresponds to the case of "common baselines" (see text for details). For display purposes, all maps are smoothed to 1 +degree resolution. The colour scale corresponds to ±1 mK for the intensity maps, and ±0.3 mK for polarization. +part of the 1/ 𝑓 noise contribution associated with long time-scale +variations, partly due to the fact that the baseline solution is better +constrained when using the full database. Differences between the +two halves ℎ1 and ℎ2 now will be entirely due to the fact that each +half uses different input data, and not to the possible uncertainties +in the determination of the baseline solution. For this reason, these +null test maps are found to be particularly useful to study those +variations in the data which can be (mainly) ascribed to calibration +uncertainties, instrument changes or to variability of the sky signal. +Thus, these maps will be used specifically in Section 5.2 to assess +the internal calibration of the wide survey. For all the remaining +analyses, and in particular, for assessing the noise levels in the wide +survey maps, we will always use the default set of null tests maps +("with independent baselines"). +As illustration, Figs. 12, 13 and 14 present few examples of +null test difference maps for horn 3 11 GHz, after smoothing to one +degree resolution. Fig. 12 shows the half mission difference map +both for the "independent baselines" and the "common baselines" +cases. Fig. 13 contains the ring, halfring and tbem null tests for the +case of independent baselines, while Fig. 14 shows the same three +cases for the "common baselines" solution. +4.1.2 +Other data splits +In addition to the null tests described above, other data splits have +been considered and generated for the MFI wide survey. In particu- +lar, we generated the four "maps per period", in correspondence to +periods 1, 2, 5 and 6, both for the case of "independent baselines", +and also with "common baselines". Although these four maps per +period do not have exactly the same sky coverage (e.g. elevation 30 +is only used in period 5) or the same format (e.g. polarization maps +are not generated in period 1), they are still very useful for valida- +tion purposes (RFI residuals, gain model, calibration), as shown in +the following sections. Moreover, these maps are also used for the +study of transients and in particular, to characterise the potential +variability of some bright point sources (see e.g. Herranz et al. +2023). +4.2 +Assessing systematic effects with null tests in power +spectra and maps +4.2.1 +Power spectra +Fig. 15 presents the binned raw power spectra (i.e. uncorrected for +the beam and pixel window functions) of the six null-test difference +maps described in the previous section and computed using eq. 13, +compared to the raw power spectra of the final maps for each horn +and frequency. For simplicity, we show only two cases, for horn +3 (11 GHz) and horn 4 (17 GHz). The equivalent figures for other +horns and frequencies provide qualitatively similar results. In this +section, the 𝐶ℓ’s are computed with the publicly available code +Xpol6, which is based on a pseudo-𝐶ℓ estimator, and accounts for +incomplete sky coverage (Tristram et al. 2005). The mask adopted +for this computation is the default one described in Section 3.1 +(NCP+sat+lowdec), using a 5◦ apodization with a cosine function, +as implemented in the NaMaster library (Alonso et al. 2019). In +all panels, we show as a reference the angular power spectrum of +the final map in black, and the spectra of the different "null test +difference maps" (eq. 13) in various colours. For completeness, +these figures also include the power spectra (as dotted lines) of the +null-test difference maps for the case of "common baselines". We +also include the ideal white noise level for each map, computed +from the normalized weights (see Sect. 4.3.2 for details). +All six null test difference maps present a similar behaviour, +being asymptotically flat at high multipoles when reaching the white +noise level, and increasing at low multipoles (large angular scales) as +expected for residual 1/ 𝑓 noise. A comparison of these six null test +power spectra provides a useful tool to identify and isolate different +sources of systematic effects or calibration errors. In polarization, +6 https://gitlab.in2p3.fr/tristram/Xpol +MNRAS 000, 1–58 (2022) + +half noise map (311, I) +-1.0 +1.0 mKhalf noise map (311, Q +-0.30 +0.30 mKhalf noise map (3l1, U) +-0.30 +0.30 mKhalf noise map (311, I) +-1.0 +1.0 mKhalf noise map (311, Q +-0.30 +0.30 mKhalf noise map (3l1, U) +-0.30 +0.30 mK20 +Rubiño-Martín et al. +Figure 13. Three examples of null test difference maps for horn 3 11 GHz, for the case of "independent baselines": ring (top), halfring (centre) and tbem +(bottom). From left to right, each row shows the Stokes I (left), Q (centre) and U (right) difference maps. For display purposes, all maps are smoothed to 1 +degree resolution. The colour scale corresponds to ±1 mK for the intensity maps, and ±0.3 mK for polarization. +all null test spectra are basically consistent among them, except the +ring case, which presents a slightly lower level of 1/ 𝑓 residuals at +low multipoles. This behaviour is expected because the ring null +test maps probe noise variations in scales of one minute, while the +others cases (half, daynight, tbem, pwv) probe longer time scales. +We also note that the halfring null test tends to be slightly above +the other noise estimates, but again this is expected as this null +test uses basically half of the possible crossings for each pixel, +and thus the baseline solution is less constrained. However, this +is not the case of halfring null test with common baselines, as +in this case the baseline solution was obtained with the complete +dataset. For the intensity maps, the qualitative behaviour is similar +to polarization, although the scatter among the null tests in the 1/ 𝑓 +residuals at low multipolesislarger, particularlyat11 GHz where the +RFI contamination due to geostationary satellites was higher. In this +case, the largest 1/ 𝑓 residuals at low multipoles correspond to the +tbem, daynight and halfring cases, as expected. By construction, the +halfring case amplifies the presence of residual RFI signals. In the +case of tbem and daynight, this might indicate some low-level RFI +residual which becomes visible when splitting the data according +to the daily gain variations. We have confirmed that this is indeed +the case, by constructing a new set of maps excluding period 1 +in intensity, which was the period most affected by RFI due to the +absence of the extended shielding in the telescope. When generating +the halfring null test for the case of no period 1, that small excess +disappears. Finally, we note that the power spectra for the null test +difference maps with "common baselines" present a significantly +lower level of 1/ 𝑓 residuals, as anticipated. +4.2.2 +Maps +Visual inspection of the null test difference maps provides comple- +mentary information to the one obtained from the power spectra +analysis, in terms of identifying localised features due to systematic +effects. For example, the halfring null test maps (see the example +for horn 3 at 11 GHz in Fig. 13 and 14) can be used to assess +the residual systematic effects due to uncertainties in the pointing +model. As described in Génova-Santos et al. (2023), the pointing +model solution for each MFI horn provides a reconstruction of the +pointing with an overall 1 arcmin accuracy. Any residual pointing +error will produce a characteristic feature in the halfring null-test +map, as each one of the two sub-maps (halfring1 and halfring2) +uses totally different ranges of local coordinates of the telescope. +Indeed, the morphology and amplitude of the features appearing in +the intensity map along the Galactic plane, both around the Galactic +centre and the Cygnus area, match the expected residual signals for +a shift of 1 arcmin between the halfring1 and halfring2 sub-maps. +Null test difference maps can also be used for assessing the +level of residuals in real space. For example, a cross correlation +analysis of each null test difference map (𝑛) with the corresponding +signal map (𝑚) can be used to trace the presence of both errors in +the overall gain model or time-dependent RFI residuals. As usual, +MNRAS 000, 1–58 (2022) + +ring noise map (311, I) +-1.0 +1.0 mKring noise map (311, Q) +-0.30 +0.30 mKring noise map (311, U) +-0.30 +0.30 mKhalfring noise map (311, I) +-1.0 +1.0 mKhalfring noise map (311, Q +-0.30 +0.30 mKhalfring noise map (311, U) +-0.30 +0.30 mKtbem noise map (311, I) +-1.0 +1.0 mKtbem noise map (311, Q) +-0.30 +0.30 mKtbem noise map (311, U) +-0.30 +0.30 mKQUIJOTE MFI wide survey +21 +Figure 14. Same as Fig. 13, but for the case of "common baselines" difference maps. The colour scale corresponds also to ±1 mK for the intensity maps, and +±0.3 mK for polarization. +Table 10. Cross-correlation in real space between the half mission difference +maps and the final signal maps. Columns 2–4 correspond to the case of half +mission maps with common baselines, while columns 5–7 show the results +for the case of independent baselines. Error bars are of the order of 0.1 in +all cases. +Channel +𝛼T +𝛼Q +𝛼U +𝛼T +𝛼Q +𝛼U +[%] +[%] +[%] +[%] +[%] +[%] +Common baselines +Indep. baselines +217 +0.2 +0.5 +0.9 +−4.8 +0.4 +0.8 +219 +0.3 +1.4 +1.3 +−1.8 +1.4 +1.3 +311 +−0.1 +0.0 +0.7 +0.1 +−0.6 +0.8 +313 +−0.2 +0.3 +0.9 +−0.1 +0.5 +1.0 +417 +0.3 +−0.2 +−0.6 +−3.0 +−0.5 +−0.6 +419 +1.2 +−0.1 +0.0 +−0.4 +−0.1 +0.2 +a cross-correlation coefficient 𝛼 can be obtained as the minimum +variance estimator that minimizes 𝑛 − 𝛼𝑚 (see e.g. Hernández- +Monteagudo & Rubiño-Martín 2004). Table 10 presents the corre- +lation coefficients 𝛼, in percent units, for the case of the half mission +null tests both for common and independent baselines. The analysis +is carried out using the standard mask NCP+sat+lowdec defined +in Sect. 3.1. These numbers are consistent with the power spectra +analyses described in the previous subsection, and lie below the +calibration uncertainty of the wide survey (see details in Sect. 5 +and Table 16). In particular, for horn 3, these values are within one +per cent, both in intensity (𝐼) and polarization (𝑄, 𝑈). Moreover, in +polarization all values are below 1.4 per cent. +4.3 +Noise characterization: 1/ 𝑓 noise and correlations +Noise parameters for the MFI instrument have been described in +(Génova-Santos et al. 2023), and are summarized in Table 3. Those +values determine some of the noise properties of the final wide sur- +vey maps. Here, we use the half-mission difference maps (hereafter +HMDM), constructed as in equation 13 and for the case of "inde- +pendent baselines", to assess the overall noise properties of the MFI +wide survey, including white noise levels, 1/ 𝑓 -type components +and correlation properties. The analyses are done both in harmonic +(Sect. 4.3.1) and real (Sect. 4.3.2) space, using the standard mask +defined as NCP+sat+lowdec in Sect. 3.1, which contains the region +in the declination range 6◦ < 𝛿 < 70◦. In addition, and due to +the MFI receiver design, there are well-known noise correlations +at the TOD level (also called "common mode 1/ 𝑓 noise") between +channels of the same horn, which are inherited by the final maps. +We use the cross-spectra of different HMDM to characterize these +noise correlations at the map level, both between the two frequen- +cies of the same horn (Sect. 4.3.3) and between the correlated and +uncorrelated channels contributing to a given map (Sect. 4.3.4). +MNRAS 000, 1–58 (2022) + +ring noise map (311, I) +-1.0 +1.0 mKring noise map (311, Q) +-0.30 +0.30 mKring noise map (311, U) +-0.30 +0.30 mKhalfring noise map (311, I) +-1.0 +1.0 mKhalfring noise map (311, Q +-0.30 +0.30 mKhalfring noise map (311, U) +-0.30 +0.30 mKtbem noise map (311, I) +-1.0 +1.0 mKtbem noise map (311, Q +-0.30 +0.30 mKtbem noise map (311, U) +-0.30 +0.30 mK22 +Rubiño-Martín et al. +Figure 15. Binned raw power spectra (Δℓ = 11) of the six null test difference maps discussed in the text, for horn 3 at 11 GHz (left) and horn 4 at 17 GHz +(right). For comparison, we also include as dashed lines the spectra of the null test difference maps for the case of "common baselines". Black solid lines depict +the spectra of the signal maps, while the horizontal dashed lines indicate the ideal white noise level for each map (see text for details). +4.3.1 +Noise properties in harmonic space +Our analysis of the noise properties in harmonic space is shown in +Fig. 16 and Table 11. The power spectra for the HMDM are com- +puted using NaMaster and then fitted with the following empirical +model: +𝐶ℓ = 𝐶w +� +1 + +�ℓk +ℓ +� 𝛼� +, +(15) +which accounts for a 1/ 𝑓 noise component projected on sky. We +fit for the three parameters in this equation in two steps. First, we +obtain the white noise level 𝐶w as the average level of the angu- +lar power spectrum at high multipoles (ℓ ∈ [700, 800] for TT, and +ℓ ∈ [600, 800] for EE and BB). Then, the knee-multipole ℓk and the +slope 𝛼 are obtained analytically after fitting for a linear relation in +log10(𝐶ℓ − 𝐶w) 𝑣𝑠 log10(ℓ), in the multipole range ℓ ∈ [20, 100] +for both intensity and polarization. To have a better fit in the high +multipole range for the EE and BB case of horn 2, we use here +the range ℓ ∈ [80, 300]. The parameter 𝐶w, which represents the +white noise level of the full maps, can be translated into the com- +monly used quantity 𝜎1-deg, the equivalent noise level (rms) of the +map for a 1-degree beam, with the relation 𝜎1-deg = √︁𝐶w/Ω1-deg, +where Ω1-deg is the solid angle of a Gaussian beam with a FWHM +of 1-degree, which corresponds to 0.345 msr= 1.133 deg2. These +numbers (third column in Table 11) can be directly compared to +those obtained with real space statistics in the next subsection7. +In summary, for the intensity spectra, horn 3 presents the low- +est noise levels both for the 1/ 𝑓 and the white noise components, +while horn 4 is the most noisy one. However, in polarization, horn +4 has a much better performance, yielding the lowest noise levels, +while horn 2 is the noisiest in this case. Although the noise levels +for horn 3 in polarization are slightly higher than those for horn 4, +7 Note that if we want to quote the map sensitivity in the usual units of +𝜇K.arcmin (or 𝜇K.deg), we can not use directly 𝜎1-deg, as we have to +account for the √︁Ω1-deg factor. For instance, the white noise level of the +MFI 311 map in polarization is 42.2 𝜇K per 1-degree beam, or equivalently, +44.9 𝜇K.deg = 2695.1 𝜇K.arcmin, consistently with the reported 𝐶w value. +MNRAS 000, 1–58 (2022) + +Horn 3 11 GHz △lbin=11.0 +half +halfring +pwv +Map +ring +daynight +tbem +10 +10-1 +[mk?] +10-2 +10 +10-4 +10-5 +10-6 +10-3 +10-4 +10 +10-6 +10-7 +10-3 +10-4 +[mk2] +10- +5 +10- +6 +101 +102Horn417GHzAlbin=11.0 +half +halfring +pwv +Map +ring +daynight +tbem +100 +10-1 +[mk2] +10-2 +10 +10-4 +10-5 +10-6 +10-3 +10-4 +[mk²] +10 +5 +10- +10-7 +10-3 +10-4 +[mk?] +10 +10- +10- +101 +102QUIJOTE MFI wide survey +23 +Figure 16. Best-fit solutions to the power spectra of the half-mission differ- +ence maps (HMDM). Using eq. 15, we obtain the best-fit models depicted +here as dotted lines. The corresponding coefficients are listed in Table 11. +given that the sky signal is significantly brighter at lower frequencies +(see Fig. 15), the wide survey polarization maps of horn 3 (11 and +13 GHz) have the better signal-to-noise ratios. Regarding the cor- +related noise component, we find that the noise spectra in intensity +are dominated by the 1/ℓ component down to scales of 1 degree, +as a consequence of the large 1/ 𝑓 noise in the intensity TODs. +In polarization, we find typical knee-multipoles of ℓk = 54–86 for +horns 3 and 4, as expected for the significantly lower correlated +noise component. +4.3.2 +Noise properties in real space +First, we normalize the HMDM by dividing each individual pixel by +the square root of its covariance as computed from the map weights +(i.e., 𝜎𝑖 = 𝑤−1/2 +𝑖 +). We recall that those weights are propagated +through the pipeline and the map-making code, and were computed +Table 11. Noise levels from the fit to the noise power spectra based on +the parametric equation 15, computed from the half-mission null tests with +independent baselines. In polarization, we show the results of the fit to the +EE spectra. Results for BB are fully consistent. +Channel +𝐶w +𝜎1-deg +𝛼 +ℓk +[mK2 sr] +[𝜇K] +Intensity (TT) +217 +6.13 × 10−6 +133.5 +1.50 +228.8 +219 +1.05 × 10−5 +174.5 +1.82 +229.3 +311 +2.56 × 10−6 +86.3 +1.27 +221.4 +313 +1.29 × 10−6 +61.3 +1.60 +192.5 +417 +1.07 × 10−5 +176.4 +1.45 +230.4 +419 +1.40 × 10−5 +201.7 +1.82 +243.6 +Polarization (EE) +217 +1.21 × 10−6 +59.4 +1.20 +145.0 +219 +1.87 × 10−6 +73.7 +1.30 +173.7 +311 +6.13 × 10−7 +42.2 +1.24 +86.0 +313 +4.95 × 10−7 +37.9 +1.35 +75.3 +417 +4.42 × 10−7 +35.8 +1.06 +53.5 +419 +5.02 × 10−7 +38.2 +1.24 +73.2 +Table 12. Recalibration factor of the noise standard deviation included in +the weight maps, based on null test maps. +Map +H2,17 +H2,19 +H3,11 +H3,13 +H4,17 +H4,19 +Half mission null test +I +4.974 +5.596 +3.424 +3.016 +4.695 +5.108 +Q +1.723 +2.001 +1.471 +1.372 +1.285 +1.292 +U +1.723 +1.999 +1.473 +1.373 +1.285 +1.292 +Ring null test +I +4.896 +5.449 +3.410 +2.993 +4.641 +4.978 +Q +1.717 +1.994 +1.471 +1.370 +1.286 +1.289 +U +1.716 +1.991 +1.473 +1.370 +1.285 +1.291 +from the variance of each individual 40 ms sample in the TOD. For +this normalized map, we fit for the standard deviation within the +reference mask. The results are shown in Table 12. As expected, +these values are reasonably close to unity for the case of the polar- +ization maps, while in intensity these factors are greater than 3 in +all cases. These deviations from unity are generally consistent with +the level of 1/ 𝑓 noise in each case (see e.g. Table 11). This set of +values could be used to renormalize the weight maps, so they would +be representative of the actual noise levels, while preserving the +underlying spatial distribution of the hit maps. Indeed, these factors +are used to estimate the ideal white noise of each map at the power +spectrum level. For example, the dashed lines in Fig. 15 are com- +puted with these rescaled weight maps. Moreover, these rescaled +weight maps can be used to produce signal-to-noise maps for each +frequency (see Appendix C1). +As a second analysis, we repeat the same procedure but now we +normalize each difference map according to the square root of the +number of hits. Taking into account that hits correspond to 40 ms +samples, we can obtain from here representative normalization val- +ues to describe the noise standard deviation as +𝜎 = +𝜎0 +√𝑁hit +. +(16) +MNRAS 000, 1–58 (2022) + +FitnoiseClAl=10.0 +10-2 +h2 17GHz +h2 19GHz +h3 11GHz +10-3 +h3 13GHz +[mk2] +h4 17GHz +h4 19GHz +10-5 +10-B +10-4 +[mk²] +10- +10-6 +10-3 +10-4 +[mk2] +l adb +10- +-5 +10-6 +101 +10224 +Rubiño-Martín et al. +Table 13. Characteristic value of the sensitivity for each channel, 𝜎0, in +units of mK s1/2. Based on the half-mission null test maps. +Map +H2,17 +H2,19 +H3,11 +H3,13 +H4,17 +H4,19 +Half mission null test +I +5.896 +7.445 +3.481 +2.422 +7.939 +8.427 +Q +1.878 +2.280 +1.371 +1.188 +1.101 +1.059 +U +1.875 +2.273 +1.372 +1.188 +1.100 +1.064 +Table 14. Mean noise figures in the final MFI maps, in units of 𝜎1-deg (𝜇K +per 1-degree beam), using real-space statistics. A variance map is estimated +based on the half-mission nulltest maps, computing the variance within a +circle of 1 degree radius. Those values are then converted into 𝜎1-deg. +Map +H2,17 +H2,19 +H3,11 +H3,13 +H4,17 +H4,19 +Half mission null test +I +136.6 +184.8 +88.3 +65.0 +184.2 +214.8 +Q +59.4 +76.4 +40.5 +35.9 +34.2 +32.7 +U +59.4 +76.1 +40.6 +35.9 +34.1 +32.9 +Our results are shown in Table 13. The values obtained for the +MFI wide survey in polarization are comparable to those obtained +for raster scan observations with the MFI in smaller regions (see +e.g. last column in Table 1 from Génova-Santos et al. 2017), and +represent the actual sensitivity of the instrument. +Finally, we can also estimate the noise variance directly from +the HMDM, using apertures of 1-degree radius across the same +mask. The average values obtained from this analysis are given in Ta- +ble 14. To facilitate the comparison with the numbers in the previous +subsection, these values are re-scaled by the factor +√︃ +Ωpix/Ω1-deg, +so they represent 𝜎1-deg. In summary, the final combined maps of +the MFI wide survey in polarization present sensitivities within the +range 35–40 𝜇K per 1-degree beam for the four frequencies. +4.3.3 +Noise correlations between frequencies of the same horn +Two MFI frequency channels from the same horn have a corre- +lated ("common mode") 1/ 𝑓 noise component, due to the fact that +they share the same LNA. This fact is particularly relevant for the +intensity maps, which are strongly dominated by correlated noise. +Because of this reason, our final wide survey maps at 11 and 13 GHz +have correlated noise between them, as is the case for the maps at +17 and 19 GHz. +In order to characterize the actual degree of correlation be- +tween two wide-survey maps obtained from the same horn, we use +the normalized cross-spectra between the corresponding null-test +difference maps. As in the previous section, we use as a reference +the HMDM for the case of independent baselines. Following the +notation in Sect. 4.1, here 𝑛ℎ, 𝑓 represents the half-mission differ- +ence map for horn ℎ and frequency 𝑓 (see eq. 13). Then, for a given +horn ℎ(= 2, 3, 4), the normalized correlation between the lowest +frequency band 𝑓1 and the highest frequency band 𝑓2, is given by +𝜌ℓ ≡ +𝐶 +𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2 +ℓ +√︃ +𝐶 +𝑛ℎ, 𝑓1 +ℓ +𝐶 +𝑛ℎ, 𝑓2 +ℓ +, +(17) +where 𝐶 +𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2 +ℓ +is the cross-spectrum between the two difference +maps, and 𝐶 +𝑛ℎ, 𝑓𝑖 +ℓ +for 𝑖 = 1, 2 represents the auto-spectra. +Figure 17 shows this normalized cross-spectrum 𝜌ℓ in the +final MFI wide survey maps for horns 2, 3 and 4, both in intensity +and polarization. In intensity, the resulting noise correlation is of +the order of 75–85 per cent for the three horns, being relatively +flat in the multipole range 20 ≲ ℓ ≲ 300. In polarization, the +correlation is found to be ∼ 20–60 per cent depending on the horn, +with a moderate dependence on the multipole, being slightly lower at +higher multipoles (smaller scales). In order to obtain a representative +value for this correlation, we compute the average (and standard +deviation) of 𝜌ℓ in the multipole range [20, 200]. For TT, we obtain +85.0 ± 0.3 %, 76.5 ± 0.4 % and 84.1 ± 0.3 % for horns 2, 3 and +4, respectively. In polarization, for EE we obtain 60.7 ± 1.0 %, +32.8 ± 1.4 % and 20.9 ± 1.2 %, and for BB we have 60.8 ± 0.9 %, +36.2 ± 1.1 % and 21.7 ± 1.2 %, again for horns 2, 3 and 4. This +high degree of correlation has to be taken into account when doing +combined analyses of the two frequency maps of the same horn. +As a consistency check, and in order to test that these inter- +frequency correlations are entirely due to instrumental (common +mode) 1/ 𝑓 noise, and not to external correlated signals produced +either by the atmosphere or by RFI, we performed the same analysis +but now comparing two frequencies coming from two different +horns. In particular, we evaluated the cross-correlation of horn 2 at +17 GHz with horn 4 at 19 GHz, obtaining −0.64 ± 2.47 %, 0.47 ± +0.96 % and −0.49 ± 0.96 % for TT, EE, and BB, respectively. In +addition, the cross-correlation of horn 4 at 17 GHz with horn 2 at +19 GHz gives −0.32 ± 2.13 %, 0.99 ± 0.83 % and 0.72 ± 0.76 %, +again for TT, EE and BB. In both cases, the results are consistent +with zero within the error bar. +4.3.4 +Noise correlations between channels +As described above, for any given horn and frequency sub-band +of MFI, we produce two versions of the intensity and polarization +maps, the so-called correlated (𝑥c) and uncorrelated (𝑥u) maps. +Due to the MFI design, we expect a high degree of correlation +between the noise affecting those two versions of the intensity maps, +due to the fact that they all share the same LNAs and there is +no cancellation of the 1/ 𝑓 noise in any of the sums of channels +contributing to 𝑥c and 𝑥u. We can use the same methodology applied +in the previous sub-section to characterize this correlation level of +the noise between correlated and uncorrelated channels maps for +a given horn and frequency. We also use the half-mission null test +maps as a reference for this analysis. But now, in the post-processing +stage, we generate two independent versions for each individual +map, using either the correlated or the uncorrelated information +only. With these maps, and using again eq. 13, for a given horn and +frequency we can produce 𝑛c and 𝑛u, the half-mission difference +maps of the correlated and uncorrelated channels, respectively. In +analogy to equation 17, we now compute +𝜌ℓ ≡ +𝐶𝑛c×𝑛u +ℓ +√︃ +𝐶𝑛c +ℓ 𝐶𝑛u +ℓ +, +(18) +where 𝐶𝑛c×𝑛u +ℓ +is the cross-spectrum between the two difference +maps, and 𝐶𝑛c +ℓ and 𝐶𝑛u +ℓ +are the auto-spectra. +Fig. 18 shows the resulting correlation level between correlated +and uncorrelated channels. As expected, we find a very high degree +of correlation (of the order of 90 per cent) in intensity, and a signal +consistent with zero in polarization (both for EE and BB spectra). +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +25 +Figure 17. Cross-correlation spectra of the half-mission difference maps between the two frequencies of the same horn, for TT (left), EE (centre) and BB +(right). +Figure 18. Cross-correlation spectra of the half-mission difference maps between the correlated and uncorrelated channels from the same horn and frequency, +for TT (left), EE (centre) and BB (right). +Table 15. Average inter-channel correlations < 𝜌ℓ > of the half-mission +difference maps between the correlated and uncorrelated channels for a +given horn and frequency. The values correspond to the mean and standard +deviation of the 𝜌ℓ displayed in Figure 18, computed in the multipole range +20–200. +Channel +TT (%) +EE (%) +BB (%) +217 +97.14 ± 0.08 +−1.36 ± 1.10 +−0.91 ± 1.37 +219 +87.91 ± 0.34 +2.40 ± 1.19 +−0.60 ± 1.04 +311 +85.82 ± 0.26 +3.81 ± 0.75 +2.38 ± 1.18 +313 +79.65 ± 0.42 +−0.32 ± 0.89 +−2.01 ± 0.91 +417 +97.95 ± 0.04 +−3.26 ± 1.07 +−1.22 ± 1.12 +419 +91.56 ± 0.22 +−2.81 ± 0.80 +−0.61 ± 1.15 +Again, as a representative value for this correlation, we compute +the average and standard deviation of 𝜌ℓ in the multipole range +[20, 200]. The results are shown in Table 15. These average corre- +lation values in intensity are used in the pipeline in order to produce +the final combinations of correlated and uncorrelated channels, as +described in Section 2.4.1. +4.4 +Impact of residuals on the power spectra: atmospheric +and RFI corrections +As described in Sect. 2, the MFI wide-survey pipeline incorporates +several steps tailored to correct for the contribution of atmospheric +and RFI signals in the final maps. Atmospheric corrections are +applied at the TOD level (see Sect. 2.2.4), and for intensity maps +only. When projected on maps, they appear as large scale patterns +with an increasing amplitude in frequency (see Fig. 7). RFI signals +are corrected both in the intensity and polarization maps, in two +stages. First, RFI signals at the TOD level are corrected using spatial +templates as described in Sect. 2.2.3. When projected on sky, they +also appear as large scale patterns with a moderate amplitude (≲ +0.5 mK) and presenting a higher amplitude in intensity (see Fig. 6). +Later, in the post-processing stage (Sect. 2.4), any residual RFI +signals emerging after co-adding all data in the map-making process +are corrected using a function of the declination. In terms of relative +amplitude, this is by far the largest correction applied to the MFI +wide-survey polarization data, with its amplitude being higher in +the 11 and 13 GHz channels due to the emission of geo-stationary +satellites entering through the far sidelobes. Indeed, the effective +transfer function of the MFI wide survey in polarization is mainly +determined by this effect (see Sect. 2.5). +In order to quantify the relative importance of these three +MNRAS 000, 1–58 (2022) + +Correlationlow/highfreguencies +TT +EE +BB +100 +80 +60 +% +40 +20 +h2 17x19 +0 +h3 11x13 +-20 +h4 17x19 +101 +102 +103101 +102 +103101 +102 +103 +l +l +lCorrelation corr/uncorr +TT +EE +BB +100 +80 +60 +% +40 +20 +217 +313 +0 +219 +417 +-20 +311 +419 +101 +102 +103101 +102 +103101 +102 +103 +l +l +l26 +Rubiño-Martín et al. +Figure 19. Raw angular power spectra of the ATMOS (red), RFI (green) and FDEC (blue) patterns removed from the MFI wide survey maps, for horn 3 at +11 GHz (top row) and horn 4 at 19 GHz (bottom row). For each case, we represent TT (left), EE (centre) and BB (right) spectra. Solid black lines correspond to +the angular power spectra of the corresponding wide survey maps, while dashed lines correspond to the half-mission difference maps. All spectra are computed +using the default analysis mask (NCP+sat+lowdec). +corrections, and to evaluate the possible impact of any residual +systematic effects due to uncorrected contamination in the final +wide survey maps, we have computed the angular power spectra +of those patterns that are removed from the maps, and we have +compared them with the spectra of the final maps and the half- +mission noise levels. Figure 19 shows the resulting power spectra +for the two extreme frequency values (11 and 19 GHz) taken here as +representative cases, with 11 GHz being the one with highest RFI +contamination, and 19 GHz the one with the highest atmospheric +contamination. In this plot, we use the notation of ATMOS, RFI +and FDEC for "atmospheric", "RFI at TOD level using a function +of azimuth", and "RFI at the map level using function of declination" +corrections, respectively. +Regarding the atmospheric contribution (ATMOS) to the in- +tensity power spectra, the removed pattern is subdominant at all an- +gular scales in the 11 GHz case when compared to the noise level. +At 19 GHz, we have a similar behaviour at small angular scales +(ℓ >∼ 20). However, the atmospheric residuals become comparable +to the noise levels for multipoles ℓ ≲ 20, as can be anticipated from +the visual inspection of Fig. 7. +For the RFI contribution at the TOD level, the removed patterns +both in intensity and polarization are always below the noise levels +at all frequencies, although they become comparable to the noise at +large angular scales ℓ ≲ 20. Thus, in this RFI case, as well as for +ATMOS, any residual systematic effect with an amplitude being a +fraction of the applied correction will have negligible impact at the +power spectrum level. +Finally, for the removed FDEC patterns, the largest amplitude +is found at 11 GHz, as anticipated from Fig. 8. At this frequency, the +removed pattern in intensity is above the correlated noise level for +multipoles ℓ ≲ 20. Moreover, in polarization, the applied correction +is found to be critical, in the sense that its amplitude is above the sky +signal for multipoles ℓ ≲ 30. When looking at the 19 GHz FDEC +patterns, in intensity the corrected amplitude is always below the +noise levels for all multipoles, while in polarization again becomes +comparable to the sky signal for ℓ ≲ 20. In this case, although the +underlying assumption for modelling residual RFI signals using a +function of the declination is very robust and well tested, it is im- +portant to keep in mind that residual contributions might have an +impact on the polarization maps of the MFI wide survey on large +angular scales. In addition, as explained in Sect. 2.5, the FDEC pro- +cedure also affects the same multipole range by introducing a signal +error in the reconstructed sky. For these reasons, in the following +sections involving scientific analyses based on power spectra of the +polarization signals in the wide survey, we adopt the conservative +choice of restricting the study to multipoles ℓ ≥ 30. +4.5 +Inter-frequency comparison of the MFI maps +As an additional validation test, here we present an inter-frequency +comparison of the MFI wide survey maps, together with a com- +parison with external data. For this test, we rely on the assump- +tion that the average spectral index of the polarized synchrotron +emission in the QUIJOTE maps is 𝛽 = −3.1 (see discussion be- +low in Sect. 8). We then rescale the MFI wide survey maps at 1 +degree resolution to the central frequency of the WMAP K-band +map, 𝜈 = 22.8 GHz, accounting for colour correction factors both +for MFI and WMAP maps. Figure 20 shows the rescaled MFI po- +larization maps at 11 and 13 GHz compared to WMAP-K, while +Figure 21 shows the differences for pairs of those maps (313−311, +311−WMAP, 313−WMAP). A visual inspection shows that there +is obvious polarized emission in the Galactic plane which is not +consistent with the 𝛽 = −3.1 spectral index, mainly towards the +Galactic centre or the Fan region (𝑙 ≈ 135◦). In the maps we can +also identify some residual intensity to polarization leakage in the +Cygnus area (around 𝑙 ≈ 80◦). However, the large scale emission +MNRAS 000, 1–58 (2022) + +TT H3, 11GHZ +TT +RFI +100 +noise +FDEC +ATMOS +10-2 +C,T [mk?] +10-4 +10-6 +10-8 +10-10 +101 +102EE H3. 11GHZ +10-2 +EE +RFI +noise +FDEC +10-4 +CEE [mK?2] +10-6 +10-8. +10-10 +101 +102BB H3, 11GHZ +10-2 +BB +RFI +noise +FDEC +10-4 +10-6 +10-8. +10-10 +101 +102TT H4,. 19GHZ +TT +RFI +100 +noise +FDEC +ATMOS +10-2 +C}T [mk?] +10-4, +10-6 +10-8, +10-10 +101 +102EE H4. 19GHZ +10-2 +EE +RFI +noise +FDEC +10-4 +CEE [mK?2] +10-6. +10-8. +10-10 +101 +102BB H4, 19GHZ +10-2 +BB +RFI +noise +FDEC +10-4 +CBB[mK2] +10-6 +10-8. +10-10 +101 +102QUIJOTE MFI wide survey +27 +Figure 20. Comparison of the rescaled polarization MFI maps at 11 and 13 GHz with the 9-yr WMAP-K band map (Bennett et al. 2013). MFI maps are +rescaled to 23 GHz using an average spectral of 𝛽 = −3.1, and accounting for colour corrections. All maps use the same colour scale, saturated at ±0.1 mK. +From left to right, we show MFI 11 GHz (rescaled), MFI 13 GHz (rescaled) and WMAP-K. Top row: Stokes Q maps. Bottom row: Stokes U maps. For display +purposes, to facilitate the comparison of the different structures near the mask edges, we applied here the QUIJOTE MFI sky mask to the WMAP map. +Figure 21. Inter-frequency comparison of the rescaled maps shown in Fig. 20. Top (bottom) row shows differences of Stokes Q (U) maps. First column shows +the difference between the rescaled 11 and 13 GHz MFI maps. Second and third column show the MFI 11 GHz minus WMAP-K, and MFI 13 GHz minus +WMAP-K maps, respectively. All maps use the same colour scale as in Fig. 20, saturated at ±0.1 mK. +far from the Galactic plane is largely suppressed in this difference, +showing a good consistency of the MFI and WMAP-K maps. The +residual emission in the difference map 313-311 is basically consis- +tent with the expected noise level for the difference of both maps, as +shown in Fig. 22. In this comparison, we use the EE power spectra +for the rescaled maps using the default QUIJOTE mask with the +Galactic cut |𝑏| > 10◦, and restricting the comparison to multipoles +ℓ ≥ 30. +5 +ACCURACY OF THE WIDE SURVEY CALIBRATION +In this section we assess the overall calibration uncertainty of the +QUIJOTE MFI wide survey maps in intensity and polarization, +using the information described in the pipeline paper (Génova- +Santos et al. 2023) to account for known systematics, and also +presenting a set of consistency checks based on the null test maps, +in order to evaluate the impact of unknown systematics. Table 16 +shows the summary of all types of uncertainties considered in this +MNRAS 000, 1–58 (2022) + +QUOTE Q H3 11GHz (1deg, rescaled) +mK +-0.1 +0.1QUOTE Q H3 13GHz (1deg, rescaled) +mK +-0.1 +0.1WMAP 23GHz Q (1deg) +mK +-0.1 +0.1QUlOTE U H3 11GHz (1deg, rescaled) +mK +-0.1 +0.1QUlOTE U H3 13GHz (1deg, rescaled) +mK +-0.1 +0.1WMAP 23GHz U (1deg) +mK +-0.1 +0.1MFl H3 13GHz - MFI311 (Q, 1deg,rescaled) +mK +-0.1 +0.1MFl H3 11GHz - WMAP (Q, 1deg, rescaled +mK +-0.1 +0.1MFl H3 13GHz - WMAP (Q, 1deg, rescaled +mK +-0.1 +0.1MFl H3 13GHz - MFI311 (U, 1deg, rescaled) +mK +-0.1 +0.1MFl H3 11GHz - WMAP (U, ldeg, rescaled) +mK +-0.1 +0.1MFl H3 13GHz - WMAP (U, 1deg, rescaled) +mK +-0.1 +0.128 +Rubiño-Martín et al. +Figure 22. EE power spectra of the inter-frequency comparison of the MFI +rescaled maps 313−311, shown in the first column of Fig. 21. Black and red +solid lines show the EE power spectra of the rescaled MFI 11 and 13 GHz +maps, respectively. Blue solid line is the power spectrum of the difference +map 313−311, while the yellow dashed line shows the expected noise level +for that difference map, assuming an average inter-frequency correlation of +32.8 % (see Sect. 4.3.3). +work, as well as the impact of each of them in the overall calibration +error budget. +5.1 +Statistical uncertainty and known systematics +5.1.1 +Calibration model +An important contribution to the global systematic uncertainty bud- +get comes from calibration uncertainties, and in particular, the cali- +brator model. As discussed in Génova-Santos et al. (2023), the two +main amplitude calibrators of QUIJOTE MFI are Tau A and Cas +A, which are amongst the brightest sources on the sky in this fre- +quency range. As explained in subsection 2.6, the wide survey maps +have been recalibrated using flux densities extracted on these maps +at the position of Tau A. These flux densities are measured with +sensitivities better than 0.3 % in all frequencies (see also Table 24 +and Sect. 9) while the internal calibration accuracy of QUIJOTE +is better than 1 % as shown below in subsection 5.2. Therefore in +our case the dominant error component is associated with the cal- +ibration models that are used as reference. As will be discussed in +detail in Génova-Santos & Rubiño-Martín in prep. (see also sub- +section 2.6), using different tests we estimate that the Tau A model +has an uncertainty of ≈ 4 % in our frequency range. We believe +this value is dominated by calibration errors of the different data +that are used to model this spectrum. In the case of Tau A there is +also an important contribution due to the modelling of its secular +decrease, which leads to errors when data taken at different epochs +are combined to model its spectrum. We decide to set a conservative +overall calibration uncertainty of 5 %. The reliability of this number +is supported by the tests on radiosources and planets presented in +Section 9, as well as other calibration tests based on the detection +of primary CMB anisotropies shown in Sect. 5.3. +5.1.2 +Colour corrections +The overall 5 % calibration uncertainty would strictly apply to any +analysis performed in our maps on sources or regions with a power- +law spectrum with index 𝛼 = −0.3, as that of Tau A (our primary +calibrator). For a different spectrum, uncertainties in the colour +corrections must be factored in. These are mainly associated with +errors in the measurement or characterisation of the instrument +bandpasses. MFI bandpasses were last measured in 2020, for the +instrumental configuration corresponding to period 6. The statisti- +cal uncertainties of these measurements are very low, such that they +lead to errors in the global calibrated antenna temperature below +0.01% for a range of spectral indices 𝛼 ∈ [−3, +3] and for all horns +and frequencies. On the other hand, MFI suffered various modifi- +cations over its lifetime (see Table 2), which may have introduced +modifications in the actual bandpass shapes of periods 1, 2 and 5 +with respect to period 6. +For the MFI wide survey, we conservatively assign errors to +the colour corrections by comparing the last bandpass measurement +from period 6 with a previous one performed in 2013 during period +1. Through comparing the colour correction coefficients obtained in +both cases we find that channel 219 presents the largest differences, +and in this case the error scales approximately as 𝜖×|𝛼+0.3| %, with +𝜖 = 1.03. Note that the error increases as the spectral index of the +observation, 𝛼, departs from that of the primary calibrator. For 311 +and 313, we obtain 𝜖 ≈ 0.01 and 𝜖 ≈ 0.53 respectively. For 217 we +have 𝜖 ≈ 0.51, while for horn 4 we have values between 𝜖 = 0.2–0.4. +We must note that these uncertainties are somewhat conservative, +as differences between the two measured bandpasses may not be +entirely real, but could also be due to shortcomings in the 2013 +measurements, which are deemed much less reliable than those of +2020 due to measurement techniques (see details in Génova-Santos +et al. (2023)). Taking this into account, and the fact that errors in the +other channels are smaller, as a conservative choice for this paper, +in Table 16 we have assigned an overall 0.5 × |𝛼 + 0.3| % error to +colour corrections for 11 and 13 GHz, and 1 × |𝛼 + 0.3| % for 17 +and 19 GHz. +Note that these errors in the colour correction coefficients +should impact the consistency checks presented in Section 9, where +we compare with models flux densities of sources with spectral in- +dices ranging between −0.3 and −1.2, and of planets with 𝛼 ≈ 2, or +those presented below in this section where we correlate our maps +with templates tracing the CMB anisotropies or the CMB dipole +that also have 𝛼 ≈ 2. In the former case, we find differences of +∼ 5% which we are confident are due to uncertainties in the source +calibration models. For the CMB anisotropies and CMB dipole, the +differences are ∼ 3 % and ∼ 10 % respectively, and are driven by +statistical noise (see Sect. 5.3). +5.1.3 +Beams +One of the instrumental aspects that are most carefully charac- +terised in CMB experiments are the beams and derived window +functions, as they have a direct impact on the amplitude of the +derived power spectrum and thence on cosmological parameters. +In QUIJOTE MFI this is even more important as its calibration is +tied to unresolved point sources. Comparison between beam radial +profiles derived from observations of bright point sources and the +numerical optical simulation based on CST software8 described in +Génova-Santos et al. (2023) demonstrates an accuracy in the deter- +mination of the intensity beam typically below the 2 % level (with +respect to the centre of the main beam). Given that the MFI maps +are (re)calibrated using a beam-fitting photometry on point sources, +errors in the beams will directly impact the global map temperature +8 https://www.3ds.com/products-services/simulia/products/ +cst-studio-suite/ +MNRAS 000, 1–58 (2022) + +Mask: default + Ibl > 10° +10-6 +311 rescaled +Difference 313-311 +313 rescaled +Expected noise 313-311 +10-7 +10-8 +102QUIJOTE MFI wide survey +29 +Table 16. Accuracy of the calibration in the QUIJOTE MFI wide survey data. Second column indicates if the type of uncertainty is applicable to intensity (I) +and/or to polarization (P) maps. +Type of uncertainty +Applies to +11 GHz +13 GHz +17 GHz +19 GHz +Method +Reference +Calibration model +I,P +5 % +5 % +5 % +5 % +Model for calibrators +Sect. 5.1.1 +Colour corrections𝑎 +I,P +0.5 % +0.5 % +1 % +1 % +Bandpass measurements +Sect. 5.1.2 +Beam uncertainty +I,P +2 % +2 % +2 % +2 % +CST beam model, Tau A +Sect. 5.1.3 +Zero level [mK] +I +−0.74 ± 0.20 +−0.59 ± 0.22 +0 +0 +Plane-parallel model +Sect. 5.4 +I→P leakage +P +0.65 % +0.4 % +0.8 % +0.9 % +Cygnus area +Sect. 5.1.4 +Polarization efficiency +P +3 % +3 % +4 % +4 % +Lab measurements, Tau A +Sect. 5.1.5 +Polarization angle (deg) +P +0.6 +0.9 +1.0 +3.2 +Tau A, WMAP/Planck +Sect. 5.5 +Unknown systematics: +Real space (𝜇K/beam) +I +< 53 +< 49 +< 118 +< 224 +Null tests at 𝑁side = 64 +Sect. 5.2.1 +Real space (𝜇K/beam) +P +< 12 +< 15 +< 10 +< 13 +Null tests at 𝑁side = 64 +Sect. 5.2.1 +Harmonic space (30 < ℓ < 200) +I +0.2 % +0.3 % +0.5 % +0.7 % +Null tests +Sect. 5.2.2 +Harmonic (30 < ℓ < 200) +P +3 % +4 % +6 % +6 % +Null tests +Sect. 5.2.2 +Overall calibration error𝑏 +I +5 % +5 % +5 % +5 % +Overall calibration error𝑏 +P +5 % +5 % +6 % +6 % +𝑎 These numbers should be multiplied by |𝛼 + 0.3|, being 𝛼 the spectral index of the source. +𝑏 Obtained as the maximum value of the following errors: for intensity, calibration, beam uncertainty and unknown systematics in harmonic space; +and for polarization, we add also I→P leakage and polar efficiency. +scale. We confidently estimate the error in this temperature scale +to be below 2 %. Note though that in extracting flux densities of +point sources using the same beam-fitting photometry that is used +for the main calibration, these errors would be largely suppressed. +In Table 16, we adopt a conservative value of 2 per cent, which cor- +responds to the maximum error associated with the determination +of the brightness of a beam-filling emission. +In polarization, a detailed description of the MFI beams can be +found in Génova-Santos et al. (2023), where we use the CST optical +simulations and the Mueller matrix formalism. Due to the MFI +optical design, the cross-polar terms are significantly smaller than +the copolar terms. For example, for horn 3, the cross-polar terms are +less than 0.05 % of the copolar beams across the band. This implies +that the diagonal components of the Mueller matrix (𝑀II and 𝑀QQ) +can be considered nearly identical (with that accuracy). Moreover, +the leakage terms 𝑀IQ and 𝑀QI are also identical in this limit, and +are given by one half of the difference of the copolar beams at 0◦ +and 90◦. As shown in Génova-Santos et al. (2023), these terms have +a quadrupolar structure with two positive and two negative lobes, +with typical peak amplitudes (relative to the copolar peak) of ≲ 1 %. +As shown below in Sect. 9, when studying bright compact sources +in the MFI wide survey, these patterns are clearly visible around Tau +A (in Stokes 𝑈 parameter, because most of the signal appears in 𝑄) +and Cas A (in this case, as the source is essentially unpolarized, they +are seen both in 𝑄 and 𝑈 maps, rotated by 45◦). When integrated on +scales larger than the beam, these patterns average to zero, and thus +have minimum impact on the photometry analyses (see also Leahy +et al. 2010, for the case of Planck beams). For example, for the MFI +311 map, the impact on a photometry measurement using either +aperture photometry in 1 deg, or beam fitting, is well below 0.05 % +across the full frequency band. Thus, we neglect this contribution +to the overall calibration error due to beam uncertainties, and in +Table 16 we adopt the same calibration uncertainty in polarization +as for intensity beams. +5.1.4 +Intensity-to-polarization leakage +Despite of the fact that the MFI is a true polarimeter, in the sense +that the polarization signal is produced directly for each individual +horn and frequency band, there are several known systematic effects +that may lead to spurious polarization signals, particularly in bright +regions in intensity. In the previous subsection we have already dis- +cussed, for bright point sources, the intensity-to-polarization leak- +age (hereafter IPL) terms due to beam non-idealities. Here, we +discuss the IPL terms arising from the bandpass mismatch between +the two pairs of channels that contribute to a given polarization +timeline. For the MFI instrument, the 𝑟-factors in equations 3 and +4 are determined using Tau A observations (see details in Génova- +Santos et al. 2023). When observing a sky region with a bright +intensity emission, the effective 𝑟-factor might change depending +on the spectral index of the sky emission, particularly if it differs +from that of Tau A (𝛼 = −0.3). Using the detailed measurements of +the bandpasses, we have estimated that for spectral indices typical +of Galactic emission (𝛼 ∈ [−1.5, 0]), the amount of signal leaked +into Stokes 𝑄 or 𝑈 due to this effect is typically below 0.2 % of +the intensity signal. For a CMB spectrum (𝛼 ≈ 2), it is still below +0.5 %. +Here, we provide an independent confirmation of the order +of magnitude of the IPL in the MFI wide survey maps using the +sky emission in the Cygnus region, located at Galactic coordinates +(𝑙, 𝑏) = (80◦, 0◦). Figure 23 shows this area in more detail. As +the intensity emission in this region is dominated by free-free, it +is expected to be almost unpolarized. We use a cross-correlation +analysis (similar to the one used in Sect. 4.2.2) to obtain the corre- +lation coefficient 𝛼 that minimizes 𝑄 − 𝛼𝐼 within a region centred +at (𝑙, 𝑏) = (80◦, 0◦) with a radius of 5◦. The values are always +below 1 per cent for all cases, as expected. For 311, we find 0.10 % +and 0.65 % for Stokes Q and U, respectively. The largest values are +found for 419, where we obtain 0.91% and −0.41% for Stokes Q +and U. This effect in the Cygnus area is clearly seen in the maps +of Fig. 21 and 23. The values reported in Table 16 correspond to +the most conservative case (Stokes Q or U) at each frequency, in +absolute value. +MNRAS 000, 1–58 (2022) + +30 +Rubiño-Martín et al. +Figure 23. Minimaps of 15◦ × 15◦ around the Cygnus region, located at +Galactic coordinates (𝑙, 𝑏) = (80◦, 0◦). We show the horn 3 11 GHz (top) +and horn 4 19 GHz maps (bottom) at their original resolution. The circle +indicates the region where the IPL is computed (see text for details). The two +bright compact objects in the polarization maps located outside the circle, +W63 and Cygnus A, are discussed in Sect. 9. +5.1.5 +Polarization efficiency +As discussed in Sect. 2.6, the calibration of the polarization ef- +ficiency of the MFI wide survey data is done in two steps. First, +we use laboratory measurements taken at the end of period 6 to +calibrate the polar efficiency of each individual MFI channel. In +addition, we use the wide survey data in period 6 to add also the +correction factors to these polar efficiencies associated with a pos- +sible error in the determination of the 𝑟-factors. These procedures +provide a determination of the polarization efficiency in period 6 +with a relative accuracy of 2 %. Then, in a second step these values +from period 6 are transferred to the other two periods that are used +in the construction of the MFI wide survey polarization maps (i.e. 2 +and 5), using beam fitting photometry (BF1d) measurements on Tau +A. The error budget for these factors is given by the accuracy of the +flux density extraction, which is found to be of the order of 1 % for +horn 3, and 2 % for horns 2 and 4. As they correspond to systematic +errors, we adopt the conservative approach of adding them linearly, +and we quote an overall 3 % error in the polar efficiency for horn 3, +and 4 % for horns 2 and 4. In the following subsections we evaluate +unknown systematic effects in the polarization maps, noting that in +those cases, the global errors include the polar efficiency error. In +addition, in Sect. 9 we also discuss the polarization fraction of Tau +A and Cyg A, and the polarized flux in W63, as further consistency +tests for this polar efficiency calibration. +5.2 +Internal calibration of the wide survey and consistency +checks: evaluating unknown systematics +Following the methodologies outlined in Planck Collaboration et al. +(2014c) and Planck Collaboration et al. (2014d), we use internal +consistency checks based on null test maps and other data splits of +the wide survey in order to estimate the impact of systematic effects +Table 17. Systematic effects in the MFI wide survey maps, evaluated in the +maps degraded to 𝑁side = 64. The excess signal (last column) is computed +as the quadratic difference between the values for half and ring null test +difference maps. See text for details. +Channel +T,Q,U +p-p (half) +rms (half) +rms (ring) +Excess rms +[𝜇K] +[𝜇K] +[𝜇K] +[𝜇K] +217 +T +1177.8 +249.8 +224.6 +109.3 +217 +Q +410.8 +88.8 +86.9 +18.5 +217 +U +417.0 +87.7 +86.7 +12.8 +219 +T +1736.3 +363.0 +297.8 +207.6 +219 +Q +552.6 +116.0 +113.2 +25.5 +219 +U +539.7 +115.1 +113.4 +19.2 +311 +T +736.7 +153.8 +144.3 +53.3 +311 +Q +283.4 +59.3 +58.5 +10.1 +311 +U +282.5 +59.5 +58.3 +12.0 +313 +T +538.5 +113.0 +101.7 +49.3 +313 +Q +241.9 +51.2 +49.2 +14.3 +313 +U +239.1 +51.0 +48.7 +15.1 +417 +T +1586.5 +332.8 +304.5 +134.3 +417 +Q +210.4 +45.0 +44.5 +6.8 +417 +U +209.8 +44.8 +44.6 +4.1 +419 +T +2053.8 +429.9 +352.1 +246.6 +419 +Q +232.9 +48.8 +48.2 +7.3 +419 +U +233.2 +49.5 +48.2 +11.3 +in the overall calibration. This is particularly useful for assessing +the impact of "unknown systematics", i.e. those for which we do +not have specific measurements or numerical simulations. For the +MFI wide survey, and given that we want to focus on the relative +calibration of the instrument, we use as a reference the set of null +test maps and data splits labelled as "with common baselines" in +Sect. 4.1. +5.2.1 +Unknown systematics in real space +Uncertainties due to (unknown) calibration or systematics effects at +the pixel scale have been calculated using the HMDM for common +baselines, degraded to 𝑁side = 64. At this resolution, each pixel +roughly corresponds to the beam size. The reference mask for the +analysis is the default one (sat+NCP+lowdec) as defined in Sect. 3.1. +Table 17 lists the rms values and peak-to-peak (p-p) variation +for the HMDM. Following Planck Collaboration et al. (2014c), the +p-p values are computed as the difference between the 99 % and +the 1 % quantiles in the pixel value distribution, in order to neglect +possible outliers9. A comparison between these numbers for the +half-mission null tests and those for the ring null tests is useful +for checking residual calibration and/or systematic effects on large +angular scales. Given that the ring null test maps cancel out possible +variations in scales longer than 30 s (i.e. the duration of one azimuth +scan), they can be used as our best estimate of the noise level, which +includes white noise and 1/ 𝑓 on degree scales. Any variation on +scales longer than one minute, due either to calibration uncertainties +in the gain model or systematic effects, will appear as a signal excess +in the HMDM. As illustration, the top panel in Fig. 14 shows the ring +null-test difference maps for the 311 (horn 3 at 11 GHz) case. The +results of this comparison are shown in Table 17. Column 5 presents +the rms value for the ring difference maps, and column 6 shows the +9 Note that for a Gaussian distribution, we should have p-p=4.65𝜎. +MNRAS 000, 1–58 (2022) + +0 +50 +100150200 +-1 +0 +1 +2 +3 +4 +-6 +-4 +-2 +0 +2 +4 +6 +4 +00 +2 +0 +b +-2 +-4 +-6 + 11 GHz +Q 11 GHz +U 11 GHz +86848280787674 +86 84 82 80 78 76 74 +86848280787674 +1 (deg) +1 (deg) +1 (deg)0 +20 +40 +60 +80 +100 +-1.0-0.50.00.51.01.5 +-1.0-0.50.00.51.0 +mK +CMB +6 +4 +0.0 +2 +0 +b +-2 +-4 +-6 +I 19 GHz + 19. GHz +U.19. GHz +86848280787674 +86848280787674 +86848280787674 +1 (deg) +1 (deg) +1 (deg)QUIJOTE MFI wide survey +31 +signal excess in the half-mission difference maps. Comparing these +values with those in Tables 11 and 14 for the noise levels for the wide +survey, we find that in polarization, the rms excess due to unknown +systematics is well below the white noise levels, with typical values +in the range 5–20𝜇K. In intensity, we find a similar situation for horn +3 and the 17 GHz frequency maps of horns 2 and 4. For the two maps +at 19 GHz (horns 2 and 4), the residuals are slightly larger than the +white noise levels, but still well below the total noise contribution in +those channels (column 5). As a reference, for horn 3, the residuals +at beam scales are of the order of ∼ 50𝜇K. These numbers are used +to complete the main table 16, appearing as "unknown systematics" +in real space. As a conservative choice, the values for horns 2 and 4 +are combined linearly instead of using a quadratic combination. +5.2.2 +Unknown systematics in harmonic space +We use the ratio of cross-power spectra of the null test maps with +some external maps, as the reference tool to validate the calibration +in harmonic space. The use of cross-spectra to external maps min- +imises the effects of noise bias on the power spectrum estimation. +In practice, given two maps 1 and 2 that we want to compare, we +compute +𝐴1,2 = +�� +𝐶1,X +ℓ +𝐶2,X +ℓ +� +ℓ +� +X +, +(19) +where 𝐶𝑖,X +ℓ +is the cross-spectrum of map 𝑖 (=1, 2) with some other +external map X, with X running over all possible uncorrelated ex- +ternal maps, and the brackets represent the (unweighted) average +in a given multipole range (< ... >ℓ) or over all external maps +(< ... >X), respectively. For completeness, we also evaluate the +uncertainty on this parameter (𝜎𝐴1,2) as the standard deviation of +those ratios over the external maps, +𝜎𝐴1,2 = +1 +√𝑛X +𝑠𝑡𝑑𝑋 +�� +𝐶1,X +ℓ +𝐶2,X +ℓ +� +ℓ +� +, +(20) +where 𝑛X is the number of external maps involved in the analysis. +In this section, all cross-spectra are obtained using Xpol. The +reference mask adopted for this computation is the default one +(sat+NCP+lowdec), which preserves the declination range 6◦ ≤ +𝛿 ≤ 70◦. This mask is apodized using a 5◦ cosine function, as +implemented in the NaMaster library (Alonso et al. 2019). All +maps have been smoothed to a common resolution of one degree. +For MFI, the ratios are evaluated and averaged within the multipole +range ℓ = 30 to ℓ = 200. The lower value of ℓ = 30 guarantees +that the pseudo-𝐶ℓ estimation is not affected by mode coupling due +to incomplete sky coverage, and constitutes a conservative choice +regarding possible large scale residuals due to RFI and atmosphere, +as discussed in the previous section. As external maps, we decided +to use low frequency maps (≤ 70 GHz) from satellites, in order to +have similar foreground components to the signal in the QUIJOTE +maps. In particular, we use the 9-year WMAP maps (Bennett et al. +2013) for bands K, Ka, Q and V, and the PR2 Planck-LFI maps +at 30, 44 and 70 GHz corrected from bandpass leakage (Planck +Collaboration et al. 2016b). +5.2.2.1 +Intra-nulltest calibration. We first evaluate the relative +calibration of the wide survey, using the six null test maps described +in Sect. 4.1, namely half (mission), rings, halfring, daynight, pwv +and tbem. For each case, we compare the relative calibration of the +Figure 24. Intra-nulltest calibration of the MFI widey survey. We show the +consistency of the null test maps, for intensity (TT, top) and polarization +(average of EE and BB, bottom). +two maps in each pair ℎ1 and ℎ2, as in equation 19, and we evaluate +the error bar using equation 20. +Fig. 24 shows the result both for intensity (TT) and polarization +(average of EE and BB) data. In intensity, we find a good consistency +of all the different data splits well within one per cent. At 11 and +13 GHz, the maximum discrepancy is found to be 0.3 %. The average +of the six null test cases is consistent with one (perfect relative +calibration) within 0.2 %. At 17 and 19 GHz, the maps from horn +4 present a maximum discrepancy of 0.7 %, and the scatter of the +six measurements stays within 0.5 %. Horn 2, which is known to be +the noisiest one, presents the larger discrepancy of −1.6 % for the +half-mission null test, and the average of the six values is consistent +with one within 1 %. +In polarization, we find larger values of the scatter, as expected +due to the lower signal-to-noise ratios of these maps, although we +remind that in this case our analysis also probes possible time vari- +ations of the polarization efficiency values on top of the global cal- +ibration. For horn 3, the maximum discrepancy is associated with +the halfring null test, which presents deviations of +7 % for 311, +and -8 % for 313. However, we note that this null test is expected to +be noisier than the others, due to the lower number of independent +crossings in each half. The average of the six measurements is fully +consistent with one, and has a scatter of 2.9 % and 3.8 % for 11 and +13 GHz, respectively. For horn 4, we find a maximum discrepancy +of 6.4 %. The average of the six measurements is again consistent +with one, and the scatter is 3.8 % and 2.8 % for 17 and 19 GHz. +Finally, for horn 2, as in intensity, we find the largest scatter of the +MNRAS 000, 1–58 (2022) + +Intra-nulltest calibration TT +1.03 +half +halfrings +pwv +rings +daynight +tbem +1.02 +1.01 +A +1.00 +0.99 +0.98 +217 +219 +311 +313 +417 +419 +Channelntra-nultestcalibrationEE+BB +half +halfrings +pwv +1.3 +rings +daynight +tbem +1.2 +A +1.1 +1.0 +0.9 +217 +219 +311 +313 +417 +419 +Channel32 +Rubiño-Martín et al. +measurements. The largest discrepancy is found to be 17 % but with +a large error bar. The average of the six measurements is slightly +biased towards positive values of 𝐴 for 219, but not significantly +(two sigmas). The scatter of the measurements is 5.9 % and 5.2 % +for 217 and 219, respectively. +In summary, the internal calibration scale of the MFI wide +survey seems to be consistent within 0.7 per cent in intensity for all +horns, reaching 0.2 % for horn 3. In polarization, we find consistency +within 3–4 per cent in for horns 3 and 4, and within 10 % for horn 2. +To put in context these values, it is useful to compare them with the +expected scatter in the 𝐴 values in the case of a perfectly calibrated +instrument with the realistic noise levels of the MFI wide survey. +For this purpose, we have repeated this analysis using simulations +including realistic 1/ 𝑓 noise levels as in Sect. 5 of Guidi et al. +(2021). According to these simulations, the expected scatter of the +six null tests in intensity is within 0.1–0.2 %, while in polarization +we expect 2 % for horn 3 and horn 4 at 17 GHz, and we could have +up to 5–6 % for horn 2 and horn 4 at 19 GHz. We stress that these +numbers are driven by the 1/ 𝑓 noise in the maps, and therefore they +represent the actual sensitivity of this method to detect calibration +errors. Any calibration uncertainty due to systematic effects in the +real data will add to these values. +When comparing these values from simulations with those +found for real sky measurements, we find that they are consistent in +intensity, but the real data produce slightly larger scatter in polar- +ization. This small excess of uncertainty in the polarization values +from the real maps can be ascribed to polarization efficiency sys- +tematic errors. As a conservative approach, we decided to quote as +calibration uncertainty in Table 16 the final numbers obtained from +this test, thus including also the 1/ 𝑓 noise contribution. +5.2.2.2 +Inter-period calibration. We now evaluate the time sta- +bility of the wide survey calibration, using the four maps per period +described in Sect. 4.1.2, again for the case of "common baselines". +We also note that period 1 only has observations at high elevations, +so in order to have a common sky coverage for this comparison in +the four maps, we restrict the analysis in this particular case to a +sky mask covering the declination range 8◦ ≤ 𝛿 ≤ 50◦. As usual, +this extended mask is apodized using a 5◦ cosine function, as im- +plemented in the NaMaster library (Alonso et al. 2019). Fig. 25 +shows the comparison of the 𝐴 factors for the four maps by period +used for the wide survey (periods 1, 2, 5 and 6), when compared to +the total final map for each horn and frequency. +In intensity, the internal consistency is found to be again better +than 1 %. The largest discrepancy in absolute value is found for the +map 419 in period 1, at the level of -1.5 %. The standard deviation +of the four 𝐴 values for each horn and frequency is found to be +∼ 0.5 % for channels in horns 2 and 3, and 0.7–1 % for horn 4. In +polarization, we recall that some periods are not used for the final +maps. In particular, period 1 is not used in polarization, period 2 +is not used for horn 4, and period 5 is not used for horn 2. The +maximum discrepancy with respect to the final map is found in 313 +for period 5, at the level of −3.7 %. Taken as a whole, these values +suggest that the calibration scale is stable within 1 per percent in +intensity, and within 2 per cent in polarization, during the six years +of observations covered by the wide survey. +5.2.2.3 +Inter-horn calibration for horns 2 and 4. +Given that +the frequencies of 17 and 19 GHz are observed with horns 2 and 4, +we also carry out an inter-horn comparison of the final wide survey +maps at these frequencies using the same methodology as above, +and where the 𝐴 factor in equation 19 now compares the ratio of the +Figure 25. Inter-period consistency checks, in intensity (TT, top) and polar- +ization (average of EE and BB, bottom). We show the 𝐴 factor computed as +in equation 19, when comparing the map per period (i.e. using the data of +that given period only) to the total final map, for each horn and frequency. +two maps of a given frequency from the two horns. In this case, we +obtain two values, 𝐴217,417 and 𝐴219,419. The results are displayed +in Fig. 26 both for intensity (TT) and polarization (EE and BB, here +plotted separately). We find that the relative calibration of the wide +survey between horns 2 and 4 is consistent within 0.2 per cent in +intensity. In polarization, this test is not providing very restrictive +results due to the high noise levels of horn 2 in comparison to horn +4. Nevertheless, we can conclude that the relative calibration of the +two 17 GHz maps is found to be consistent within 2 per cent, while +for 19 GHz we find consistency within 4 per cent if we average the +values for EE and BB. In this later case, our simulations show that +the separated values for EE or BB alone might differ by more than +4 per cent in the ideal case of a perfect calibration, due to the (white +plus 1/ 𝑓 ) noise levels. +5.2.3 +Summary of the internal calibration tests +The overall calibration uncertainty quoted for the QUIJOTE MFI +wide survey maps is 5 % in intensity for all frequency maps, 5 % +in polarization for 11 and 13 GHz, and 6 % in polarization for the +combined 17 and 19 GHz maps (see last two rows in Table 16). +These values are mainly limited by the physical modelling of the +point-sources (Tau A, Cas A) used to calibrate the experiment. In +intensity, all the tests in this section show that the internal consis- +tency of the calibration and gain model, which spans 6 years of +measurements, is within the one per cent level. In polarization, the +MNRAS 000, 1–58 (2022) + +Inter-period calibration TT +1.020 +p1 +p2 +p5 +p6 +1.015 +1.010 +1.005 +A 1.000 +0.995 +0.990 +0.985 +0.980 +217 +219 +311 +313 +417 +419 +ChannelInter-periodcalibrationEE+BB +1.03 +p2 +p5 +p6 +1.02 +1.01 +1.00 +0.99 +0.98 +0.97 +0.96 +217 +219 +311 +313 +417 +419 +ChannelQUIJOTE MFI wide survey +33 +Figure 26. Inter-horn consistency check between horns 2 and 4, in intensity +(top) and polarization (bottom). +internal consistency tests show that the calibration is controlled at +the 2–3 per cent level for frequencies 11, 13 and 17 GHz, while for +19 GHz, and particularly for horn 2, this uncertainty could be up to +6 %. However, we note that in this later case, the quoted uncertainty +includes calibration errors, polarization efficiency uncertainties and +1/ 𝑓 noise contributions. +5.3 +Other calibration tests +5.3.1 +CMB anisotropies +CMB anisotropies in intensity can be measured in the QUIJOTE +MFI wide survey maps using a cross-correlation with an external +CMB template. We follow the methodology described and validated +in Section 6.5 of Guidi et al. (2021), and use a template fitting +method with two templates: a reference CMB map (mCMB), and +a "foreground" map to account for chance alignments between the +CMB and the Galactic foregrounds (f). The basic assumption is that +the QUIJOTE map (mMFI) can be written as a linear combination +of these two maps as +mMFI = 𝐴mCMB + 𝐵f + n, +(21) +where 𝐴 and 𝐵 are the parameters of the linear combination, and n +represents a noise component. Using the cross spectra of the QUI- +JOTE maps with both external templates, 𝐶MFI,CMB +ℓ +and 𝐶MFI,f +ℓ +, +we can extract both 𝐴 and 𝐵 parameters. As shown in Guidi et al. +(2021), this method produces unbiased results for the CMB recon- +struction (𝐴 = 1), provided that there is a perfect consistency with +Figure 27. Relative amplitude of the CMB signal in the QUIJOTE MFI +maps, using cross-correlations with the Planck SMICA map. Error bars are +obtained using rotations of the CMB map. For consistency, we show that the +average signal of the cross-correlation with rotated CMB maps is consistent +with zero, as expected. +the calibration of the CMB map. Thus, the method can be used as +an additional calibration test. +Here, we use as a reference the SMICA 2018 map (Planck Col- +laboration et al. 2020d), but we have checked that consistent values +are obtained using other versions of the Planck CMB map (NILC, +COMMANDER, SEVEM). As foreground template, we use the +WMAP 9-year K-band map (Bennett et al. 2013), after subtracting +the CMB component. The analysis mask is the same as in Guidi +et al. (2021), which combines the default QUIJOTE analysis mask +(NCP+sat+lowdec) with the Planck common confidence mask for +temperature analyses (Planck Collaboration et al. 2020d), apodized +with a simple 2-degree smoothing. All cross-spectra in this section +are computed using Xpol. Error bars are obtained using rotations +of the CMB map in steps of Δ𝑙 = 18◦, as in Guidi et al. (2021). +The analysis is carried out in the multipole range [100, 200], but +consistent results are obtained in other ranges (e.g. we also tested +[30, 200], although the overall significance is lower in this case due +to the larger 1/ 𝑓 contribution of lower multipoles). The final results +are shown in Figure 27 and Table 18. The CMB signal is detected +in all channels, with a significance larger than 10-sigma in all cases. +These error bars are consistent with the level of 1/ 𝑓 noise in the +QUIJOTE maps (see Table 4 in Guidi et al. (2021)). We note that, +due to the strongly correlated noise in the MFI intensity maps, es- +timates from the same horn tend to deviate in the same direction. +All values are consistent with 𝐴 = 1, providing an independent +confirmation of the calibration scale of the maps. Finally, we also +provide a combined measurement of the CMB signal present in the +QUIJOTE MFI maps, using a weighted average combination of all +channels and accounting for the noise correlation between frequen- +cies of the same horn. The overall result (1.02 ± 0.03) provides +a 35-sigma detection of the CMB anisotropies in the QUIJOTE +MFI intensity maps, and shows a consistent calibration with Planck +within three per cent. +5.3.2 +CMB dipole +As an additional calibration test, we present here the detection of the +CMB dipole in the MFI wide survey maps, using a cross-correlation +technique similar to the one used in the previous subsection for the +CMB anisotropies. For this analysis, specific MFI wide survey maps +are generated excluding the dipole removal and the atmospheric cor- +MNRAS 000, 1–58 (2022) + +nter-hornscalibrationTT +1.002 +1.001 +A +1.000 +0.999 +17 +19 +Frequency[GHz]Inter-horns calibration EE,BB +EE +1.02 +BB +1.00 +0.98 +0.96 +0.94 +0.92 +17 +19 +Frequency[GHz]CMB Cross-correlations l E[100, 20o] +1.2 +1.0 +0.8 +A +0.6 + +0.4 +0.2 +0.0 +217 +219 +3i1 +3i3 +417 +419 +Channels34 +Rubiño-Martín et al. +Table 18. Relative amplitude (𝐴) of the CMB component in the QUIJOTE- +MFI wide survey maps with respect to the SMICA Planck map, obtained with +cross-correlations in the multipole range 100–200. Error bars are obtained +using rotations of the CMB map. +Channel +A +Uncertainty +217 +1.080 +0.068 +219 +1.086 +0.086 +311 +1.010 +0.037 +313 +1.005 +0.033 +417 +1.030 +0.086 +419 +0.974 +0.097 +Combined +1.019 +0.029 +Figure 28. MFI wide survey 311 (horn 3 at 11 GHz) map, with the dipole +component not removed from the map. For display purposes, the map has +been downgraded to resolution 𝑁side = 256. +rection steps in the post-processing stage of the pipeline. Figure 28 +shows one example of these maps, for the case of horn 3 at 11 GHz. +We use a template fitting method in real space with three tem- +plates: a reference CMB dipole template map (mdip), a "foreground" +map to account for the Galactic component (f), and a constant map +accounting for a residual monopole term (𝐶). As in the previous +section, we assume that the MFI wide survey maps (mMFI) can be +written as a linear combination of those three templates as +mMFI = 𝐴mdip + 𝐵f + 𝐶 + n, +(22) +where 𝐴, 𝐵 and 𝐶 are the three coefficients to be obtained and +n represents the noise component. The dipole template map mdip +is prepared following the methodology outlined in Sect. 4.4.2 of +Guidi et al. (2021), including both the solar and orbital CMB dipole +terms with the measured amplitudes by the Planck collaboration. +The dipole prediction is generated at the TOD level, and then this +is projected into a sky map using the PICASSO map-making algo- +rithm. For the Galactic template, we use again the WMAP 9-year +K-band map after subtracting the CMB component. For this analy- +sis, all maps are degraded to a common resolution of one degree. +The analysis mask combines the default QUIJOTE analysis mask +(NCP+sat+lowdec), the Planck confidence CMB mask for temper- +ature analyses (Planck Collaboration et al. 2020d), and a Galactic +mask |𝑏| < 30◦, in order to avoid a possible bias in the dipole +determination due to the Galactic emission. +We first validate the methodology using end-to-end simula- +tions of the MFI wide survey including the dipole component and +realistic 1/ 𝑓 noise levels as in Guidi et al. (2021). We find that +our approach provides unbiased estimates of the dipole amplitude +Table 19. Fitting for the CMB dipole in the MFI wide survey maps. We +present the relative amplitude with respect to the expected CMB dipole, and +the associated uncertainty. See text for details. +Channel +Relative amplitude +Uncertainty +217 +1.04 +0.22 +219 +0.97 +0.47 +311 +0.88 +0.09 +313 +0.92 +0.12 +417 +0.99 +0.30 +419 +1.23 +0.67 +Combined +0.92 +0.09 +(i.e. 𝐴 = 1) for all MFI frequency maps, with typical errors of +few percent. We have also tested the impact of the three different +corrections that are applied to the maps (RFI, FDEC and ATMOS) +on the reconstructed dipole amplitude 𝐴. In summary, we find that +including or not the RFI and FDEC corrections does not bias the +recovered 𝐴 value. However, the ATMOS correction significantly +affects the recovered amplitude, especially in the high frequency +MFI bands. This is expected because the atmospheric templates are +built on approximately one hour timescales, and on those scales the +CMB dipole component is a very stable signal in the azimuth scans +(rings). Because of this, the ATMOS correction is not applied for +this analysis. +The measured values in real data are presented in Table 19, for +each one of the MFI wide survey maps separately. Error bars have +been estimated using the following methodology. We rely on the +null test maps for independent baselines as the most representative +method to capture large angular scale noise in the maps. Thus, we +repeat the analysis and detect the CMB dipole in the half1/2, pwv1/2, +tbem1/2 and daynight1/2 maps. The reported values correspond to +the average dipole of the 8 cases, and the error bar is the scatter of the +8 measurements, taken to be a representative error of the method. +We have tested that we obtain almost identical results if we carry +out the analysis on maps with no FDEC and/or RFI corrections. +Finally, we also present the weighted average combination of +all channels, accounting for the correlation between frequencies of +the same horn. The value is 𝐴 = 0.92 ± 0.09, which corresponds to +a 10-sigma detection of the CMB dipole, and it is consistent with +the Planck calibration within nine per cent. +5.3.3 +Bright point sources and planets +Bright radio sources and planets have been used extensively as a +basic calibration test for MFI wide survey maps in several stages +of the pipeline. Indeed, the maps in each period are recalibrated in +order to match the Tau A model in intensity (Sect. 2.6). Below in +Sect. 9 we present a detailed study of few bright objects (Tau A, +Cas A, Cyg A, 3C274, W63, Jupiter and Venus), which could be +seen as a further validation test of the overall calibration scale of +the experiment. +5.4 +Setting the zero levels +The QUIJOTE MFI wide survey intensity maps produced by our +default pipeline are insensitive to the true absolute zero level +(monopole) of the sky emission. A monopole signal is essentially +unconstrained for QUIJOTE MFI, as a global constant added to +MNRAS 000, 1–58 (2022) + +H3, 11GHz (with dipole) +-10 +mK +10QUIJOTE MFI wide survey +35 +the full TOD database is not changing the map-making solution af- +ter the basic TOD processing. Indeed, in the post-processing stage +maps are corrected of any residual monopole and dipole signals. +In order to estimate the zero levels of these maps in intensity, +we follow a methodology similar to the one adopted by WMAP +(Bennett et al. 2003), and we assume a plane-parallel model for the +Galactic emission. In that case, the zero level of the maps can be +estimated by fitting a cosecant model of the form: +Δ𝑇 = 𝐴 csc(|𝑏|) + 𝐵. +(23) +For this analysis, we use the smoothed maps at 1◦ angular reso- +lution, and degrade them to 𝑁side = 64 in order to have approx- +imately independent pixels. We carry out the fit independently in +both hemispheres, using the Galactic latitude ranges 15◦ < 𝑏 < 90◦ +and −90◦ < 𝑏 < −15◦ for the northern and southern hemispheres, +respectively. We mask the satellite band, and in the case of the +northern sky, our analysis also excludes the region in Galactic lon- +gitude corresponding to the North Polar Spur (0◦ ≤ 𝑙 ≤ 35◦). Error +bars are computed using the scatter of the results around the mean +value, when adding realistic noise simulations. For this analysis, +we use 100 of the simulations described in Sect. 6.2. The reference +results adopted here correspond to the northern hemisphere, due to +the larger sky fraction covered by the QUIJOTE MFI footprint. For +QUIJOTE MFI 11 GHz (horn 3), we have 𝐵 = −0.74 ± 0.20 mK, +where the error bar includes both the effect of varying sky emission +and the noise variance contained in the simulations. Similarly, for +QUIJOTE MFI 13 GHz (horn3) we have 𝐵 = −0.59 ± 0.22 mK. +The results for the southern hemisphere are consistent with those +(−0.59 ± 0.27 mK and −0.42 ± 0.26 mK for 11 and 13 GHz, re- +spectively), although they have larger error bars. For the other two +frequency bands (17 and 19 GHz), and both for horns 2 and 4, the +zero levels are statistically consistent with zero in both hemispheres +(with typical error bars of 1.2–1.3 mK). These values are inserted +in Table 16. Finally, we note that there are other methods in the +literature for deriving the zero levels of radio maps (see e.g. Wehus +et al. 2017), which could be applied here. However, we emphasize +that those analyses should be done carefully, due to the special filter- +ing of large angular scales (FDEC) applied to the MFI wide survey +maps. +5.5 +Polarization angle +As described in Génova-Santos et al. (2023), the reference angle for +each MFI observation is calibrated using daily Tau A observations. +Our calibration scheme provides a reference angle for each period +and channel, as this value changes across the spectral band, from +horn to horn, and also with the instrument configuration. As this +daily calibration might suffer from 1/ 𝑓 noise uncertainties, the final +QUIJOTE MFI wide survey maps are recalibrated again using Tau +A in each period (see Sect. 2.6). Here, we can evaluate the error +budget associated with the polarization angle in the wide survey +maps using Tau A. As a reference method, we use aperture pho- +tometry in the polarization maps smoothed to 1 degree. We adopt +an integration radius of 𝑟1 = 1.5◦ for the primary aperture, and +an outer annulus between 𝑟1 and 𝑟2 = +√ +2𝑟1 to correct for the lo- +cal background contribution. The photometry results are described +in Table 24 and Sect. 9. Table 20 presents the error budget in the +polarization angle obtained using two methodologies. First, col- +umn 2 presents the scatter (standard deviation) of the Tau A angle +measurements obtained from the null test maps with independent +baselines (half1/2, pwv1/2, ring1/2, daynight1/2 and halfring1/2). +On the other hand, column 3 presents the statistical error obtained +Table 20. Error budget for the polarization angle in the wide survey, based +on Tau A photometry. We include the error budget from the scatter of the +measurements in the different null tests (column 2) and the statistical error +obtained from the photometry method (column 3). +Channel +Error (null tests) +Error (stat.) +(deg) +(deg) +217 +0.71 +0.91 +219 +0.98 +0.96 +311 +0.44 +0.50 +313 +0.34 +0.67 +417 +1.18 +0.64 +419 +1.70 +0.59 +Comb. 17 GHz +0.96 +0.53 +Comb. 19 GHz +1.43 +0.51 +Table 21. Comparison of the reconstructed angles in the QUIJOTE MFI +wide survey data to WMAP-K (column 2), LFI30 (column 3) and MFI 311 +(column 4). See text for details. +Channel +WMAP-K +LFI30 +MFI-311 +(deg) +(deg) +(deg) +217 +−2.8 ± 1.5 +−3.3 ± 1.5 +−4.2 ± 1.5 +219 +0.8 ± 3.0 +0.4 ± 3.0 +−0.4 ± 2.9 +311 +0.6 ± 0.6 +−0.5 ± 0.6 +– +313 +−1.2 ± 0.6 +−2.0 ± 0.6 +−2.2 ± 0.6 +417 +−1.2 ± 1.0 +−1.6 ± 1.0 +−2.3 ± 1.0 +419 +0.5 ± 3.6 +0.0 ± 3.6 +−0.9 ± 3.5 +Comb. 17 GHz +−1.6 ± 0.9 +−2.2 ± 0.9 +−2.8 ± 0.9 +Comb. 19 GHz +0.9 ± 3.2 +0.3 ± 3.2 +−0.5 ± 3.1 +from the propagation of the errors from the photometry measure- +ment in the final maps. As a conservative approach, we keep the +highest value of each pair as representative of the error budget in +the angle determination from Tau A. We see that the uncertainty +changes from 0.5◦ for 311, to 1.7◦ for 419. +As a further consistency check for the polarization angle cal- +ibration, we compare the measured MFI wide survey polarization +angle maps with those from WMAP 9-year K-band map (Bennett +et al. 2013) and Planck PR4 LFI30 data (Planck Collaboration et al. +2020f). Table 21 presents the results of this comparison, including +also an internal comparison to the MFI 311 map. The analysis is car- +ried out smoothing all maps to 1 degree resolution, and degrading +them to 𝑁side = 64, in order to match approximately the beam scale +in one pixel. We use the standard analysis mask (NCP+sat+lowdec), +but in addition, we keep only those high signal-to-noise pixels with +a nominal uncertainty in the MFI 311 angle 𝜎𝜙311 ≤ 2◦. In order +to avoid bright regions that might bias the comparison, pixels that +have an absolute value in 𝑄 or 𝑈 that is greater than 2 mK in the +WMAP K-band after being rescaled to 11.1 GHz using a spectral +index of −3.0 are also flagged. Finally, we also exclude the bright +Cygnus area removing all pixels within 5 degrees around the loca- +tion (𝑙, 𝑏) = (80◦, 0◦). The resulting analysis area has 𝑓sky = 0.124. +In order to correct for residual zero level differences between +the MFI and the WMAP/Planck maps (e.g. due to unresolved point +sources), we use a TT plot technique between WMAP-K and each +MFI Stokes Q and U map within the analysis mask, and we remove +the fitted zero levels from the MFI maps. We note that the resulting +values are basically consistent with zero (within the error), but of +MNRAS 000, 1–58 (2022) + +36 +Rubiño-Martín et al. +the order of 20 𝜇K for 311 and 313. Although small, they might +introduce measurable differences (at the level of a degree) in our +analysis. For each MFI map, we compute the weighted mean of the +difference between the two angles (e.g. 𝜙MFI − 𝜙WMAP for the first +case), using as weights the inverse variance of the angle, which in +turn is derived from the 𝑄 and𝑈 weight maps. Error bars in Table 21 +are generated with a Monte Carlo method using 100 of the noise +simulations described in Sect. 6.2. We add each noise simulation to +the corresponding MFI map, and repeat the same procedure. The +error bar corresponds to the standard deviation of the 100 values. +In general, all the measured differences are statistically consistent +with zero given the noise uncertainty. MFI 311 (horn 3 at 11 GHz) +is consistent with both WMAP-K and LFI30 within the quoted +uncertainty of 0.6◦. The situation is similar for the 19 GHz maps +(both horns 2 and 4). However, we note that there is a moderate +tension with the MFI 313, which deviates in the case of LFI30 up +to 3.3 sigmas, and the 17 GHz cases, which deviates 2.4 sigmas for +the combined map of horn 2 and horn 4. In order to investigate this +possible discrepancy, we have repeated the analysis but using all the +different null test maps with independent baselines (half1/2, pwv1/2, +ring1/2, daynight1/2, halfring1/2). The error is now computed as +the standard deviation of all those values. The result for the MFI +313 comparison with LFI30 now gives −2.0 ± 0.9, showing that +maybe the error in this case is slightly underestimated. While we +are still finding a discrepancy, the significance is now reduced to 2.2 +sigmas. Another point that we have studied is the possible impact +of Faraday rotation in this comparison. Using the Galactic Faraday +depth maps from Hutschenreuter & Enßlin (2020), we estimate that +in our analysis region the mean rotation measure is −11.9 rad m−2. +This would introduce differences of the order of approximately +−0.4◦ between MFI311/MFI313 and LFI30. Although this value is +not enough to explain the discrepancy, it helps to further decrease +the tension below the 2 sigma level. +The final results in Table 16 contain the worst case value based +on the three values reported in this section (two values for Tau +A in Table 20, and the standard deviation of the comparison with +WMAP/Planck in Table 21). +6 +SIMULATIONS +6.1 +Sky signal +Some of the analyses in this paper make use of sky simulations. +Our reference sky simulations were developed within the context +of the RADIOFOREGROUNDS project10, and are described in +detail in Sect. 5.2 of Guidi et al. (2021). They contain different +foreground components from the Planck FFP10 sky model (Planck +Collaboration et al. 2020b,c), a CMB realization, and the CMB +dipole contribution. For some applications, these sky simulations +are projected into the MFI wide survey TODs, and the PICASSO +map-making code is used to generate synthetic maps with the same +flagging and number of hits as in the real wide survey data. These +simulated data can also include a noise contribution, injected at +the TOD level. As explained in this paper, this approach has been +extensively used to validate some aspects of the pipeline (map- +making, transfer function, null tests, determination of the CMB +dipole, etc.). These sky simulations are also used below to evaluate +the statistical errors associated with the power spectra (see Sect. 7). +10 www.radioforegrounds.eu +6.2 +Simulated noise maps +In addition to the end-to-end noise simulations that have been pro- +duced as explained in the previous subsection, we also construct +noise simulations for the different channels (i.e., pair frequency- +horn) maps, starting from the HMDM of totally independent splits. +The simulations aim to account for the measured anisotropic be- +haviour, spatial correlations and the correlations between the two +frequency channels of the same horn in the wide survey maps (∼ 60– +80% in intensity, and ∼ 20% in polarization, as seen in Sect. 4.3.3). +The anisotropic behaviour follows the properties of the cor- +responding Local Variance (LV) maps per channel and per Stokes +parameter. These LV maps are estimated from the HMDM maps, by +assigning, at each pixel at resolution 𝑁side = 512, the variance com- +puted from the surrounding pixels at a given distance (39 arcmin; +this value has been chosen as a compromise to have enough pixels to +provide an accurate estimation and, at the same time, to preserve as +much information at small scale as possible). The estimation of the +variance takes into account only those pixels which are within the +observed sky at each channel. Each one of the HMDM are normal- +ized by dividing them by the square root of the corresponding LV +maps. The non-observed pixels of the normalized HMDM are filled +with a Gaussian random realization with unit dispersion, building +in this manner extended-normalized HMDM maps. +We now compute the noise spatial correlation by computing +the TT, TE and BB angular power spectra (APS) of the extended- +normalized HMDM maps, and from there, we derive a model of +these APS. This is done by estimating a smoothed version of the +observed APS of the extended-normalized HMDM maps for each +channel, using a polynomial fit (of order 4), and by defining the max- +imum multipole that provides a variance (at the map level) as close +as possible to the one of the corresponding extended-normalized +HMDM maps. Following this process, we end-up with a model for +the noise correlations that provides the right power level. +A noise simulation is now generated by drawing a Gaussian +random map in harmonic space, following the corresponding mod- +els of the noise APS for each frequency map. The maps (T, Q and +U) are further multiplied by the square root of the corresponding LV +map. We use the correlation coefficient between frequency maps of +the same horn to further modify the simulated map of the second +member of the pair (e.g., the 13 GHz frequency channel in the case +of horn 3). In particular, we construct the final version of the simu- +lated map of the second member of the pair as a linear combination +of the first member of the pair and the initial version of the second +map, taking into account the correlation coefficient. In this way, +all the pixel-based statistics are maintained for the two members of +the pair, as well as the correlation. The APS of the first member +are also maintained, but, eventually, we modify the APS properties +of the second member and the cross-correlation. The correlation at +pixel level is imposed for T, Q and U. Notice that for the Stokes +parameters this is done as if they were scalars. Nevertheless, the +properties of the polarization intensity are preserved, although we +are not able to reproduce the observed cross-correlation in 𝑃. We +find that this approximation is the most adequate for our further +analyses, since most of them are addressed in the pixel domain. As +illustration, Figure 29 shows the power spectra for a subset of 100 +noise simulations for horn 3 at 11 GHz. +7 +POWER SPECTRA OF THE WIDE SURVEY MAPS +In this section we study the main properties of the auto- and cross- +spectra of the MFI wide survey maps. We consider three masks, +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +37 +Figure 29. Power spectra (TT, EE, and BB) for 100 noise simulations of the +311 map (horn 3 at 11 GHz). The red line shows the reference noise power +spectrum for the half mission difference maps (labelled as HD) which was +used to generate the simulations. The individual power spectrum for each +simulation is shown in light grey, and the average of those 100 simulations +in light blue. +corresponding to different Galactic latitude cuts (|𝑏| > 5◦, 10◦ +and 20◦), which are always combined with the default QUIJOTE +analysis mask (NCP+sat+lowdec). As usual, each of these three +masks is apodized with a five degree apodization kernel and the +cosine function implemented in Alonso et al. (2019). All spectra +have been computed with the NaMaster code, enabling for the +option of "purification" of E and B modes, which allows a better +reconstruction of the E and B mixing matrix for cut-sky spectra. In +Appendix E we discuss the validity of the use of this pseudo-Cℓ +approach for the wide survey maps. +Throughout this section, all power spectra have been corrected +by the MFI beam window functions, as well as the pixel window +function (which in this case corresponds to a HEALPix map with +𝑁side = 512). Noise levels (𝑁ℓ) are estimated from the half-mission +difference maps (with independent baselines), and then subtracted +from the corresponding power spectra of the maps, in order to +obtain the spectrum of the sky signal, 𝐶sky +ℓ += 𝐶map +ℓ +− 𝑁ℓ. We have +tested that using another estimate of the noise power spectra (e.g. +the average of several null test difference maps) produces consistent +results to those presented in this section. All spectra are binned +using Δℓ = 10. In all figures in this section, we represent band +power values 𝐷ℓ = ℓ(ℓ + 1)𝐶𝑋𝑌 +ℓ +/2𝜋, where 𝑋,𝑌 ∈ {𝑇, 𝐸, 𝐵}. +Uncertainties in the power spectra of the maps 𝜎(𝐶map +ℓ +) are +estimated using 100 simulations including sky signal (Sect. 6.1) and +realistic noise simulations (Sect. 6.2). The same noise simulations +Figure 30. TT, EE and BB spectra for |𝑏| > 5◦, and for all frequencies +(11, 13, 17, 19 GHz), represented as solid circles with their corresponding +uncertainties. As a reference, dashed lines depict the noise spectra 𝑁ℓ for +each case, using the same colour scheme. +are also used to estimate the uncertainties in the noise level, 𝜎(𝑁ℓ). +The quoted uncertainties in 𝜎(𝐶sky +ℓ +) are obtained as the quadratic +sum of both 𝜎(𝐶map +ℓ +) and 𝜎(𝑁ℓ). +Figure 30 shows the (auto) power spectra (TT, EE, BB) of +the wide survey maps, for the particular case of the Galactic mask +with |𝑏| > 5◦, combined with the default QUIJOTE analysis mask +(NCP+sat+lowdec). We have a high significance detection of TT, +particularly for the two lowest frequencies. At these frequencies, +MNRAS 000, 1–58 (2022) + +Noise simulations 11GHz, H3 +Sims +10-3 +[mk2] + +HD +10-4 +10-5 +10-4 +[mk?] +10-5 +10-6 +CBB [mK2] +10-5 +10-6 +101 +102TT, Ibl>5° +1.000 +0.100 +[mk?] +D +0.010 +11 GHz +13 GHz +17 GHz +19GHz +0.001 +50 +100 +150 +200 +Multipole lEE, lbl>5° +11 +GHz +13 GHz +17 GHz +19 GHz +mk²] +10 +D +10 +10- +50 +100 +150 +200 +Multipole lBB, +Ibl>5° +11( +GHz +13 GHz +17 GHz +19 GHz +D +10 +50 +100 +150 +200 +Multipole l38 +Rubiño-Martín et al. +Table 22. Best fit results obtained after fitting the model in equation 24 to +the wide survey EE and BB power spectra at 11 GHz, in the multipole range +30 < ℓ < 200. No colour corrections were applied when fitting the spectra. +Mask +|𝑏| > 5◦ +|𝑏| > 10◦ +|𝑏| > 20◦ +𝑓sky +0.38 +0.34 +0.27 +EE and BB fitted separately +𝐴EE [𝜇K2] +1.52 ± 0.15 +1.05 ± 0.18 +0.81 ± 0.19 +𝐴BB [𝜇K2] +0.52 ± 0.15 +0.20 ± 0.12 +0.18 ± 0.13 +𝛼EE +−3.00 ± 0.16 +−2.72 ± 0.26 +−2.96 ± 0.36 +𝛼BB +−3.08 ± 0.42 +−3.13 ± 0.87 +−3.12 ± 1.03 +𝑐EE [𝜇K2] +0.07 ± 0.09 +−0.13 ± 0.11 +−0.09 ± 0.12 +𝑐BB [𝜇K2] +0.10 ± 0.09 +−0.06 ± 0.09 +−0.09 ± 0.09 +𝐴BB/𝐴EE +0.34 ± 0.10 +0.19 ± 0.12 +0.22 ± 0.18 +Joint EE and BB analysis +𝐴EE [𝜇K2] +1.49 ± 0.12 +0.97 ± 0.13 +0.78 ± 0.14 +𝛼EE (= 𝛼BB) +−3.04 ± 0.13 +−2.83 ± 0.21 +−3.03 ± 0.29 +𝑐EE (= 𝑐BB) [𝜇K2] +0.09 ± 0.06 +−0.08 ± 0.06 +−0.08 ± 0.07 +𝐴BB/𝐴EE +0.36 ± 0.04 +0.26 ± 0.07 +0.26 ± 0.08 +the polarized emission is dominated by Galactic synchrotron. The +EE synchrotron signal is clearly detected at large angular scales +(ℓ ≲ 100) for 11 and 13 GHz, and the BB signal is also significantly +detected in that range for 11 GHz. In the next three subsections +we discuss the angular and frequency dependence of these spectra. +A multi-frequency analysis of the power spectra of the MFI wide +survey maps, in combination with WMAP and Planck data, will be +presented in a separate paper (Vansyngel et al. 2023). +7.1 +Fitting the EE and BB auto-spectra at 11 GHz +Figure 31 shows the TT, EE and BB auto-spectra at 11 GHz, for +three masks with Galactic latitude cuts |𝑏| > 5◦, 10◦ and 20◦. +We focus here on the polarization spectra, EE and BB. Following +Krachmalnicoff et al. (2018), we fit for these spectra in the multipole +range 30 < ℓ < 300, using the following parameterization +𝐶XX +ℓ += 𝐴XX +� +ℓ +80 +� 𝛼XX ++ 𝑐XX, +(24) +where 𝑋 ∈ {𝐸, 𝐵}, 𝐴XX is the amplitude of the spectrum at the pivot +multipole ℓ = 80, 𝛼XX is the slope of the multipole dependence, +and 𝑐XX is a global constant which represents the contribution of +unresolved (Poisson distributed) radio sources. +The power spectra are fitted using the EMCEE ensemble sam- +pler (Foreman-Mackey et al. 2013), and using a standard Gaus- +sian likelihood function. Our best-fit results, obtained from the +marginalised posterior distributions for each parameter, are given +in Table 22. First, we fit for the EE and BB power spectra sep- +arately. In all three cases, the global constants 𝑐EE and 𝑐BB are +statistically consistent with zero, as expected given the noise lev- +els of the wide survey maps, and the expected contribution from +radio sources at these frequencies, estimated to be ≲ 30 𝜇K.deg at +11 GHz (Puglisi et al. 2018; Herranz et al. 2023). Both the EE and +BB spectra present similar values of the slope, and no dependence +on the Galactic latitude cut is observed. When combining the ratios +of the EE and BB signals, we find that 𝐴BB/𝐴EE is of the order +of 0.2 for the two higher Galactic cuts (|𝑏| > 10◦ and |𝑏| > 20◦), +and we obtain 0.34 ± 0.10 for the lowest cut (|𝑏| > 5◦). In order to +increase the significance of this measurement, and based on these +Figure 31. TT, EE and BB spectra for QUIJOTE MFI 11GHz, as a function +of the Galactic cut. Dashed lines represent the corresponding noise spectra +𝑁ℓ for each case, using the same colour scheme. +results, we repeat the analysis now assuming that both EE and BB +spectra have the same slope (𝛼EE = 𝛼BB) and Poissonian terms +contributions (𝑐𝐸𝐸 = 𝑐𝐵𝐵). In this case, we can fit simultaneously +for the EE and BB spectra using four parameters (𝐴EE, 𝛼EE, 𝑐EE +and 𝐴BB/𝐴EE). The results for the amplitudes and slopes are con- +sistent with the values obtained in the previous case. Regarding the +ratio of the amplitudes, we have now a higher significance, with +𝐴BB/𝐴EE = 0.26 ± 0.08 for the |𝑏| > 20◦ case. In summary, the +MFI wide survey data at 11 GHz show more power in the EE spectra +MNRAS 000, 1–58 (2022) + +TT + 11GHZ +1.00 +[mk²] +0.10 +D +Ibl> 5° +Ibl>10° +Ib1>20° +0.01 +50 +100 +150 +200 +250 +Multipole lEE 1 1GHz +Ibl> +lbl>10° +lb/>20° +mk²] +10 +D +10- +50 +100 +150 +200 +Multipole lBB +3 11GHz +Ibl> 5° +lb/>10° +lb/>20° +mk²] +10° +D +10 +50 +100 +150 +200 +Multipole lQUIJOTE MFI wide survey +39 +Table 23. Best fit results obtained after fitting a constant model to the +wide survey EB and TB power spectra at 11 GHz, in the multipole range +30 < ℓ < 150. No colour corrections are applied. +Mask +|𝑏| > 5◦ +|𝑏| > 10◦ +|𝑏| > 20◦ +𝐴EB [𝜇K2] +−0.014 ± 0.037 +0.002 ± 0.038 +0.043 ± 0.041 +𝐴EB/𝐴EE (ℓ = 80) +−0.010 ± 0.025 +0.002 ± 0.038 +0.057 ± 0.059 +𝐴TB [𝜇K2] +−0.17 ± 0.24 +−0.15 ± 0.20 +−0.21 ± 0.19 +𝐴TB/𝐴EE (ℓ = 80) +−0.11 ± 0.16 +−0.15 ± 0.20 +−0.28 ± 0.28 +than BB, with a typical BB/EE ratio of a factor of 0.26. This value +is approximately half of the equivalent BB/EE ratio for thermal dust +emission, as derived from Planck observations at 353 GHz (Planck +Collaboration et al. 2016a, 2020e). +Our numbers for the synchrotron emission at 11 GHz can be +compared with others in the literature. Planck Collaboration et al. +(2020d) found 𝛼EE = −2.84 ± 0.05, 𝛼BB = −2.76 ± 0.09 and +𝐴BB/𝐴EE = 0.34 for the synchrotron map at 30 GHz obtained with +Commander (Eriksen et al. 2008), and analysing a sky area of +𝑓sky = 0.78 and a multipole range ℓ = 4-140. Following a similar +methodology to the one used here, Martire et al. (2022) carried +out a combined analysis of WMAP-K band and Planck LFI30 data, +finding very stable values for the slopes and BB/EE ratios as a +function of the sky mask. For the case of a mask preserving 50 % of +the sky, they obtain 𝛼EE = −2.79 ± 0.05, 𝛼BB = −2.77 ± 0.15, and +𝐴BB/𝐴EE = 0.22 ± 0.02. In both cases, the values are consistent +with our results at 11 GHz. On the other hand, using S-PASS data +at 2.3 GHz, Krachmalnicoff et al. (2018) find significantly larger +values of the BB/EE ratio for similar Galactic cuts in the southern +sky, with values of 0.87 ± 0.02 for |𝑏| > 20◦, and 0.64 ± 0.03 for +|𝑏| > 30◦. +7.2 +TE, TB and EB spectra at 11 GHz +Figure 32 shows the TE, EB and TB power spectra for the 11 GHz +map, evaluated in the same sky masks as in the previous subsection +(see also Fig. 31). Given that the power spectra of the HMDM is +statistically consistent with zero in all three cases (TE, EB and TB), +we do not apply the 𝑁ℓ correction in this subsection. Error bars are +computed using the same methodology described above. However, +for the EB spectra, we also add in quadrature the uncertainty on +the power spectrum due to the polarization angle (Table 16), using +equation 5 in Minami et al. (2019), and assuming that the underlying +EB power spectrum is zero. +We detect a positive cross-correlation between the total inten- +sity T and the E-mode polarization (TE> 0) at large angular scales +for the three considered Galactic cuts (up to ℓ ≲ 80 for |𝑏| > 5◦, +and ℓ ≲ 50 for |𝑏| > 10◦ and |𝑏| > 20◦). Beyond ℓ >∼ 150, this +TE cross spectrum becomes very noisy. We also find a null corre- +lation in TB and EB in the range 30 ≲ ℓ ≲ 150, as expected for +a parity-invariant emission process and an accurate calibration of +the polarization angle. Beyond this multipole range, the error bars +increase significantly, in particular for the TB case. +We provide a quantitative measurement of the TB/EE and +EB/EE ratios by fitting these spectra to a constant value (i.e. 𝐶TB +ℓ += +𝐴TB, and 𝐶EB +ℓ += 𝐴EB), in the range 30 ≲ ℓ ≲ 150. The results are +presented in Table 23, where we have used the EE fits from Table 22. +For the synchrotron emission, the MFI 311 maps provide upper +limits on the EB signal at the level of 4 per cent of the EE component +at ℓ = 80 for the |𝑏| > 10◦ cut. These results are consistent with +Figure 32. TE, EB and TB spectra for QUIJOTE MFI 11GHz, as a function +of the Galactic cut. +those found in Martire et al. (2022) for WMAP/Planck. Similarly, +for the TB component we provide upper limits at the level of 20 % of +the EE component. We recall that for the thermal dust emission, the +Planck satellite found a positive TE signal at large scales, a weakly +positive TB, and a EB statistically consistent with zero (Planck +Collaboration et al. 2016a, 2020e). +7.3 +Frequency dependence of the EE and BB signal +We carry out a simultaneous fit of all the power spectra shown in +Figure 30, using the parameterization from eq. 24, but assuming that +the amplitudes are related via a power law dependence in frequency +with a temperature spectral index 𝛽s,EE. In practice, the amplitude +MNRAS 000, 1–58 (2022) + +TB +3 11GHZ +0.010 +0.005 +[mk²] +0.000 +D +-0.005 + lbl> 5° +/b/>10° +Ibl>20° +-0.010 +50 +100 +150 +200 +Multipole lTE 11GHz +0.010 +0.005 +D. [mk"] +0.000 +-0.005 + lbl> 5° +/b/>10° +Ibl>20° +-0.010 +50 +100 +150 +200 +Multipole lEB 11GHz +0.006 +Ibl> 5° +Ibl>10° +Ibl>20° +0.004 +0.002 +[mk?] +0.000 +D +-0.002 +-0.004 +-0.006 +50 +100 +150 +200 +Multipole l40 +Rubiño-Martín et al. +at a given frequency channel 𝜈 is computed as: +𝐴EE(𝜈) = 𝐴EE +� +𝜈 +11.1 GHz +�2𝛽s,EE +(25) +where 𝐴EE represents the EE amplitude in the MFI 311 map. There- +fore, for this fit, we have seven parameters, namely 𝐴EE and 𝛼EE for +the amplitude and angular dependence of the synchrotron signal at +11 GHz; the spectral index 𝛽s,EE describing the frequency depen- +dence, and four constant coefficients 𝑐11 +EE, 𝑐13 +EE, 𝑐17 +EE and 𝑐19 +EE, ac- +counting for the unresolved source contributions at each frequency. +For this analysis, we also introduce the colour correction term based +on the fitted spectral index, using values reported in Table 4. For +the |𝑏| > 5◦ mask, we obtain 𝐴EE = 1.48 ± 0.13 𝜇K2, 𝛼EE = +−2.97 ± 0.13 and 𝛽s,EE = −2.99 ± 0.14. Similarly, we repeat the +analysis for the BB power spectra, finding 𝐴BB = 0.47 ± 0.12 𝜇K2, +𝛼BB = −3.14 ± 0.33, and 𝛽s,BB = −2.79 ± 0.35. +In both cases, the first two parameters are in agreement with +the values reported in Table 22, taking into account that the colour +correction term for the 11 GHz map and for a spectral index of +𝛽 ≈ −3 is 0.967. The power spectrum of the synchrotron emission +detected in the MFI wide survey maps scales with an average index +of −2.99 for EE. The BB analysis is consistent with this value. The +weighted average of the two values is −2.96 ± 0.13, consistent with +the result of −2.96 ± 0.09 for the 50 per cent mask obtained in +Martire et al. (2022) for the combination of WMAP-K and LFI30 +data. Our value also agrees with the study carried out in the next +section for a real space analysis. A more detailed analysis on the +reconstruction of the synchrotron spectral index with QUIJOTE +MFI wide survey data is presented in two accompanying papers (de +la Hoz et al. 2023a; Vansyngel et al. 2023). +8 +BASIC PROPERTIES OF THE WIDE SURVEY MAPS +8.1 +Spectral index of the MFI sky emission +8.1.1 +Intensity +We first investigate the spectral dependence of the intensity emission +in the MFI wide survey maps. We use as a reference the MFI 11 GHz +map, which presents the largest signal-to-noise, and we evaluate the +spectral index of the sky emission when comparing it to the Haslam +408 MHz (Haslam et al. 1982) and WMAP-K 9-year maps (Bennett +et al. 2013). The version of the Haslam map used here corresponds +to the destriped map from Remazeilles et al. (2015). For this spectral +analysis in real space, all external maps are filtered using the FDEC +procedure, degraded to 2◦ angular resolution, and then downgraded +to 𝑁side = 64 resolution. Zero levels of all maps are corrected as +in Sect. 5.4. Colour corrections for MFI-311 and WMAP-K are +taken into account. The analysis region is restricted to the sky area +covered by MFI 11 GHz, but excluding the satellite band (satband) +as described in Sect. 3.1. For each 𝑁side = 64 pixel 𝑝 within the +allowed mask, we solve for the spectral index 𝛽(𝑝) using a standard +gaussian likelihood function L, which for the case of Haslam and +MFI 11 GHz reads +−2 ln L(𝑝) = +� +𝐼408(𝑝) +� +11.1 +0.408 +�𝛽( 𝑝) +− 𝑐𝑐11(𝛽(𝑝))𝐼11(𝑝) +�2 +𝜎(𝑝)2 +, +(26) +where 𝑐𝑐11 is the colour correction for MFI 11 GHz, and the noise +term 𝜎 is evaluated using 1000 noise simulations for MFI (see +Sect. 6.2), and accounting for a 10 per cent calibration error in the +Figure 33. Spectral index of the intensity emission in the QUIJOTE 11 GHz +map. Top: Spectral index of 𝛽408MHz−11GHz. The average index is 𝛽 = −2.9. +As expected, the Galactic plane regions have a flatter index, while the regions +off the plane have steeper values. Bottom: Spectral index of 𝛽11GHz−23GHz. +The average spectral index in this case is 𝛽 ≈ −2.6. In this colour scale, +dark red corresponds to AME dominated regions. +Haslam map. Similarly, for the spectral index in intensity between +MFI 11 GHz and WMAP-K, we use the same approach, accounting +for the WMAP noise levels in the evaluation of the noise term. +Figure 33 shows the results for the case of 𝛽408MHz−11GHz +(top panel) and 𝛽11GHz−23GHz (bottom panel), both for the inten- +sity emission. Figure 35 shows a histogram with the distribution of +spectral indices in both maps. The median intensity spectral index +𝛽408MHz−11GHz in the full analysis mask is −2.90, with a standard +deviation of the values across the map of 0.20. This value is con- +sistent with the expectation for the average synchrotron emission at +these frequencies (see e.g. Platania et al. 1998; de Oliveira-Costa +et al. 1999; Fernández-Cerezo et al. 2006). Moreover, the spatial de- +pendence confirms the well-known steepening of the spectral index +at high Galactic latitudes (see e.g. the 408 MHz–23 GHz spectral +index map in Bennett et al. 2003). +The 𝛽11GHz−23GHz intensity spectral index presents a much +broader distribution of values, due to the presence of multiple spec- +tral components (AME, free-free and synchrotron). In order to avoid +extreme values for low signal-to-noise (high Galactic latitude) pix- +els, in this case we also add a broad gaussian prior 𝛽 = −3.1±0.5 to +the likelihood in equation 26. We have checked that this has a mini- +mal impact in the final histogram. The median spectral index in this +case is −2.59, and the standard deviation of the values is 0.43. Some +of the bright AME dominated regions (Perseus, Lambda Orionis and +rho Ophiucus) are clearly visible in dark red colour, while free-free +dominated regions (e.g. Cygnus area) appear as light red. A more +detailed study of the spectral properties of the sky emission in in- +tensity along the Galactic plane (|𝑏| ≤ 10◦) in the MFI wide survey +MNRAS 000, 1–58 (2022) + +Spectral index in intensity (Haslam to MFll1) +-3.4 +β408MHz - 11GHz +-2.2Spectral index in intensity (MFIl1 to WMAP-K) +-3.4 +β11GHz - 23GHzQUIJOTE MFI wide survey +41 +Figure 34. Top: Spectral index map of the polarized emission between +QUIJOTE 11 GHz and WMAP 23 GHz. Bottom: the associated error map. +maps is carried out in an accompanying paper (Fernandez-Torreiro +et al. 2023). We also present a component separation analysis of the +full MFI maps in de la Hoz et al. (2023b). +8.1.2 +Polarization +In polarization, the 𝛽11GHz−23GHz spectral index presents a cleaner +interpretation in this case, as we are dominated by synchrotron +emission only. Figure 34 presents the recovered polarization spec- +tral index map, following the same methodology as for the intensity. +The fit is carried out simultaneously in Stokes Q and U parameters, +and in order to obtain a stable solution for high Galactic latitude +pixels, we add a Gaussian prior 𝛽 = −3.1 ± 0.3 to the likelihood +in equation 26. The bottom panel in that figure shows the asso- +ciated error map, derived from the posterior distribution. Fig. 35 +includes also the histogram of these polarization 𝛽11GHz−23GHz +values, showing that the median value is −3.09, and the standard +deviation is 0.14. For comparison, we also include in this figure +the histogram of spectral index values for the PySM synchrotron +model 1 (Thorne et al. 2017), which in turn corresponds to "Model +4" of Miville-Deschênes et al. (2008) calculated from a combi- +nation of Haslam and WMAP 23 GHz polarization data using a +model of the Galactic magnetic field. We find that in the same sky +mask, the PySM spectral index map peaks at a higher value and +presents a much narrower distribution (−2.99 ± 0.06). As a further +consistency check, Appendix F presents the results for the same +analysis carried out in this section, but using the MFI 13 GHz map +as reference. We can see that both the mean values and widths of +the distributions discussed here are consistently reproduced in this +case. These values for 𝛽11GHz−23GHz in polarization are consistent +with those measured in the range 22.8–100 GHz (𝛽s ∼ −3.1) by +Figure 35. Histogram of spectral index values obtained from Figures 33 and +34. We show in dashed lines the mean of the prior adopted in the determina- +tion of the spectral index in polarization. For comparison, we also include +the histogram of spectral index values from the PySM synchrotron model 1 +(Thorne et al. 2017). We recall that in the intensity case for 𝛽11GHz−23GHz +(blue line), the 11 GHz map contains free-free and AME in addition to syn- +chrotron, and thus the histogram presents a different shape with a broader +distribution (see text for details). +other authors (Dunkley et al. 2009; Fuskeland et al. 2014, 2021; +Harper et al. 2022). A more detailed study of the spectral properties +of the sky emission in polarization using the MFI wide survey maps +in combination with WMAP and Planck, including a discussion on +synchrotron spectral curvature, is carried out in an accompanying +paper (de la Hoz et al. 2023a). +8.2 +E- and B-mode maps +As a complementary view of the relative power distribution in the E- +and B-mode components for the synchrotron emission traced by the +QUIJOTE MFI wide survey map, we have obtained in this section +E- and B-mode maps. We use the full QUIJOTE observed area, but +we mask the satellite band (satband) as described in Sect. 3.1. In +order to minimize the impact of E/B mixing (Lewis et al. 2001), +we apodize this analysis mask using a Gaussian kernel of 2◦. E- +and B-mode maps are then generated using the standard HEALPix +routines anafast and synfast, as +𝐸( ˆ𝑛) = +∞ +∑︁ +ℓ=2 +ℓ +∑︁ +𝑚=−ℓ +𝑎E +ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛) +𝐵( ˆ𝑛) = +∞ +∑︁ +ℓ=2 +ℓ +∑︁ +𝑚=−ℓ +𝑎B +ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛), +(27) +where 𝑎E +ℓ,𝑚 and 𝑎B +ℓ,𝑚 are the corresponding harmonic coefficients. +Figure 36 shows the derived maps for MFI 11 GHz. As ex- +pected from the power spectrum analysis in Sect. 7, there is sig- +nificantly more power in the E-mode than in the B-mode map. +Moreover, most of the brightest synchrotron features in the polar- +ized intensity map (North Polar Spur, Fan region, Galactic centre) +appear mostly in the E-mode, as expected due to the underlying +magnetic field structure. Strongly polarized radio sources (Tau A, +Cyg A) appear in these E- and B-mode maps with the characteristic +quadrupole patterns with two positive and two negative lobes, and +with the B-mode profile rotated by 45◦ with respect to the E-mode +map (see e.g. Diego-Palazuelos et al. 2021). +MNRAS 000, 1–58 (2022) + +-3.4 +β11GHz - 23GHz +-2.7Error in the spectral index in polarization (MFll1 to WMAP-K) +0 +(β11GHz - 23GHz) +0.41.2 +Intensity（408MHz-11GHz） +Intensity +(11GHz-23GHz +1.0 +Polarization +（11GHz-23GHz +Prior β=-3.1±0.3 +(normalized) +PySM +0.8 +0.6 +count +0.4 +ixel +0.2 +0.0 +-4 +-3 +-2 +Spectral index β42 +Rubiño-Martín et al. +Figure 36. E and B-mode maps at 11 GHz. Most of the brightest features in +the QUIJOTE map (North Polar Spur, Fan region, Galactic plane) appear in +the E-mode map. +8.3 +Bright structures in the polarized intensity maps +The MFI wide survey polarized intensity maps are dominated by +several bright and extended structures (see Figure 5). We discuss +some of them in four accompanying papers: the Fan region (Ruiz- +Granados et al. 2023), the Haze and Galactic center (Guidi et al. +2023), the North Polar Spur (Watson et al. 2023), and other syn- +chrotron loops and spurs (Peel et al. 2023). +8.4 +AME in the MFI wide survey maps +The MFI wide survey maps can be used to characterize the spec- +tral properties of the AME, both in intensity and polarization. In +particular, in Fernandez-Torreiro et al. (2023) we present a study +of the diffuse AME emission in intensity along the Galactic plane +(|𝑏| ≤ 10◦), while Poidevin et al. (2023) characterizes the SED in +intensity for 52 compact sources with AME. Finally, two additional +papers update the constraints in intensity and polarization of the +AME in several Galactic regions (Tramonte et al. 2023; Lopez- +Caraballo et al. 2023). +9 +BRIGHT COMPACT SOURCES AND PLANETS IN THE +WIDE SURVEY +Despite its coarse angular resolution a high number of point sources +are detected to high significance in the QUIJOTE MFI wide survey +data. In a companion paper, where we discuss radio source de- +tectability in these maps and derived statistical properties (Herranz +et al. 2023), we show that we detect 235 point sources at S/N> 3 +at 11 GHz, while 85 are detected at S/N> 5. As a further consis- +tency check of the global amplitude calibration, in this section we +compare with models the recovered flux densities on four of the +brightest sources having well characterised spectra (Tau A, Cas A, +Cyg A and 3C274), and in two planets (Jupiter and Venus). We +also calculate polarization flux densities in three bright polarized +sources (Tau A, Cyg A and W63) to assess the accuracy of the +polarization calibration. +9.1 +Compact sources in intensity +Tau A (also known as the Crab nebula), Cas A and Cyg A are +amongst the brightest compact sources in the microwave range, and +hence they have traditionally been used to calibrate experiments op- +erating in this frequency range, including CMB experiments (Baars +et al. 1977). Using WMAP data, Weiland et al. (2011) presented +updated spectrum models in the range ∼ 1–300 GHz of these three +sources and of 3C274 (also known as Virgo A or M87) and 3C58. +Here we will focus on Tau A, Cas A, Cyg A and 3C274, while 3C58 +will be discussed in detail in Ruiz-Granados et al. (2023). +Figure 37 shows the MFI wide survey maps on the positions of +these four sources (Tau A, Cas A, Cyg A and 3C274) at 11 GHz and +19 GHz, smoothed to a common angular resolution of 1◦. We note +that Tau A and Cas A are the two main calibrators of QUIJOTE MFI, +and thus we have much more sensitive data on these two sources +obtained in raster mode. However we focus here on the wide survey +maps only, in order to provide another consistency check for the +calibration scheme. +We extracted total-intensity flux densities on these maps us- +ing a beam-fitting photometry (BF1d), consisting in fitting a 1◦- +FWHM Gaussian beam superimposed on a flat background. We +applied colour corrections following the methodology described in +Génova-Santos et al. (2023), and using for each source a spectral +index derived from the model. We compare these flux densities with +spectral emission models that we have specifically derived for these +sources, and which will be presented in a separate paper (Génova- +Santos & Rubiño-Martín, in preparation). While in that paper we +discuss models extracted with different photometry techniques, here +we compare with models derived from WMAP and Planck maps +convolved to a common resolution of 1◦ and using the same BF1d +technique that we applied to QUIJOTE MFI. In particular, the Tau A +model was used in Sect. 2.6 to recalibrate the wide survey maps. As +it will be discussed in depth in Génova-Santos & Rubiño-Martín (in +prep.), the uncertainties of these models are of the order of 3–5 %, +and are driven not by the statistical noise of the individual obser- +vations which is well below this value, but by systematic effects +and calibration uncertainties of the fitted data, which lead to higher +model-fitting residuals than would be expected in the presence of +just statistical errors. In the cases of Tau A and Cas A, modelling of +their secular decrease also introduces significant uncertainty. +Final QUIJOTE MFI flux densities, for each horn and fre- +quency, and relative deviation with respect to the fitted intensity +models, are quoted in Table 24. All values are referred to date +2016.3 (1 April, 2016), which roughly corresponds to the middle +of the wide survey observations. It can be seen that in most cases +the measured flux densities deviate less than 3–5 % with respect +to the models, while in the case of Tau A, which is the main am- +plitude calibrator, the deviations are within 1 % (the difference is +not exactly zero due to the way the different periods are calibrated +and combined; see section 2.6). The level of these deviations is +expected given the typical model uncertainties, and therefore these +results give full confidence to our global calibration strategy and the +MNRAS 000, 1–58 (2022) + +MFI 11GHz - E modes +mKMFI 11GHz - B modes +mKQUIJOTE MFI wide survey +43 +Figure 37. Minimaps of 5◦ × 5◦ size around four bright radiosources: Tau A (first row), Cas-A (second row), Cygnus-A (third row), and 3C274 (bottom row) +at 11 GHz (first three columns are I, Q, and U), and 19 GHz (columns 4 to 6 are I, Q, U, respectively). For display purposes, we use the MFI maps degraded to +a common angular resolution of one degree. +quoted uncertainty (see Table 16). A detailed discussion on the vari- +ability of these four sources (and others in the wide survey maps) +can be found in Herranz et al. (2023). +9.2 +Planets in intensity +Venus and Jupiter are also detected to high significance in the QUI- +JOTE MFI wide survey data. Owing to its orbital motion Venus +declination varies roughly between ±27◦. Given Tenerife’s latitude +(28.3◦ N), when its declination is close to 27◦ it is always visible in +any of the elevations considered in the wide-survey. On the contrary, +when it reaches its minimum declination of −27◦ it culminates at +elevation 35.5◦, and therefore it is only picked up in observations at +elevations 30 or 35◦. The distance between this planet and the Earth +changes between 0.27 and 1.74 A.U., meaning that there is a factor +≈ 42 variation between its minimum and maximum brightness. At +19 GHz its flux density is expected to vary between 10.9 and 445 Jy. +Then, during its inferior conjunction it is amongst the brightest +sources on the sky at the QUIJOTE MFI frequencies. In the case +of Jupiter, being an external planet, this variation is much smaller. +Its distance to Earth varies between 4.1 and 6.4 A.U., producing a +variation of its flux density at 19 GHz between 26.1 and 61.1 Jy. +Between 2012 and 2016 its declination was always positive, reach- +ing 23◦, meaning that it was picked up in most of the wide survey +data. Between 2016 and 2018 its declination dropped below zero, +reaching −22◦, and therefore during this period it was only visible +on the wide survey observations performed at low elevations. +While further details will be given in a future paper where +we will discuss planets and other bright astronomical sources, here +we briefly describe the procedure we have developed to estimate +planets’ brightness temperatures. We implemented a specific map- +making in which we rotate the coordinates of QUIJOTE MFI wide +survey data to planet-centred coordinates, to produce planet-centred +maps. We use the same final calibrated data that were used to pro- +MNRAS 000, 1–58 (2022) + +50100150200250300 +-20 +-15 +-10 +-5 +0 +0.5 +0.0 +0.5 +1.0 +mKcMB +mK +CMB +-4 +(deg) +-5 +-6 +b +-7 +-8 +Tau A I 11 GHz +Tau A Q 11 GHz +Tau A U 11 GHz +187186185184183 +187186185184183 +187186185184183 +1 (deg) +1 (deg) +1 (deg)0 +20 +40 +60 +80 +100 +-6 +-4 +-2 +0 +0.0 +0.1 +0.2 +0.3 +mKcMB +-4 +(deg) +-5 +-6 +b +-7 +-8 +Tau A I 19 GHz +Tau A Q 19 GHz +Tau A U 19 GHz +187186185184183 +187186185184183 +187186 185 184183 +1 (deg) +1 (deg) +1 (deg)50 +100 +150 +200 +250 +0.5 +0.0 +0.5 +1.0 +0.5 +0.0 +0.5 +1.0 +mK +CMB +mK +CMB +0 +-1 +(deg) +2- +b +-3 +-4 +Cas A I 11 GHz +Cas A Q 11 GHz +Cas A U 11 GHz +114 113 112 111110 +114 113 112 111 110 +114 113 112 111 110 +1 (deg) +1 (deg) +1 (deg)10 +20 +30 +40 +50 +60 +-0.2 +-0.1 +0.0 +0.1 +0.050.000.050.100.15 +mK, +CMB +mK, +CMB +0 +-1 +(deg +2- +b +-3 +-4 +Cas A I 19 GHz +Cas A Q 19 GHz +Cas A U 19 +GHz +114 113 112 111 110 +114113112111110 +114 113 112111110 +1 (deg) +1 (deg) +1 (deg)20 +40 +60 +80 +100 +0 +1 +2 +3 +-4 +-3 +-2 +-1 +0 +mKcMB +8 +7 +00 +(deg +6 +b +5 +4 +Cyg A I 11 GHz +Cyg A Q 11 + GHz +Cyg A U. 11 GHz +78 +76 +75 +74 +78 +76 +75 +74 +78 +76 +75 +74 +1 (deg) +1 (deg) +1 (deg)0 +5 +10 +15 +20 +25 +30 +0.0 +0.10 +0.20 +0.3 +-0.2 +0.1 +0.0 +8 +7 +80 +(deg +6 +b +5 +4 +Cyg A I 19 GHz +Cyg. A.. +Q +19 +GHz +Cyg A U 19 GHz +78 +76 +75 +74 +78 +76 +75 +74 +78 +76 +75 +74 +1 (deg) +1 (deg) +1 (deg)0 +5 +10 +15 +20 +25-0.3-0.2-0.10.00.10.2 +0.4 +-0.2 +0.0 +0.2 +mKcMB +76 +80 75 +(deg +b +74 +73 +3C274 I 11 GHz +3C274 +11 GH: +C274 U 11 GHz +72 +286 285 284 283 282 +286 285 284 283 282 +286 285 284 283 282 +1 (deg) +1 (deg) +1 (deg)0 +1 +2 +3 +4 +5 -0.10 +-0.05 +0.00 +0.05 +0.10 +0.0 +0.10 +mK +CMB +mK, +CMB +76 +a0 75 +(deg +b +74 +73 +3C274 I 19 GHz +3C274 Q 19 GHz +3C274 U 19 GHz +72 +286285284283282 +286285284283282 +286 285 284 283 282 +1 (deg) +1 (deg) +1 (deg)44 +Rubiño-Martín et al. +Table 24. Flux densities (Jy), in intensity and in polarization, extracted from the QUIJOTE MFI wide survey maps at one degree resolution on Tau A, Cas A, +Cyg A and 3C274. Intensity measurements are based on BF1d photometry, while the polarization measurements used AP1d. For the intensity measurements, +inside parentheses we quote the percent deviation of flux densities with respect to predictions from spectral models. Tau A and Cas A values are referred to an +effective date corresponding to 1 April 2016. All flux densities include colour corrections. +Source +Stokes +311 (11.1 GHz) +313 (12.9 GHz) +217 (16.7 GHz) +417 (17.0 GHz) +219 (18.7 GHz) +419 (19.0 GHz) +Tau A +I +440.0 ± 0.9 (-0.8) +427.4 ± 0.8 (+0.7) +391.2 ± 0.8 (-0.5) +393.4 ± 0.8 (+0.6) +377.9 ± 0.7 (-0.6) +378.8 ± 0.8 (+0.2) +Q +−29.27 ± 0.51 +−31.20 ± 0.51 +−28.00 ± 0.83 +−28.12 ± 0.43 +−26.36 ± 1.52 +−28.42 ± 0.66 +U +0.63 ± 0.51 +0.90 ± 0.73 +1.43 ± 0.89 +1.05 ± 0.63 +0.34 ± 0.89 +1.87 ± 0.59 +Cas A +I +340.9 ± 1.8 (-1.1) +309.7 ± 1.8 (-0.5) +255.8 ± 1.9 (-2.4) +256.3 ± 1.9 (-1.0) +236.2 ± 2.1 (-2.8) +235.7 ± 1.9 (-2.0) +Q +−1.18 ± 0.62 +−0.01 ± 0.53 +0.32 ± 0.57 +−0.93 ± 0.32 +−0.34 ± 0.65 +−1.25 ± 0.64 +U +0.15 ± 0.34 +−0.90 ± 0.39 +0.26 ± 0.47 +−0.28 ± 0.45 +1.18 ± 0.72 +0.29 ± 0.51 +Cyg A +I +129.3 ± 1.0 (-4.1) +108.7 ± 1.0 (-3.5) +79.2 ± 1.0 (-4.3) +78.1 ± 1.0 (-3.5) +69.5 ± 0.9 (-3.8) +67.5 ± 1.0 (-4.8) +Q +3.93 ± 0.61 +1.69 ± 0.64 +−0.55 ± 0.54 +0.41 ± 0.45 +−1.24 ± 0.66 +0.59 ± 0.38 +U +−5.95 ± 0.44 +−4.64 ± 0.39 +−2.23 ± 0.59 +−1.60 ± 0.45 +−1.52 ± 0.98 +−1.26 ± 0.44 +3C274 +I +34.2 ± 0.1 (-5.3) +30.9 ± 0.1 (-3.8) +25.6 ± 0.2 (-3.1) +25.9 ± 0.2 (-0.5) +22.3 ± 0.3 (-8.1) +24.0 ± 0.3 (+0.2) +Q +−0.26 ± 0.48 +0.39 ± 0.52 +−0.54 ± 0.77 +−0.19 ± 0.38 +0.45 ± 1.10 +−0.24 ± 0.48 +U +−0.74 ± 0.44 +−0.81 ± 0.56 +−2.14 ± 0.72 +−0.97 ± 0.45 +−2.63 ± 1.19 +−1.35 ± 0.47 +duce the final maps that are presented in this paper. In order to ac- +count for the 1/𝑑2 effect we define distance bins (3 bins for Jupiter +and 6 for Venus), and produce individual maps for each bin. We +have verified in the final map that the (symmetrized) beam shape +is well preserved, this being a health check both for the tailored +map-making that we use here as well as for the pointing model. On +these maps we apply a beam-fitting photometry to derive flux den- +sities for each distance bin and for each horn/frequency. These flux +densities are then colour-corrected using a Rayleigh-Jeans spectrum +(spectral index 𝛼 = 2). In addition to data maps for each redshift bin +we produce maps of 1/𝑑2 using the same noise weights and flags +that are applied to the data. These maps are later used to calculate +an effective distance at the position of the planet. Using this infor- +mation we fit the flux densities measured in each bin to a 1/𝑑2 law +in order to derive the final brightness temperatures. +Our Venus and Jupiter brightness temperatures derived from +QUIJOTE MFI are listed in Table 25 and plotted in Figure 38, in +comparison with other data at similar frequencies, as well as with +various models giving the spectral dependency of the brightness +temperatures of these planets. In both cases we have corrected for the +planet absorption of the CMB monopole, and therefore the quoted +values represent the intrinsic brightness temperature of the planets. +In the case of Venus, it is seen that the ancillary measurements seem +a bit high with respect to the Bellotti (2015) and Fahd (1992) models +and therefore we performed a power-law fit to the data in the range +7–100 GHz (dashed line in the figure) and use this fit as a reference +to compare with the QUIJOTE MFI values. In the case of Jupiter we +use as reference the model of Karim et al. (2018), which seems to +trace better the ancillary data, and in particular the VLA data from +de Pater et al. (2019) below the ammonia absorption at 23 GHz. As +can be seen in Table 25, both for Venus and Jupiter the QUIJOTE +MFI measurements deviate always less than 5 % from the models +(note that in some cases the statistical error bar is larger than this +value), which bestows confidence to our calibration strategy. +9.3 +Polarized sources +Figure 37 also shows wide-survey polarization maps of Tau A, +Cas A, Cyg A and 3C274, projected in Galactic coordinates and +convolved to an angular resolution of one degree. Clear polarized +emission is seen in Tau A, mainly concentrated in the 𝑄 map, +as expected due to its polarization angle (see e.g. Weiland et al. +2011). The 𝑈 map shows the typical cloverleaf pattern (with the +expected peak-to-peak amplitude of ∼ 1 % with respect to the total +intensity) arising from the differences between the two co-polar +beams (Génova-Santos et al. 2023). This pattern is also visible in the +𝑄 and 𝑈 maps of Cas A, more notably at 11 GHz. Due to it being a +very young shell-type supernova remnant (SNR), the magnetic field +of Cas A is expected to be radial (Anderson et al. 1995). Being ∼ 5′ +across, this source is unresolved by the QUIJOTE MFI beam and +therefore we expect zero integrated polarization. Clear polarized +emission is also seen in Cyg A. A rotation of the polarization angle +is apparent between 11 and 19 GHz, which is due to the two jets of +this radio galaxy having different rotation measures, the so-called +Laing-Garrington effect (Laing 1988). In the case of 3C274, we +only have a marginal polarization detection in the 𝑈 maps. This is +expected given our noise levels (between 0.5–1 Jy), and the fact that +the measured polarization fraction at 23 GHz is approximately 4 % +(Weiland et al. 2011). +In order to minimize systematic effects introduced by differ- +ences between the two co-polar beams, we extract flux densities +in polarization through an aperture photometry technique on maps +smoothed to one-degree angular resolution (AP1d). The circular +aperture radius (𝑟1) is taken to be 𝑟1 = 1.5◦ for Tau A and 3C274, +and 𝑟1 = 1.3◦ for Cas A and Cyg A, due to the larger foreground +contamination in the surroundings of the latter two sources. In the +case of Cas A, we also mask the region centred at Galactic co- +ordinates (𝑙, 𝑏) = (111.11◦, −0.53◦), using an exclusion radius of +0.7◦. The background emission in all four cases is corrected using +the mean of the signal in the annulus between 𝑟1 and 𝑟2 = +√ +2𝑟1. +Table 24 shows the Stokes 𝑄 and 𝑈 flux densities measured on Tau +A, Cas A, Cyg A and 3C274. +We now discuss the first three cases in detail, as well as the +bright polarized emission in W63. For this discussion, we also ap- +ply the same methodology (i.e. AP1d for polarization and BF1d for +intensity) to derive the photometry values for these sources using +WMAP 9-year data (Bennett et al. 2013) and Planck 2018 maps +(Planck Collaboration et al. 2020a) at the common one-degree res- +olution. Specially for the cases of Tau A and Cas A, and for Planck +LFI, we correct for the intensity-to-polarization leakage due to band- +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +45 +Table 25. Brightness temperatures (in Kelvin) of Jupiter and Venus extracted from the QUIJOTE MFI wide survey data. Inside parentheses we quote the +percent deviation with respect to predictions from spectral models. +Planet +311 (11.1 GHz) +313 (12.9 GHz) +217 (16.7 GHz) +417 (17.0 GHz) +219 (18.7 GHz) +419 (19.0 GHz) +Jupiter +176.3 ± 0.4 (+0.8) +170.3 ± 1.7 (+2.6) +153.9 ± 8.7 (+1.0) +148.7 ± 2.2 (-1.9) +145.7 ± 14.5 (-0.9) +141.0 ± 8.0 (-3.6) +Venus +578.3 ± 14.4 (-4.6) +568.2 ± 4.7 (-2.5) +546.6 ± 5.7 (+0.4) +533.1 ± 9.5 (-1.6) +526.7 ± 10.9 (-0.4) +518.0 ± 11.5 (-1.6) +Figure 38. Venus (top) and Jupiter (bottom) brightness temperatures derived +from the QUIJOTE MFI wide survey data (red and yellow) in comparison +with ancillary data, and with various models. Venus data have been obtained +from Bellotti (2015); Hafez et al. (2008); Dahal et al. (2021), while the +plotted models are from Bellotti (2015); Fahd (1992). We also show a +power-law fit to the observed data in the range 7–100 GHz that we use to +compare with the QUIJOTE MFI measurements. Jupiter data come from +Hafez et al. (2008); Gibson et al. (2005); de Pater et al. (2019); Karim et al. +(2018); Weiland et al. (2011); Planck Collaboration et al. (2016e, 2017b). +We plot the model by Karim et al. (2018) and the ESA1 model. +pass mismatch following the methodology described in Appendix C +of Planck Collaboration et al. (2016h), using the maps of projection +factors described in Planck Collaboration et al. (2016c), and the +spectral index of each source derived from the intensity SED. +9.3.1 +Tau A and Cyg A +Figure 39 shows our results for the polarization fractions in Tau A +and Cyg A at one degree resolution. We include also our WMAP (for +both sources) and Planck (only for Tau A) measurements, as well +as ancillary measurements both for Tau A (Kuz’min & Udal’Tsov +1959; Mayer & Sloanaker 1959; Mayer et al. 1962; Davies & Ver- +schuur 1963; Hollinger et al. 1964; Mayer et al. 1964; Morris & +Berge 1964; Boland et al. 1966; Gardner & Whiteoak 1966; Hobbs +& Haddock 1967a; Sastry et al. 1967; Satoh et al. 1967; Hobbs +1968; Hollinger & Hobbs 1968; Mayer & Hollinger 1968; Seielstad +& Weiler 1968; Johnston & Hobbs 1969; Dmitrenko et al. 1970; +Wright 1970; Green et al. 1975; Hafez et al. 2008; Aumont et al. +2010) and for Cyg A (Mayer et al. 1962; Hollinger et al. 1964; Sobol- +eva 1966; Boland et al. 1966; Mezger & Schraml 1966; Hobbs & +Haddock 1967b). +For Tau A, we show in solid grey lines Monte Carlo realiza- +tions of a simple model for the spectral dependency of its polar- +ization fraction that accounts for the Faraday depolarization (Burn +1966), and that is consistent with both the ancillary measurements +at low frequencies (𝜈 ≲ 10 GHz) and existing measurements of +the Faraday dispersion in the region (e.g. Bietenholz & Kronberg +1991). This model will be described in detail in a separate paper +(Génova-Santos & Rubiño-Martín, in preparation). We note that in +our recalibration strategy of the MFI wide survey maps, we use +Tau A for fixing the intensity calibration scale and the polarization +angle, while the polarization amplitude is essentially given by inde- +pendent polarization efficiency measurements. Thus, this analysis +provides a consistency test on the MFI polarization calibration. +Cyg A data points clearly show the effect of the Faraday de- +polarization produced by the Laing-Garrington effect. It is evident +from this plot that the maximum alignment between the polarization +directions of the two jets occurs at ≈ 10 GHz, and then the measured +polarization fractions decrease in both sides of the spectrum. Mod- +elling this effect is complicated and beyond the scope of this paper. +The QUIJOTE MFI measurements are in good agreement with the +other measurements, again providing confidence in our calibration +strategy. +9.3.2 +Cas A +Figure 40 shows the polarization fraction measured in Cas A with +QUIJOTE MFI. We also include our photometry results for WMAP +and Planck, and ancillary data from the literature (Mayer et al. 1962; +Hollinger et al. 1964; Hobbs & Haddock 1967a; Sastry et al. 1967; +Seielstad & Weiler 1968; Vinyaikin 2014). All values are noise- +debiased using the PMAS estimator (Plaszczynski et al. 2014). The +intensity-to-polarization leakage due to the co-polar beam asym- +metry is almost cancelled in the integrated flux densities thanks to +the positive and negative structure of this pattern, leading to inte- +grated polarization fractions in QUIJOTE of around ∼ 0.3 %. Note +that similar levels are detected in WMAP, and could also be due to +beam effects as discussed in Weiland et al. (2011). At face value, +these numbers can be considered as a conservative upper limit on +the overall intensity-to-polarization leakage in the MFI wide survey +maps. +MNRAS 000, 1–58 (2022) + +46 +Rubiño-Martín et al. +Figure 39. Consistency checks on polarized sources detected on the QUI- +JOTE MFI the wide survey. We show polarization fractions measured in +Tau A and in Cyg A, in comparison with our WMAP and Planck results +obtained using the same methodology, and with ancillary measurements +(see the complete list of references in the main text). In the case of Tau +A we overplot in grey models for the polarization fraction that account for +Faraday rotation. The Cyg A data show the Laing-Garrington effect arising +from different rotation measures in the two lobes of this galaxy. +9.3.3 +W63 region +As an additional test of the polarization calibration of the MFI, +we also investigate the polarized intensities of W63, another SNR +which appears as a very bright extended structure in the polarization +maps at these frequencies. The top panel in Fig. 41 shows the MFI +11 GHz Stokes I, Q and U maps for this object. The total-intensity +emission of W63 is practically embedded inside the emission of +the Cygnus X star-forming complex, so it is difficult to extract re- +liable total-intensity flux density estimates in this case. However, +the polarization signal is reasonably isolated. Thus, we only discuss +its polarized flux density here. In order to capture all the flux in +the region, we use an aperture radius of 𝑟1 = 2◦. As in the pre- +vious cases, we carry out this analysis in the smoothed maps at +one degree resolution (i.e. AP1d photometry). The bottom panel in +Fig. 41 shows the SED in polarized intensity 𝑃 = +√︁ +𝑄2 + 𝑈2 derived +from our photometry measurements, including also our results for +WMAP and Planck applying the same methodology. All values are +noise-debiased using the PMAS estimator. Error bars account for +the photometry error plus the corresponding calibration uncertain- +ties added in quadrature. For MFI, we use the values reported in +Figure 40. Polarization fractions measured on Cas A in QUIJOTE MFI wide +survey data, in comparison with other measurements. WMAP and Planck +results are obtained using the same methodology as for the MFI maps values. +The complete list of ancillary measurements is given in the main text. Upper +limits are represented with arrows. At degree-beam scales, the polarized +emission of Cas A is expected to be zero, so these measurements serve as +a consistency check for the overall intensity-to-polarization leakage of the +MFI wide survey maps. +Table 16, while for WMAP and Planck data we adopt the conser- +vative value of 3 %, as done for similar analyses (see e.g. Planck +Collaboration et al. 2014a; Poidevin et al. 2019; Cepeda-Arroita +et al. 2021). The WMAP and Planck polarized intensity flux in +W63 can be fitted to a power-law 𝑃 = 6.97(𝜈/22.8GHz)−0.68 Jy, +that is depicted by the dashed line. The QUIJOTE MFI data are con- +sistent within 1-sigma with the fitted model, which gives additional +confidence to our calibration strategy in polarization. +10 +DATA RELEASE AND DESCRIPTION OF THE DATA +PRODUCTS +Together with this paper, there is a series of further publications +containing scientific results derived from the QUIJOTE-MFI wide +survey maps presented here. The titles of all the papers in the series +begin with "QUIJOTE scientific results", and comprise: +IV. A northern sky survey at 10–20 GHz with the Multi- +Frequency Instrument (this paper). +V. The microwave intensity and polarization spectra of the +Galactic regions W49, W51 and IC443 (Tramonte et al. 2023). +VI. The Haze as seen by QUIJOTE (Guidi et al. 2023). +VII. Galactic AME sources in the QUIJOTE-MFI North Hemi- +sphere Wide-Survey (Poidevin et al. 2023). +VIII. Diffuse polarized foregrounds from component separation +with QUIJOTE-MFI (de la Hoz et al. 2023a). +IX. Radio sources in the QUIJOTE-MFI wide survey maps (Her- +ranz et al. 2023). +X. AME variability along the Galactic Plane in the QUIJOTE- +MFI wide survey (Fernandez-Torreiro et al. 2023, in prep.). +XI. Polarized synchrotron loops and spurs in the QUIJOTE-MFI +wide survey (Peel et al. 2023, in prep.). +XII. Analysis of the polarized synchrotron emission at the power +spectrum level in the MFI wide survey (Vansyngel et al. 2023, in +prep.). +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +47 +Figure 41. Top: Minimaps of 6◦ × 6◦ size around W63 for the MFI 11 GHz +I, Q, and U maps at one degree angular resolution. A circle with radius of 2◦ +indicates the integration area for the photometry analysis. Bottom: Polarized +intensity measurements on W63 with the QUIJOTE MFI wide survey data, +in comparison with WMAP and Planck measurements. We overplot with +a dashed line a power-law fit representing the spectrum of the synchrotron +emission fitted to the WMAP and Planck data. +XIII. SNRs in the QUIJOTE-MFI wide survey (Lopez-Caraballo +et al. 2023, in prep.). +XIV. The FAN region as seen by QUIJOTE-MFI (Ruiz- +Granados et al. 2023, in prep.). +XV. The North Polar Spur as seen by QUIJOTE-MFI (Watson +et al. 2023, in prep.). +XVI. Diffuse intensity foregrounds from component separation +with QUIJOTE-MFI (de la Hoz et al. 2023b, in prep.). +In addition, we have a dedicated paper describing the MFI data +processing pipeline (Génova-Santos et al. 2023). The distribution +of released data products associated with the QUIJOTE-MFI wide +survey papers contain the following items: +• Four frequency maps (11, 13, 17, 19 GHz) in intensity and +polarization, both at native and one degree resolution. Maps at 11 +and 13 GHz correspond to those produced from MFI horn 3. Maps +at 17 and 19 GHz correspond to the weighted average of horns 2 +and 4, as described in Sec. 3. +• The associated weight and hit maps for each frequency map at +native resolution. +• One set of null tests maps (half1/2 for independent baselines). +• Instrument Model (IMO), containing central frequencies, +beams properties, beam profiles and window functions for each +MFI horn, bandpasses and colour corrections. +• The default analysis mask (sat+NCP+lowdec), as well as the +satellite mask (sat). The later is applied to all the released maps. +11 +CONCLUSIONS +This paper presents and characterizes the properties of the QUI- +JOTE wide survey maps of the northern sky carried out with the +MFI instrument. They result from approximately 9 000 h of obser- +vations spread over six years between 2013 and 2018, and include +four frequency maps at 11.1, 12.9, 16.8 and 18.8 GHz, with angu- +lar resolutions between 55 and 39 arcmin. The maps cover around +29 000 deg2 with sensitivities in linear polarization (Stokes Q and +U parameters) within 35–40 𝜇K per 1-degree beam. Although the +MFI instrument is not optimized for intensity measurements, we +also present the corresponding intensity maps at those four fre- +quencies, with sensitivities in the range 65–200 𝜇K per 1-degree +beam. +Together with the description of the specific aspects of the MFI +pipeline related to the production of the wide survey maps, we have +presented a detailed validation of the maps, a characterization of +residual systematic effects (Sect. 4), and an extensive study of their +calibration accuracy (Sect. 5 and Table 16). The overall calibration +uncertainty of the polarization maps is 5 % for the two lowest fre- +quency channels, and 6 % for the highest ones. These final maps +and other derived data products are part of a public data release +associated with this paper. +Although a full description of the science results obtained +from these maps are given in the accompanying papers listed in +Sect. 10, this paper presents some global properties of the Galactic +foregrounds at these frequencies, and in particular, the polarized +synchrotron emission. The average synchrotron spectral index in +polarization between 11 GHz and the WMAP 23 GHz is found to +be 𝛽 = −3.07 ± 0.16, showing a much broader distribution (by +a factor ∼ 2.7) than the one adopted in current synchrotron sky +models (e.g. Miville-Deschênes et al. 2008). Most of the large- +scale polarized synchrotron features in the MFI maps appear in +the E-mode map, which shows significantly more power than the +B-mode at these frequencies. Based on the analysis of the angular +power spectra of the measured polarized signal, we find that the +BB/EE ratio at multipole scales of ℓ = 80 is 0.26 ± 0.07 for a +Galactic cut |𝑏| > 10◦. This value is consistent with that found +for WMAP/Planck low frequency maps (Martire et al. 2022), but +it is significantly different from the values obtained for the S-PASS +polarized signal at 2.3 GHz (Krachmalnicoff et al. 2018), suggesting +that probably there is some contribution of Faraday rotation and/or +depolarization at lower frequencies than those probed by QUIJOTE +MFI. We also find a positive correlation in the TE spectrum for +11 GHz at large angular scales (ℓ ≲ 80), while the EB and TB signals +are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150, as +expected for the synchrotron emission, as its polarization orientation +is dictated by the Galactic magnetic field lines. +The MFI instrument was decommissioned in 2018. At this +moment, QT2 is operating with a combination of the TGI and FGI +instruments in a single cryostat. In addition to QUIJOTE, there +are two other CMB polarization experiments at the Teide Observa- +tory and providing a similar sky coverage: GroundBird and LSPE- +STRIP. GroundBird (Honda et al. 2020) is a MKIDs array with two +bands centered at 145 and 220 GHz, installed back in 2019. STRIP +(Addamo et al. 2021) is part of LSPE, a combined programme of +ground-based and balloon-borne polarization observations. STRIP +will operate in the 42 and 90 GHz bands, and will be installed at the +Teide Observatory in 2023. The QUIJOTE collaboration is devel- +oping a new instrument at these frequencies, called MFI2, with an +expected sensitivity three times better than the former MFI (Hoyland +et al. 2022). The new MFI2 is now in the final integration phase, and +MNRAS 000, 1–58 (2022) + +10 +20 +30 +40 +0.0 +0.51.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +mK, +CMB +mK +CMB +8 +7 +00 +6 +(deg +5 +b +4 +3 +V63 I 11 GHz +W63Q +Q11GHz +W63 U 11 GHz +85 +84 +83 +82 +81 +80 +85 +84 +83 +82 +81 +80 +85 +84 +83 +82 +81 +80 +1 (deg) +1 (deg) +1 (deg)48 +Rubiño-Martín et al. +it is using a digital back-end based on Field Programmable Gate Ar- +rays (FPGAs), that will allow us to identify and filter the RFI signals +from geostationary satellites directly in the data processing stage. A +new wide survey at these frequencies (10–20 GHz) will be carried +out with MFI2 at the first QUIJOTE telescope (QT1) starting 2023. +DATA AVAILABILITY +All data products described in Sect. 10 can be freely downloaded +from the QUIJOTE web page11, as well as from the RADIOFORE- +GROUNDS platform12. They include also an Explanatory Supple- +ment describing the data formats. Maps will be submitted also to +the Planck Legacy Archive (PLA) interface and the LAMBDA site. +Any other derived data products described in this paper (null test +maps, simulations, etc) are available upon request to the QUIJOTE +collaboration. +ACKNOWLEDGEMENTS +We thank the staff of the Teide Observatory for invaluable +assistance in the commissioning and operation of QUIJOTE. +The QUIJOTE experiment is being developed by the Instituto +de Astrofisica de Canarias (IAC), the Instituto de Fisica de +Cantabria (IFCA), and the Universities of Cantabria, Manch- +ester and Cambridge. Partial financial support was provided +by the Spanish Ministry of Science and Innovation under +the projects AYA2007-68058-C03-01, AYA2007-68058-C03-02, +AYA2010-21766-C03-01, AYA2010-21766-C03-02, AYA2014- +60438-P, ESP2015-70646-C2-1-R, AYA2017-84185-P, ESP2017- +83921-C2-1-R, +AYA2017-90675-REDC +(co-funded +with +EU +FEDER funds), PGC2018-101814-B-I00, PID2019-110610RB- +C21, PID2020-120514GB-I00, IACA13-3E-2336, IACA15-BE- +3707, EQC2018-004918-P, the Severo Ochoa Programs SEV- +2015-0548 and CEX2019-000920-S, the Maria de Maeztu Pro- +gram MDM-2017-0765, and by the Consolider-Ingenio project +CSD2010-00064 (EPI: Exploring the Physics of Inflation). We +acknowledge support from the ACIISI, Consejeria de Economia, +Conocimiento y Empleo del Gobierno de Canarias and the European +Regional Development Fund (ERDF) under grant with reference +ProID2020010108. This project has received funding from the Eu- +ropean Union’s Horizon 2020 research and innovation program un- +der grant agreement number 687312 (RADIOFOREGROUNDS). +This research made use of computing time available on the +high-performance computing systems at the IAC. We thankfully +acknowledge the technical expertise and assistance provided by the +Spanish Supercomputing Network (Red Española de Supercom- +putación), as well as the computer resources used: the Deimos/Diva +Supercomputer, located at the IAC. This research used resources of +the National Energy Research Scientific Computing Center, which is +supported by the Office of Science of the U.S. Department of Energy +under Contract No. DE-AC02-05CH11231. The PWV data used in +the tests presented in Section 4 comes from the Izaña Atmospheric +Observatory (IZO), and have been made available to us by the Izaña +Atmospheric Research Center (AEMET). SEH and CD acknowl- +edge support from the STFC Consolidated Grant (ST/P000649/1). +FP acknowledges support from the Spanish State Research Agency +11 http://research.iac.es/proyecto/quijote +12 http://www.radioforegrounds.eu/ +(AEI) under grant number PID2019-105552RB-C43. DT acknowl- +edges the support from the Chinese Academy of Sciences (CAS) +President’s International Fellowship Initiative (PIFI) with Grant N. +2020PM0042. Some of the presented results are based on observa- +tions obtained with Planck (http://www.esa.int/Planck), an +ESA science mission with instruments and contributions directly +funded by ESA Member States, NASA, and Canada. We acknowl- +edge the use of the Legacy Archive for Microwave Background +Data Analysis (LAMBDA). Support for LAMBDA is provided by +the NASA Office of Space Science. Some of the results in this pa- +per have been derived using the HEALPix (Górski et al. 2005) and +healpy (Zonca et al. 2019) packages. 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D, 55, 1830 +MNRAS 000, 1–58 (2022) + +50 +Rubiño-Martín et al. +Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., Gorski K., +2019, The Journal of Open Source Software, 4, 1298 +APPENDIX A: DATA FLAGGING IN THE MFI WIDE +SURVEY +Tables A1, A2, A3 and A4 show the percentage of data used (and +flagged) for each period, elevation and horn in the MFI wide survey. +APPENDIX B: IMPACT OF FDEC FILTERING ON +POLARIZATION MAPS +In this appendix, we investigate the impact of the FDEC filtering +on some of the scientific analyses carried out in this paper and +in other papers of the associated release (Sect. 10). In particular, +we consider here a photometry method (aperture photometry) and +correlation method (the so called TT plot). +For this study, we use the sky signal simulations presented in +Sect. 6.1. Figure B1 shows the simulated (noiseless) sky maps in +polarization at 11 GHz used as reference. We apply the FDEC filter +to these maps, and show in the same figure the resulting filtered +maps, as well as the residual maps (i.e. difference between the +original and the filtered map). As shown in Sect. 2.5, the FDEC +filtering effectively corresponds to a high-pass filter, which removes +the zero mode for any line of constant declination on a map in +local (equatorial) coordinates. The image illustrates again that the +effective transfer function of the FDEC filter leaves unaltered all +scales with ℓ >∼ 30, because the residual maps only contain large +scale features. +B1 +Impact of FDEC on photometry methods: aperture +photometry +From Fig. B1, one would expect that all analyses in real space +involving "local" analyses (e.g. the photometry extraction of a com- +pact source with a local determination of the background), should +be unaffected by the FDEC filtering. To test this hypothesis, we +take as a reference one of the photometry methods used in this pa- +per: the aperture photometry method (AP1d) described in Sect. 9. +Then, we apply AP1d to all possible pixels in the simulated maps +within the MFI wide survey sky mask, both to the original and to +the filtered maps. For this analysis, we use a reference aperture of +𝑟1 = 1◦, and the background is estimated in the annulus between +𝑟1 and 𝑟2 = +√ +2𝑟1. We find that the maximum difference between +the photometry on both Stokes Q and U parameters obtained in the +original map and the filtered one is 0.06 Jy, while the standard de- +viation of the difference of the two photometry methods is 0.007 Jy. +Both values are significantly smaller than the typical error in the +photometry (see e.g. the results presented in Table 24, in Sect. 9), +thus confirming that we can safely neglect any impact on the pho- +tometry due to the FDEC filtering. For completeness, we repeat the +analysis for a larger aperture of 𝑟1 = 2◦, and find that in this case the +maximum difference is 0.4 Jy, with a standard deviation of 0.037 Jy. +B2 +Impact of FDEC on correlation analyses: recovery of the +spectral index +We now evaluate the impact of the FDEC filtering on the recovery +of spectral index of the sky emission. To this end, we use the same +simulation set described above, taking as a reference the simulated +maps at 11 and 23 GHz. We now apply two different methods to +reconstruct the spectral index 𝛽 of the sky emission between 11 and +23 GHz. +First, we carry out a direct evaluation of the spectral index at +the pixel level (𝑁side = 512) in the original (unfiltered) maps, and +also in the filtered ones. This methodology is similar to the one used +in Sect 8. It is important to emphasize that both maps (the simulated +MFI 11 GHz and the simulated WMAP 23 GHz) have to be filtered +with the FDEC, in order to have consistent scales between the two. +The results are shown in Fig. B2. The reconstructed spectral index is +fully consistent with the input one. If we restrict the comparison to +pixels with high emission (polarized intensity at 11 GHz greater than +0.1 mK), and we compare the reconstructed maps after degrading to +2◦ (to be consistent with Sect 8), we find that the median difference +Δ𝛽 between the reconstructed and original spectral index is 0.0005, +while the standard deviation of the difference is 0.02. +Second, we use a correlation analysis method (also called TT +plot) to recover the spectral index of the emission between 11 and +23 GHz. For this analysis, we degrade the simulated maps at 1 +degree resolution to 𝑁side = 64, in order to have approximately +independent pixels. Then, we divide the observed sky in patches of +∼ 7.3◦, using as a reference the pixels of a 𝑁side = 8 HEALPix +map. Within each patch, we carry out a TT plot analysis assuming +a typical error in each map corresponding to 3 per cent of the sky +signal, and accounting for errors in both axes (see e.g. Fuskeland +et al. 2014). Fig. B3 shows the obtained results from the original +maps (top panel), and the FDEC filtered maps (bottom panel). As +expected, the spatial distribution of the reconstructed index has a +good correspondence with the maps shown in Fig. B2. A numerical +comparison of both maps in Fig. B3 gives that median difference +Δ𝛽 between the two maps is -0.0007, and the standard deviation of +the difference is 0.02. +Summarising, we can obtain an unbiased reconstruction of the +spectral index of the sky signal, provided that both maps are filtered +in the same way using FDEC. In practice, this means that when +doing these type of analyses using QUIJOTE MFI wide survey +maps and external ancillary data, we must filter first the external +maps using the same procedure as for the MFI maps. If the FDEC +filtering is not applied to the external ancillary data, we find that +for these simulations the standard deviation of the reconstructed +spectral index can be as large as 0.3. This issue is further discussed +in other papers in the series (see e.g. Appendix C in de la Hoz et al. +2023a). +APPENDIX C: QUIJOTE MFI WIDE SURVEY MAPS PER +HORN AT ORIGINAL RESOLUTION +Figures C1, C2 and C3 show the final MFI wide survey maps at their +original resolution (quoted as beam FWHM in Table 3), obtained +for horns 2, 3 and 4 respectively. The intensity maps of horns 2 +and 4 show some large angular-scale residual patterns, particularly +visible in the highest frequency map (19 GHz). These are due to a +combination of residual instrumental and atmospheric 1/ 𝑓 noise. +Figures C4, C5 and C6 show the corresponding weight maps at +the original resolution. Figures C7, C8 and C9 show the maps with +the number of individual TOD samples in each pixel (the so called +"hit maps", 𝑁hit). They correspond to the total number of 40 ms +samples in each HEALPix pixel of 𝑁side = 512 resolution. The ring +structures correspond to lines of constant declination, and indicate +the edges of the declination limits of observations performed at +different elevations. Due to projection effects, the number of hits +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +51 +Table A1. Fraction of data used in period 1 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Column 1 indicates +the elevation, columns 2 and 3 show the horn and frequency (0 for low and 1 for high). Columns 4 and 5 show the percentage of used data in correlated and +uncorrelated channels, respectively. Columns 6 and 7 show the percentage of flagged data during the post-processing stage, and columns 8 and 9 show the +percentage of flagged data due to Sun, Moon and planets (Mars, Venus, Jupiter). Last column indicates the range of dates when each elevation was observed. +Elevation +Horn +Freq +Used c +Used u +Flag1 c +Flag1 u +Flag2 c +Flag2 u +Range of Dates +(deg) +(%) +(%) +(%) +(%) +(%) +(%) +60 +2 +0 +71.7 +73.9 +16.8 +14.1 +1.4 +1.4 +5/2013–3/2014 +60 +2 +1 +60.7 +49.3 +29.5 +42.6 +1.4 +1.4 +5/2013–3/2014 +60 +3 +0 +33.1 +58.7 +52.8 +16.7 +6.6 +6.6 +5/2013–3/2014 +60 +3 +1 +32.8 +62.7 +54.6 +13.8 +6.6 +6.6 +5/2013–3/2014 +60 +4 +0 +73.4 +74.2 +6.7 +5.7 +1.5 +1.5 +5/2013–3/2014 +60 +4 +1 +56.9 +63.1 +27.8 +19.9 +1.5 +1.5 +5/2013–3/2014 +65 +2 +0 +81.6 +84.5 +13.8 +10.7 +2.0 +2.0 +5/2013–3/2014 +65 +2 +1 +73.8 +55.8 +22.1 +40.9 +2.0 +2.0 +5/2013–3/2014 +65 +3 +0 +39.3 +70.8 +54.7 +18.5 +1.8 +1.8 +5/2013–3/2014 +65 +3 +1 +30.9 +72.8 +62.2 +11.5 +1.8 +1.8 +5/2013–3/2014 +65 +4 +0 +86.3 +87.1 +4.5 +3.6 +1.9 +1.9 +5/2013–3/2014 +65 +4 +1 +70.6 +77.7 +22.0 +14.1 +1.9 +1.9 +5/2013–3/2014 +Table A2. Fraction of data used in period 2 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in +Table A1. +Elevation +Horn +Freq +Used c +Used u +Flag1 c +Flag1 u +Flag2 c +Flag2 u +Range of Dates +(deg) +(%) +(%) +(%) +(%) +(%) +(%) +30 +2 +0 +57.1 +56.9 +37.3 +37.5 +2.0 +2.0 +8/2014–3/2015 +30 +2 +1 +53.1 +46.1 +41.7 +49.4 +2.0 +2.0 +8/2014–3/2015 +30 +3 +0 +31.0 +37.9 +64.5 +56.4 +1.7 +1.7 +8/2014–3/2015 +30 +3 +1 +29.4 +41.6 +66.8 +53.0 +1.7 +1.7 +8/2014–3/2015 +30 +4 +0 +57.4 +58.2 +37.4 +36.5 +1.8 +1.8 +8/2014–3/2015 +30 +4 +1 +51.2 +53.0 +44.1 +42.1 +1.8 +1.8 +8/2014–3/2015 +40 +2 +0 +49.2 +51.8 +43.2 +40.1 +1.9 +1.9 +8/2014–1/2015 +40 +2 +1 +46.7 +38.5 +46.1 +55.5 +1.9 +1.9 +8/2014–1/2015 +40 +3 +0 +28.6 +51.5 +65.4 +38.0 +5.2 +5.2 +8/2014–1/2015 +40 +3 +1 +22.0 +41.1 +73.5 +50.6 +5.2 +5.2 +8/2014–1/2015 +40 +4 +0 +55.1 +56.1 +36.9 +35.8 +2.0 +2.0 +8/2014–1/2015 +40 +4 +1 +52.4 +51.7 +40.0 +40.8 +2.0 +2.0 +8/2014–1/2015 +50 +2 +0 +64.7 +65.7 +22.7 +21.4 +2.0 +2.0 +8/2014–10/2015 +50 +2 +1 +61.2 +60.9 +27.0 +27.4 +2.0 +2.0 +8/2014–10/2015 +50 +3 +0 +41.9 +55.4 +47.3 +30.2 +7.0 +7.0 +8/2014–10/2015 +50 +3 +1 +35.8 +55.5 +54.3 +29.2 +7.0 +7.0 +8/2014–10/2015 +50 +4 +0 +62.4 +62.8 +20.2 +19.7 +2.1 +2.1 +8/2014–10/2015 +50 +4 +1 +53.1 +53.1 +31.4 +31.3 +2.1 +2.1 +8/2014–10/2015 +60 +2 +0 +58.3 +59.4 +31.7 +30.3 +2.0 +2.0 +6/2014–9/2014 +60 +2 +1 +58.6 +30.5 +31.4 +64.3 +2.0 +2.0 +6/2014–9/2014 +60 +3 +0 +8.8 +50.4 +87.6 +29.0 +7.0 +7.0 +6/2014–9/2014 +60 +3 +1 +0.0 +50.8 +100.0 +29.8 +7.0 +7.0 +6/2014–9/2014 +60 +4 +0 +55.5 +54.5 +28.9 +30.1 +2.0 +2.0 +6/2014–9/2014 +60 +4 +1 +52.8 +53.0 +32.4 +32.0 +2.0 +2.0 +6/2014–9/2014 +65 +2 +0 +73.8 +79.3 +21.4 +15.5 +3.3 +3.3 +8/2014–10/2014 +65 +2 +1 +76.0 +44.8 +19.0 +52.5 +3.3 +3.3 +8/2014–10/2014 +65 +3 +0 +36.9 +66.4 +56.9 +21.6 +3.2 +3.2 +8/2014–10/2014 +65 +3 +1 +36.1 +67.2 +56.3 +17.6 +3.2 +3.2 +8/2014–10/2014 +65 +4 +0 +71.6 +76.9 +20.0 +14.2 +3.2 +3.2 +8/2014–10/2014 +65 +4 +1 +65.3 +68.4 +27.2 +23.7 +3.2 +3.2 +8/2014–10/2014 +is significantly larger in those boundaries. In the low declination +band of the maps, particularly for negative declinations, the number +of hits is significantly lower due to the combined effect of smaller +number of observations at low elevations (mainly 30◦, 35◦ and 40◦) +and projection effects. We recall that the number of hits in intensity +is larger than in polarization, due to the fact that some intensity +data are not used in polarization, as shown in Table 1 (period 1 +is not used for any polarization maps, data from period 2 are not +used in polarization for horn 4, and data from period 5 are not +used in polarization for horn 2). Finally, Fig. C10 shows the 𝑟cond +maps in polarization, and Fig. C11 shows the normalized covariance +𝑐𝑜𝑣(𝑄,𝑈), both at original resolution. +MNRAS 000, 1–58 (2022) + +52 +Rubiño-Martín et al. +Table A3. Fraction of data used in period 5 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in +Table A1. +Elevation +Horn +Freq +Used c +Used u +Flag1 c +Flag1 u +Flag2 c +Flag2 u +Range of Dates +(deg) +(%) +(%) +(%) +(%) +(%) +(%) +40 +2 +0 +57.6 +57.7 +33.9 +33.7 +2.0 +2.0 +8/2016–10/2016 +40 +2 +1 +50.9 +49.6 +41.6 +43.1 +2.0 +2.0 +8/2016–10/2016 +40 +3 +0 +60.8 +61.4 +27.6 +26.9 +5.1 +5.1 +8/2016–10/2016 +40 +3 +1 +46.1 +45.0 +45.2 +46.5 +5.1 +5.1 +8/2016–10/2016 +40 +4 +0 +59.5 +59.2 +32.4 +32.7 +2.0 +2.0 +8/2016–10/2016 +40 +4 +1 +43.4 +49.5 +50.7 +43.8 +2.0 +2.0 +8/2016–10/2016 +50 +2 +0 +65.6 +66.1 +21.6 +21.1 +2.3 +2.3 +8/2016–10/2016 +50 +2 +1 +62.0 +61.7 +26.1 +26.5 +2.3 +2.3 +8/2016–10/2016 +50 +3 +0 +57.3 +56.6 +28.7 +29.5 +6.9 +6.9 +8/2016–10/2016 +50 +3 +1 +57.8 +56.3 +26.8 +28.6 +6.9 +6.9 +8/2016–10/2016 +50 +4 +0 +62.5 +62.6 +20.6 +20.5 +2.1 +2.1 +8/2016–10/2016 +50 +4 +1 +45.5 +54.0 +41.4 +30.6 +2.1 +2.1 +8/2016–10/2016 +60 +2 +0 +79.4 +79.5 +7.1 +6.9 +2.0 +2.0 +8/2016–9/2016 +60 +2 +1 +77.0 +76.6 +9.8 +10.3 +2.0 +2.0 +8/2016–9/2016 +60 +3 +0 +61.9 +62.3 +13.2 +12.6 +7.0 +7.0 +8/2016–9/2016 +60 +3 +1 +67.2 +64.5 +7.3 +11.1 +7.0 +7.0 +8/2016–9/2016 +60 +4 +0 +72.7 +73.4 +7.2 +6.3 +1.9 +1.9 +8/2016–9/2016 +60 +4 +1 +60.3 +67.9 +23.0 +13.2 +1.9 +1.9 +8/2016–9/2016 +Table A4. Fraction of data used in period 6 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument. Same format as in +Table A1. +Elevation +Horn +Freq +Used c +Used u +Flag1 c +Flag1 u +Flag2 c +Flag2 u +Range of Dates +(deg) +(%) +(%) +(%) +(%) +(%) +(%) +35 +2 +0 +58.0 +56.6 +33.7 +35.4 +1.9 +1.9 +12/2017–6/2018 +35 +2 +1 +47.7 +47.2 +45.5 +46.1 +1.9 +1.9 +12/2017–6/2018 +35 +3 +0 +63.5 +62.0 +26.1 +27.8 +4.2 +4.2 +12/2017–6/2018 +35 +3 +1 +51.6 +51.4 +39.9 +40.1 +4.2 +4.2 +12/2017–6/2018 +35 +4 +0 +61.9 +62.6 +33.0 +32.3 +1.8 +1.8 +12/2017–6/2018 +35 +4 +1 +42.2 +24.5 +54.5 +73.5 +1.8 +1.8 +12/2017–6/2018 +50 +2 +0 +68.2 +68.3 +18.3 +18.1 +2.8 +2.8 +3/2017–4/2017 +50 +2 +1 +60.3 +59.9 +28.0 +28.4 +2.8 +2.8 +3/2017–4/2017 +50 +3 +0 +61.9 +61.6 +22.5 +23.0 +7.3 +7.3 +3/2017–4/2017 +50 +3 +1 +59.6 +56.6 +24.1 +27.9 +7.3 +7.3 +3/2017–4/2017 +50 +4 +0 +67.6 +67.6 +13.7 +13.8 +2.6 +2.6 +3/2017–4/2017 +50 +4 +1 +55.4 +52.7 +28.6 +32.1 +2.6 +2.6 +3/2017–4/2017 +60 +2 +0 +73.9 +73.9 +14.3 +14.3 +0.7 +0.7 +12/2016–2/2017 +60 +2 +1 +72.1 +72.0 +16.4 +16.5 +0.7 +0.7 +12/2016–2/2017 +60 +3 +0 +55.6 +55.6 +22.3 +22.3 +5.8 +5.8 +12/2016–2/2017 +60 +3 +1 +61.8 +61.6 +15.4 +15.6 +5.8 +5.8 +12/2016–2/2017 +60 +4 +0 +68.4 +68.5 +13.4 +13.2 +0.7 +0.7 +12/2016–2/2017 +60 +4 +1 +66.6 +65.8 +15.6 +16.6 +0.7 +0.7 +12/2016–2/2017 +65 +2 +0 +85.3 +85.2 +8.8 +8.9 +3.0 +3.0 +3/2017–4/2017 +65 +2 +1 +83.5 +83.2 +10.8 +11.1 +3.0 +3.0 +3/2017–4/2017 +65 +3 +0 +59.5 +59.5 +29.1 +29.0 +3.2 +3.2 +3/2017–4/2017 +65 +3 +1 +71.3 +69.3 +12.4 +14.9 +3.2 +3.2 +3/2017–4/2017 +65 +4 +0 +81.9 +81.9 +8.2 +8.2 +3.1 +3.1 +3/2017–4/2017 +65 +4 +1 +80.2 +76.9 +10.1 +13.8 +3.1 +3.1 +3/2017–4/2017 +70 +2 +0 +85.1 +85.1 +9.4 +9.4 +2.2 +2.2 +2/2017–4/2017 +70 +2 +1 +84.3 +84.3 +10.4 +10.4 +2.2 +2.2 +2/2017–4/2017 +70 +3 +0 +67.4 +67.5 +28.8 +28.7 +2.5 +2.5 +2/2017–4/2017 +70 +3 +1 +83.8 +82.6 +10.7 +12.0 +2.5 +2.5 +2/2017–4/2017 +70 +4 +0 +86.5 +86.5 +7.8 +7.8 +2.4 +2.4 +2/2017–4/2017 +70 +4 +1 +85.6 +84.8 +8.9 +9.7 +2.4 +2.4 +2/2017–4/2017 +MNRAS 000, 1–58 (2022) + +QUIJOTE MFI wide survey +53 +Figure B1. Example of application of FDEC filtering in simulations of the polarized MFI signal. Top (bottom) row corresponds to Stokes Q (U) parameters. +Left column shows the simulated MFI 11 GHz map at 1 deg resolution; middle column corresponds to the same map, after applying the FDEC filtering; and +last column shows the difference of the previous two maps. All maps use the same colour scale, saturated at ±1 mK. +Figure B2. Impact of the application of FDEC filtering in the reconstruction +of the spectral index in real space. We use simulations of the polarized sky +signal in MFI 11 GHz and WMAP 23 GHz. Top panel shows the (true) +underlying spectral index of the simulated signal between 11 and 23 GHz, +within the MFI observing mask. Bottom panel shows the reconstructed +spectral index after applying the FDEC filtering to both simulated maps (11 +and 23 GHz). +Figure B3. Impact of the application of FDEC filtering in the reconstruction +of the spectral index using correlation analysis (TTplot). We carry out the +correlation analysis in regions defined by HEALPix pixels of 𝑁side = 8, and +extract the spectral index of the polarized sky signal between MFI 11 GHz +and WMAP 23 GHz. Top panel shows the (true) underlying spectral index +of the simulated signal within the MFI observing mask. Bottom panel shows +the reconstructed spectral index after applying the FDEC filtering to both +simulated maps (11 and 23 GHz). +MNRAS 000, 1–58 (2022) + +Simulated MFl 11GHz Q (1deg) - original +mkSimulated MFI 11GHz Q (1deg) - no FDEC +mkSimulated MFl 11GHz Q (1deg) - FDEC +mk +一Simulated MFl 11GHz U (ldeg) - original +mkSimulated MFI 11GHz U (ldeg) - no FDEC +mkSimulated MFl 11GHz U (1deg) - FDEC +mkSimulated spectral index in polarization (MFil1 to WMAP-K) +-3.4 +β11GHz - 23GHz +-2.7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC +-3.4 +β11GHz - 23GHz +-2.7Simulated spectral index in polarization (MFil1 to WMAP-K) +-3.4 +β11GHz - 23GHz +-2.7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC +-3.4 +β11GHz - 23GHz +2.754 +Rubiño-Martín et al. +Figure C1. Original resolution QUIJOTE MFI wide survey maps for horn 2. Maps are shown in Galactic coordinates. All figures use the same linear colour +scale, saturated at 20 mKCMB for intensity (first column) and 2 mKCMB in polarization for Stokes Q (second column) and Stokes U (third column) parameters. +For display purposes, maps are downgraded to 𝑁side = 256. +Figure C2. Same as Fig. C1, but for QUIJOTE MFI wide survey maps for horn 3. +C1 +Signal-to-noise of the QUIJOTE MFI maps +From the maps at original resolution shown in Figs. C1–C3, and +the noise variance maps estimated from the inverse of the weights +presented in Figs. C4–C6 and rescaled by the factors reported in +Table 12, we can produce signal-to-noise maps for the MFI wide +survey. To this end, we downgrade these maps to a HEALPix reso- +lution of 𝑁side = 64, which roughly corresponds to the beam size of +the maps. Table C1 presents some basic statistics about the fraction +of 𝑁side = 64 pixels in the maps observed about a certain signal-to- +noise significance. As a reference, the 11 GHz polarized intensity +map has 52 % of its pixels with a signal-to-noise ratio larger than 3. +APPENDIX D: IMPACT IN THE POLARIZATION TOD OF +AN ERROR IN THE DETERMINATION OF THE +𝑟-FACTOR +We illustrate this effect using the particular case of uncorrelated +channels in the first MFI configuration, but the result is equivalent +for correlated channels and for all MFI configurations. We follow +the notation introduced in Génova-Santos et al. (2023), and used in +equation 1. Following the notation of Jarosik et al. (2003), the MFI +response for the two uncorrelated channels, 𝑥 and 𝑦, in the first MFI +MNRAS 000, 1–58 (2022) + +QUIJOTE 1 H2 17GHZ +5 +mK +20QUIJOTE Q H2 17GHz +mK +2QUIJOTE U H2 17GHZ +2 +mKQUIJOTE 1 H2 19GHZ +5 +mK +20QUIJOTE Q H2 19GHz +mK +2QUIJOTE U H2 19GHzZ +mK +2QUIJOTE I H3 11GHZ +5 +mK +20QUIJOTE Q H3 11GHz +mK +2QUIJOTE U H3 11GHzZ +2 +mKQUIJOTE I H3 13GHZ +5 +mK +20QUIJOTE Q H3 13GHZ +2 +mKQUIJOTE U H3 13GHZ +2 +mK +2QUIJOTE MFI wide survey +55 +Figure C3. Same as Fig. C1, but for QUIJOTE MFI wide survey maps for horn 4. +Figure C4. QUIJOTE MFI wide survey weight maps for horn 2. Top row is 17 GHz, and bottom row is 19 GHz. Each row shows, from left to right, the weight +maps for Stokes I, Q and U. +configuration is given by +𝑉x = +𝑠x𝑔2 +1 +2 +� +𝐼 + 𝜌x(𝑄 cos 𝜃 − 𝑈 sin 𝜃) +� +(D1) +𝑉y = +𝑠y𝑔2 +2 +2 +� +𝐼 + 𝜌y(−𝑄 cos 𝜃 + 𝑈 sin 𝜃) +� +(D2) +where 𝜃 stands for the argument of the cosine and sine in MFI +receivers (i.e. 𝜃 = 4𝜃pm + 2𝛾p, as in equations 3 and 4), 𝑠x and 𝑠y +represent the responsivities of the detectors in the two branches, 𝑔1 +and 𝑔2 represent the voltage gains of the two amplifiers in each MFI +polarimeter, and 𝜌x and 𝜌y are the polar efficiencies in each branch. +The 𝑟-factor is defined as +𝑟u ≡ +𝑠x𝑔2 +1 +𝑠y𝑔2 +2 +. +(D3) +If we are using an incorrect 𝑟-factor 𝑟′u = 𝑟u + 𝜖, where 𝑟u is +the correct underlying value, then we have +𝑉x − 𝑟′ +u𝑉y = 𝑠x𝑔2 +1 +�� 𝜌x + 𝜌y +2 ++ 𝜖 +2𝑟u +𝜌y +� +(𝑄 cos 𝜃 − 𝑈 sin 𝜃) − 𝜖 +2𝑟u +𝐼 +� +. +(D4) +We find that this error on the 𝑟-factor translates into an effective +modification of the polar efficiency, and the appearance of a constant +MNRAS 000, 1–58 (2022) + +QUIJOTE 1 H4 17GHZ +5 +mK +20QUIJOTE Q H4 17GHZ +mK +2QUIJOTE U H4 17GHzZ +2 +mKQUIJOTE 1H4 19GHZ +5 +mK +20QUIJOTE Q H4 19GHz +2 +mKQUIJOTE U H4 19GHzZ +2 +mK +2QUIJOTE WEIGHTS 1 H2 17GHZ +0 +mk-2 +30QUIJOTE WEIGHTS Q H2 17GHz +0 +mk-2 +30QUIJOTE WEIGHTS U H2 17GHZ +0 +mk-2 +30QUIJOTE WEIGHTS 1 H2 19GHz +0 +mk-2 +30QUIJOTE WEIGHTS Q H2 19GHz +0 +mk-2 +30QUIJOTE WEIGHTS U H2 19GHZ +0 +mk-2 +3056 +Rubiño-Martín et al. +Figure C5. QUIJOTE MFI wide survey weight maps for horn 3. +Figure C6. QUIJOTE MFI wide survey weight maps for horn 4. +offset factor in the polarization timeline. For the particular case of +𝜌x = 𝜌y, then the effective polar efficiency is rescaled by the factor +𝜌x → 𝜌x +� +1 + +𝜖 +2𝑟u +� +. +(D5) +Finally, we note that for the intensity timeline, the same effect gener- +ates an overall calibration shift, and a small polarization-to-intensity +leakage term. The first term is absorbed once we carry our a re- +calibration of the instrument, while the second one can be safely +ignored, as the polarization fraction of the sky emission is already +small (typically well below 10 per cent). +APPENDIX E: POWER SPECTRUM ESTIMATORS FOR +MFI WIDE SURVEY MAPS +Throughout this paper, we have been using two power spectrum esti- +mation codes, both based on a pseudo-Cℓ approach: Xpol (Tristram +et al. 2005) and NaMaster (Alonso et al. 2019). In this appendix, +we show that both methods produce consistent results for the typical +sky masks adopted in this paper. For this comparison, we take as +a reference case the MFI 11 GHz wide survey map and the default +QUIJOTE mask (NCP+sat+lowdec) combined with the Galactic +cut |𝑏| > 10◦. In addition, we justify the use of the pseudo-spectra +approach by comparing these results with those from an optimal +estimator based on a fast implementation of a quadratic maximum- +likelihood (QML) estimator (ECLIPSE, Bilbao-Ahedo et al. 2021). +MNRAS 000, 1–58 (2022) + +QUIJOTE WEIGHTS I H3 11GHz +0 +mk-2 +30QUIJOTE WEIGHTS Q H3 11GHz +0 +mk-2 +30QUIJOTE WEIGHTS U H3 11GHZ +0 +mk-2 +30QUIJOTE WEIGHTS 1 H3 13GHZ +0 +mk-2 +30QUIJOTE WEIGHTS Q H3 13GHZ +0 +mk-2 +30QUIJOTE WEIGHTS U H3 13GHZ +0 +mk-2 +30QUIJOTE WEIGHTS I H4 17GHz +0 +mk-2 +30QUIJOTE WEIGHTS Q H4 17GHz +0 +mk-2 +30QUIJOTE WEIGHTS U H4 17GHZ +0 +mk-2 +30QUIJOTE WEIGHTS 1 H4 19GHZ +0 +mk-2 +30QUIJOTE WEIGHTS Q H4 19GHz +0 +mk-2 +30QUIJOTE WEIGHTS U H4 19GHZ +0 +mk-2 +30QUIJOTE MFI wide survey +57 +Figure C7. QUIJOTE MFI wide survey hit maps for horn 2. They show the total number of 40 ms samples in each HEALPix pixel of 𝑁side = 512 resolution. +Figure C8. QUIJOTE MFI wide survey hit maps for horn 3. +Running this QML code is computationally very expensive, so the +comparison is limited to this case only. +Figure E1 shows the (binned) low multipoles points of the an- +gular power spectra and cross-spectra (30 ≤ ℓ ≤ 80) computed with +those three codes using the same mask. For the case of NaMaster +we use the "purification" option. The conclusion is that, within the +multipole range used in this paper (ℓ ≥ 30), all methods provide +consistent results, so it is justified to use the pseudo-Cℓ approach for +our computations. In this work, we use equally Xpol or NaMaster +for TT, EE and BB. For the cross-spectrum analysis in Sect. 7, we +use the NaMaster code, as it provides slightly closer results to the +(optimum) QML solution. +APPENDIX F: SPECTRAL INDEX OF THE MFI 13 GHZ +SKY EMISSION +In this appendix we repeat the same analysis carried out in Sect. 8.1, +but using now as a reference the MFI 13 GHz map. Figure F1 shows +the result for the intensity spectral index in 𝛽408MHz−13GHz (top +panel) and 𝛽13GHz−23GHz (bottom panel), while Fig. F2 presents the +polarization spectral index map 𝛽13GHz−23GHz. Again, in Fig. F3 +we show the histogram with the distribution of spectral indices in +both maps. +In general, all results are consistent with those obtained using +MFI 11 GHz as the reference map. In intensity, the median spectral +index 𝛽408MHz−13GHz in the full analysis mask is −2.83, with a +standard deviation of the values across the map of 0.19; and the +MNRAS 000, 1–58 (2022) + +QUIJOTE NHITS I H2 17GHZ +50 +nhits +1000QUIJOTE NHITS Q H2 17GHZ +50 +nhits +500QUIJOTE NHITS U H2 17GHZ +50 +nhits +500QUIJOTE NHITS I H2 19GHz +50 +nhits +1000QUIJOTE NHITS Q H2 19GHZ +50 +nhits +500QUIJOTE NHITS U H2 19GHZ +50 +nhits +500QUIJOTE NHITS I H3 11GHZ +50 +nhits +1000QUIJOTE NHITS Q H3 11GHZ +50 +nhits +500QUIJOTE NHITS U H3 11GHZ +50 +nhits +500QUIJOTE NHITS I H3 13GHZ +50 +nhits +1000QUIJOTE NHITS Q H3 13GHZ +50 +nhits +500QUIJOTE NHITS U H3 13GHZ +50 +nhits +50058 +Rubiño-Martín et al. +Figure C9. QUIJOTE MFI wide survey hit maps for horn 4. +Figure C10. QUIJOTE MFI wide survey 𝑟cond maps for all four horns. During the post-processing stage, all pixels with 𝑟cond > 3 are removed from the final +wide survey polarization maps. +𝛽13GHz−23GHz spectral index has a median of −2.83 and a standard +deviation of 0.46. We note that in this latter case, there is a peak +around −3.1, which is due to the adopted prior. In polarization, the +𝛽13GHz−23GHz spectral index presents a median value −3.09, and +the standard deviation is 0.13. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–58 (2022) + +QUIJOTE NHITS I H4 17GHZ +50 +nhits +1000QUIJOTE NHITS Q H4 17GHZ +50 +nhits +500QUIJOTE NHITS U H4 17GHZ +50 +nhits +500QUIJOTE NHITS I H4 19GHZ +50 +nhits +1000QUIJOTE NHITS Q H4 19GHZ +50 +nhits +500QUIJOTE NHITS U H4 19GHZ +50 +nhits +500QUIJOTE RCOND H2 17GHZ +0 +rcond +3QUIJOTE RCOND H2 19GHZ +0 +rcond +3QUIJOTE RCOND H3 11GHZ +0 +rcond +3QUIJOTE RCOND H3 13GHZ +0 +rcond +3QUIJOTE RCOND H4 17GHZ +0 +rcond +3QUIJOTE RCOND H4 19GHZ +0 +rcond +3QUIJOTE MFI wide survey +59 +Figure C11. QUIJOTE MFI wide survey normalized covariance (𝑐𝑜𝑣 (𝑄, 𝑈)) maps for all four horns. +Figure E1. Comparison of the angular power spectrum estimators used in this work. Using the default QUIJOTE analysis mask together with the Galactic cut +|𝑏| > 10◦, we evaluate the TT, EE, BB auto-spectra and the TE, EB and TB cross-spectra of the MFI 11 GHz map, using Xpol and NaMaster (both based +on pseudo-Cℓ formalism), and ECLIPSE (based on a QML approach). As a reference, in the first three panels we also show the corresponding noise power +spectrum. For display purposes, the different data points have been shifted by Δℓ = 1. See text for details. +MNRAS 000, 1–58 (2022) + +QUIJOTE COV(Q,U) H2 17GHz +-0.0001 +cov(Q,U)/(QQ Qu) +0.0001QUIJOTE COV(Q,U) H2 19GHz +-0.0001 +cov(Q,U)/(QQ Qu) +0.0001QUIJOTE COV(Q,U) H3 11GHz +-0.0001 +cov(Q,U)/(Qo Qu) +0.0001QUIJOTE COV(Q,U) H3 13GHz +-0.0001 +cov(Q,U)/(QQ Qu) +0.0001QUIJOTE COV(Q,U) H4 17GHz +-0.0001 +cov(Q,U)/(QQ Qu) +0.0001QUIJOTE COV(Q,U) H4 19GHz +-0.0001 +cov(Q,U)/(QQ Qu) +0.0001TT. Ibl>10° +EE.Ibl>10° +BB.Ibl>10° +100.000 +10- +10 +TT Nmt pure · +EE Nmt pure · +BB Nmt pure · +TT QML O +EE QML O +BB QML +10.000 +TT Xpol · +EE Xpol · +BB Xpol ● +10-2E +10-2 +1.000 +[mk'] +[mk'] +a +a +0.100 +10~4 +10-* +0.010E +0.001L +10-5 +10-5 +30 +40 +50 +60 +70 +80 +30 +40 +50 +60 +70 +80 +30 +40 +50 +60 +70 +80 +Multipole t +Multipole t +Multipole l +EB +TE +TB +0.0010 +0.010 +0.010 +EB Nmt pure · +TE Nmt pure · +TB Nmt pure · +EB QML O +TE QML +TB QML +EB Xpol +TE Xpol · +TB Xpol +0.0005 +0.005 +0.005 +[mk'] +[mk'] +[eyw] +0.0000 +0.000 +0.000 +6 +0.0005 +0.005 +0.005 +0.0010LI +0.010L +30 +40 +50 +60 +70 +80 +30 +40 +50 +60 +70 +80 +30 +40 +50 +60 +70 +80 +Multipole t +Multipole t +Multipole t60 +Rubiño-Martín et al. +Table C1. Fraction of 𝑁side = 64 pixels with signal-to-noise ratio (SNR) +above a certain threshold in the four QUIJOTE-MFI frequency maps (horns +2 and 4 have been combined). We report the SNR both for the intensity (I) +and the (noise debiased) polarized intensity (𝑃 = +√︁ +𝑄2 + 𝑈2) maps. +11 GHz +13 GHz +17 GHz +19 GHz +Intensity (I) +SNR> 1 +0.88 +0.90 +0.86 +0.82 +SNR> 2 +0.78 +0.81 +0.72 +0.64 +SNR> 3 +0.70 +0.73 +0.59 +0.49 +SNR> 4 +0.64 +0.66 +0.48 +0.36 +SNR> 5 +0.58 +0.60 +0.39 +0.26 +Polarized intensity (P) +SNR> 1 +0.87 +0.82 +0.69 +0.70 +SNR> 2 +0.70 +0.60 +0.38 +0.38 +SNR> 3 +0.52 +0.42 +0.19 +0.16 +SNR> 4 +0.38 +0.29 +0.10 +0.06 +SNR> 5 +0.29 +0.21 +0.06 +0.02 +Figure F1. Spectral index of the intensity emission in the QUIJOTE 13 GHz +map. Top: Spectral index of 𝛽408MHz−13GHz. The average index is approxi- +mately −2.8. Bottom: Spectral index of 𝛽13GHz−23GHz. The average spectral +index is also 𝛽 = −2.8. In this colour scale, dark red corresponds to AME +dominated regions. +Figure F2. Top: Spectral index map of the polarized emission between +QUIJOTE 13 GHz and WMAP 23 GHz. Bottom: error map. +Figure F3. Histogram of spectral index values obtained from Figures F1 +and F2. We show in dashed lines the mean of the prior adopted in the +determination of the spectral index in polarization. For comparison, we also +include the histogram of spectral index values from the PySM synchrotron +model 1 (Thorne et al. 2017). +MNRAS 000, 1–58 (2022) + +Spectral index in intensity (Haslam to MFl13) +-3.4 +β408MHz - 13GHz +-2.2Spectral index in intensity (MFI13 to WMAP-K) +-3.4 +β13GHz - 23GHz-3.4 +β13GHz - 23GHz +-2.7Error in the spectral index in polarization (MFl13 to WMAP-K) +0 +(β13GHz - 23GHz) +0.41.2 +Intensity（408MHz-13GHz +Intensity（13GHz-23GHz +1.0 +Polarization（13GHz-23GHz） +Prior β=-3.1±0.3 +(normalized) +PySM +0.8 +0.6 +count +Pixel +0.4 +0.2 +0.0 +-4 +-3 +-2 +Spectral index β \ No newline at end of file diff --git a/GNE4T4oBgHgl3EQfgA3B/content/tmp_files/load_file.txt b/GNE4T4oBgHgl3EQfgA3B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..331929450320a067041ee0c8e6feea80216aca68 --- /dev/null +++ b/GNE4T4oBgHgl3EQfgA3B/content/tmp_files/load_file.txt @@ -0,0 +1,5283 @@ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Guidi,1,2,3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Génova-Santos,1,2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Harper,4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Herranz,5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Vansyngel,1,2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Vielva,5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Watson,4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Artal,10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Ashdown,6,7 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Barreiro,5 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bilbao-Ahedo,5,11 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Casas,5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Casaponsa,5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cepeda-Arroita,4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de la Hoz,5,11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Dickinson,4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fernández-Cobos,5,12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fernández-Torreiro,1,2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' González-González,1,2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hernández-Monteagudo,1,2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' López-Caniego,13,14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' López-Caraballo,1,2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Martínez-González,5 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Peel,1,2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Peláez-Santos,1,2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Perrott,6,15 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Piccirillo,4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Razavi-Ghods,6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Scott,6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Titterington,6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Tramonte,1,2,16,17 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Vignaga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1,2 1Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain 2Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain 3Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4Jodrell Bank Centre for Astrophysics, Alan Turing Building, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Oxford Road, Manchester M13 9PL, Manchester, UK 5Instituto de Física de Cantabria (IFCA), CSIC-Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' los Castros, s/n, E-39005 Santander, Spain 6Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK 7Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 8Consejo Superior de Investigaciones Científicas, E-28006 Madrid, Spain 9Departamento de Física.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Facultad de Ciencias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Universidad de Córdoba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Campus de Rabanales, Edif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planta Baja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' E-14071 Córdoba, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10Departamento de Ingenieria de COMunicaciones (DICOM), Laboratorios de I+D de Telecomunicaciones, Plaza de la Ciencia s/n, E-39005 Santander, Spain 11Departamento de Física Moderna, Universidad de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de los Castros s/n, 39005 Santander, Spain 12Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' los Castros, s/n, E-39005 Santander, Spain 13Aurora Technology for the European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain 14Universidad Europea de Madrid, 28670, Madrid, Spain 15School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand 16Purple Mountain Observatory, CAS, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 Yuanhua Road, Qixia District, Nanjing 210034, China 17NAOC-UKZN Computational Astrophysics Center (NUCAC), University of Kwazulu-Natal, Durban 4000, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Accepted 2022 November 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Received 2022 November 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' in original form 2022 July 29 ABSTRACT We present QUIJOTE intensity and polarization maps in four frequency bands centred around 11, 13, 17 and 19 GHz, and covering approximately 29 000 deg2, including most of the Northern sky region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These maps result from 9 000 h of observations taken between May 2013 and June 2018 with the first QUIJOTE instrument (MFI), and have angular resolutions of around 1◦, and sensitivities in polarization within the range 35–40 𝜇K per 1-degree beam, being a factor ∼ 2–4 worse in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We discuss the data processing pipeline employed, and the basic characteristics of the maps in terms of real space statistics and angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A number of validation tests have been applied to characterise the accuracy of the calibration and the residual level of systematic effects, finding a conservative overall calibration uncertainty of 5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also discuss flux densities for four bright celestial sources (Tau A, Cas A, Cyg A and 3C274) which are often used as calibrators at microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The polarization signal in our maps is dominated by synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The distribution of spectral index values between the 11 GHz and WMAP 23 GHz map peaks at 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The measured BB/EE ratio at scales of ℓ = 80 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 for a Galactic cut |𝑏| > 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find a positive TE correlation for 11 GHz at large angular scales (ℓ ≲ 50), while the EB and TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The maps discussed in this paper are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Key words: cosmology: observations – cosmic microwave background ★ E-mail: jalberto@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='es © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05113v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='GA] 12 Jan 2023 2 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1 INTRODUCTION Measurements of the Cosmic Microwave Background (CMB) anisotropies provide one of the most powerful tools in modern cos- mology, playing a fundamental role in our current understanding of the physics of the early Universe and structure formation (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, CMB polarization observations open a window to probe the amplitude of primordial gravitational waves generated during the inflationary epoch (Kamionkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Zaldarriaga & Seljak 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fol- lowing this scientific motivation, observations of B-modes at large angular scales have progressed substantially over the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Current best upper limits on the tensor-to-scalar ratio come from the BICEP/Keck 2018 CMB polarization data (Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021), and give 𝑟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='036 at 95% confidence level, which improves to 𝑟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='032 when adding the latest Planck PR4 data (Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Upcoming ground-based experiments like Simons Observa- tory (Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019) or CMB-S4 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022), and space missions like LiteBIRD (LiteBIRD Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022) will improve these constraints in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to the low amplitude of this primordial B-mode signal, the control and removal of diffuse Galactic foreground contamina- tion in polarization is becoming a key challenge for current and future CMB experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Basically there are two main Galactic foregrounds that are known to emit linearly polarized radiation: the synchrotron emission resulting from cosmic ray electrons ac- celerated around the Galactic magnetic field lines, and the thermal radiation from interstellar dust grains also aligned with the mag- netic field (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016g, 2020d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Anomalous microwave emission (AME) has been also de- tected in intensity, but no polarization has been measured up to date (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although there are theoretical motivations to expect negligible polarization levels if AME is produced by spinning dust grains (Draine & Hensley 2016), improved low frequency observations will be needed to consolidate our understanding of this physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The Planck satellite (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020a) pro- duced seven full sky polarization maps covering the frequency range between 30 and 353 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The Wilkinson Microwave Anisotropy Probe (WMAP) satellite (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013) scanned the full sky in polarization in five bands between 23 and 94 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis of these data shows that, for a B-mode signal with amplitude 𝑟 = 10−3 (which is the target of the LiteBIRD space mission), there is no frequency domain or sky region where the sum of the synchrotron and thermal dust foregrounds is subdominant with respect to the expected CMB B-mode signal (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, further analyses of these and other datasets show increasing evidence of complexity in the spectral and spatial behaviour of the Galactic dust and synchrotron emissions (Choi & Page 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de Belsunce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The situation is particularly complex for the polarized syn- chrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The sensitivity of the low frequency channels from Planck and WMAP does not allow the detection of polarized synchrotron signal at intermediate and high Galactic latitudes, and therefore we are lacking a detailed spectral modelling of this emis- sion precisely in the regions of cosmological interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this context, there is a need for complementing the existing satellite observations with measurements at lower frequencies in order to improve our de- scription of the foregrounds at the required level for B-mode studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' There are only a limited number of radio surveys that preserve the large-scale structure of Galactic emission, and most of them pro- vide only intensity measurements (Haslam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Berkhuijsen 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Jonas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1998), but this situation is now changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The S-band Polarization All-Sky Survey (S-PASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Carretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019) recently provided the first map of the polar- ized radio emission over the southern sky at declinations below −1◦ taken with the Parkes radio telescope at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The C-Band All Sky Survey (C-BASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2018) will cover the full sky at 5 GHz, and the maps of the northern sky will be soon available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' With the aim of providing spectral coverage complementary to WMAP and Planck at intermediate frequencies, the Q-U-I JOint Tenerife Experiment (QUIJOTE, Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2010) is a sci- entific collaboration between the Instituto de Astrofisica de Canarias (IAC), the Instituto de Fisica de Cantabria (IFCA), the Universities of Cantabria, Manchester and Cambridge, and the IDOM company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It has the goal of characterising the polarization of the CMB and other Galactic and extragalactic physical processes in the frequency range 10–40 GHz and at large angular scales ( >∼ 1◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE has been designed to have the required sensitivity to detect a primordial gravitational-wave component if the tensor-to-scalar ratio is larger than 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The experiment is located at the Teide Observatory (altitude of 2,400 m a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='l) in Tenerife (Canary Islands), and con- sists of two telescopes equipped with three instruments: the Multi- Frequency Instrument (hereafter, MFI), operating at 10–20GHz, the Thirty-GHz Instrument (TGI) and the Forty-GHz Instrument (FGI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The two QUIJOTE telescopes, QT-1 (Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2010) and QT-2 (Sanquirce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Sanquirce-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016), are based on an offset crossed-Dragone design with projected apertures of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 m for the primary and secondary mirrors respectively, and provide optimal polarization properties (polarization leakage ≤ −25dB), low sidelobes (≤ −40 dB) and highly symmetric beams (ellipticity ≤ 2 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI is a multi-channel instrument that has been operating between November 2012 and October 2018 mounted on the first QUIJOTE telescope, QT-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI consists of four polarimeters (also called here "horns").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horns 1 and 3 operate in the band 10–14 GHz, while horns 2 and 4 operate at 16–20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using frequency filters in the back-end module (hereafter BEM) of the instrument, each horn provides outputs in two frequency sub-bands, each one with an approximate bandwidth of Δ𝜈 = 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' There are a total of 8 outputs for each polarimeter, and these are then fed into the Data Acquisition Electronics (DAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In total, the MFI provides four fre- quency bands centred around 11, 13, 17 and 19 GHz, with each band covered by two independent horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The approximate angular resolu- tion, given in terms of the full width at half-maximum, is 52 arcmin for the low-frequency bands (11 and 13 GHz), and 38 arcmin for the 17 and 19 GHz channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' During the lifetime of the instrument, we had basically two instrumental configurations for the MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The main difference of the second configuration with respect to the first one is the integration of 90◦ hybrid couplers in each polarimeter, giving correlated outputs in all four detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A more detailed de- scription of the instrument can be found in Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Pérez-de-Taoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2016), and will be included in a future paper (Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A complete description of the MFI instru- ment characteristics, as well as the MFI data processing pipeline, is included in an accompanying paper (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As described in Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2010), most of the QUIJOTE-MFI observing time was dedicated to two main surveys: a shallow Galactic survey (hereafter the "wide survey") covering all the visible sky from Tenerife at elevations larger than 30◦, and a deep cosmological survey covering approximately 3 000 deg2 in three separated sky patches in the northern sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition to those MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 3 two main surveys, a fraction of the MFI observing time was dedi- cated to raster scan observations in some selected Galactic regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Data from some of those MFI raster scan observations were already presented in three QUIJOTE collaboration publications (Génova- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019), where we charac- terised the presence of AME towards several Galactic molecular complexes, as the Perseus region, W43, W47 or Taurus, and to- wards a supernova remnant, W44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, the study of W43 provides the strongest upper limits to date on the polarization frac- tion of the AME (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Additional raster scan observations were carried out in W51, IC443, rho-Ophiucus, and M31, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A preliminary version of the MFI wide survey maps, in com- bination with C-BASS North data, were used in the study of the 𝜆-Orionis region (Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This paper presents the final maps of the QUIJOTE-MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Section 2 de- scribes the observations and the data processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final maps are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The validation and characterisation of these maps is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' An assessment of the overall calibration uncertainty of the maps is discussed in Section 5, while Section 6 describes the generation of specific noise simulations for the QUIJOTE MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Sections 7, 8 and 9 discuss some of the basic properties of the maps both in real and harmonic space, including the photometry results of some bright radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fi- nally, Section 10 describes the data products and associated scien- tific papers accompanying this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All of them are devoted to the understanding of the low frequency Galactic foregrounds in in- tensity and polarization, either in the full QUIJOTE MFI footprint or in localised regions, and using various analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The conclusions of this work are presented in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2 THE QUIJOTE-MFI WIDE SURVEY DATA The QUIJOTE wide survey is a shallow survey which covers all the visible sky from the Teide Observatory (latitude +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3◦) with elevations greater than 30◦ (more than 29 000 deg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This was one of the main scientific objectives of QUIJOTE (Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2012b), and in particular, of the MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This paper presents the QUIJOTE MFI wide survey maps, which were obtained with approximately 9 000 h of observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The four final maps at nominal frequencies 11, 13, 17 and 19 GHz, smoothed to 1 degree resolution, are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1, 2, 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps were generated using the HEALPix1 pixelization scheme (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2005) with 𝑁side = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In HEALPix the sphere is divided into 12𝑁side2 pixels of equal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, 𝑁side = 512 corresponds to a pixel size of approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 arcmin on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 5 also shows the polarized intensity (𝑃 = √︁ 𝑄2 + 𝑈2), the polarization angle direction2 (𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 arctan(−𝑈/𝑄)), and the direction of mag- netic field lines for the 11 GHz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the following subsections we describe the observations, the data processing pipeline, the map- making and the specific post-processing and recalibration applied to these maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1 https://healpix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='io 2 QUIJOTE polarization maps use the COSMO convention from HEALPix, so we use a minus sign in the definition of 𝛾 to recover the IAU convention for the angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Observations The maps described in this paper are based on MFI observations carried out between May 2013 and June 2018 using the so-called "nominal mode", which consists of continuous (360◦) azimuth scans at a constant telescope elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The default azimuth scan speed was 𝑣AZ = 6 deg s−1 from the beginning of the survey until January 9th 2014, but this was increased to 𝑣AZ = 12 deg s−1 after this date, in order to reduce the 1/ 𝑓 noise contribution in the intensity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this observing mode, every day each MFI horn covers a continuous band of 360◦ in right ascension, and a certain declination range specified by the elevation of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As in all QUIJOTE- MFI observations, and in order to minimize systematic effects in the polarization parameters, observations are carried out in four discrete positions of the polar modulators 𝜃pm =(0◦, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦, 45◦ and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the wide survey, each observation at a given elevation and modulator angle position has a typical duration of 24 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The combination of multiple elevations allows us to obtain a more homogeneous sampling of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 1 contains the final set of telescope elevations considered here to produce the maps, to- gether with the total number of hours observed and used in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In total, there are approximately 9 200 h of observations, equivalent to 383 observing days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Almost all of this observing time was suit- able for use in the preparation of the intensity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, the final polarization maps only use of the order of 5 700 h, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Observations are also separated in periods of several months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The definition of each period is usually associated with changes either in the MFI instrument configuration, telescope configuration, or simply to new observing cycles after instrument maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A complete description of those periods, as well as the associated instrument changes, can be found in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that for the MFI wide survey, we conducted observations only during periods 1, 2, 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The global dates and effective epoch (year) for each of those periods are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As noted in this table, an extended shielding was installed in the first QUIJOTE telescope (QT-1) at the beginning of period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The main reason for this was to minimize the impact of far sidelobes due to the emission of geo-stationary satellites, which were particularly important for horn 1 (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, during the operations horn 1 was either not operative (periods 5 and 6) or had problems with the positioning of the polar modulator (period 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Because of these reasons, although wide-survey maps of horn 1 have been produced for internal consistency tests, they have not been used for this paper because they are significantly affected by systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Data processing pipeline A complete description of the MFI data processing pipeline can be found in the MFI pipeline paper (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we summarize the basic characteristics of the MFI data, and we discuss those aspects which are specific of the MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Each MFI polarimeter is divided into a lower and upper band of approximately 2 GHz bandwidth which is defined by the band- pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Each sub-band has four outputs, which are labelled as (𝑉x+y,𝑉x−y,𝑉x,𝑉y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The first two outputs are called "correlated" channels because in the first (original) configuration of the instru- ment they passed through a 180◦-hybrid, and therefore they have correlated (common) 1/ 𝑓 noise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The second pair is called "uncorrelated" channels, and in the original configuration provided two outputs with independent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The first instrument configu- MNRAS 000, 1–58 (2022) 4 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI maps at 11 GHz in Galactic coordinates, smoothed to 1 degree resolution and using 𝑁side = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: intensity 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle: polarization 𝑄 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: polarization 𝑈 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUJOTE1H311GHz(1deg) 5 mk 20QUJOTEQH311GHz(1deg) mKQUOTEUH311GHz（1deg） mKQUIJOTE MFI wide survey 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI maps at 13 GHz smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: intensity 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle: polarization 𝑄 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: polarization 𝑈 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUJOTE1H313GHz(1deg) 5 mk 20QUJOTEQH313GHz(1deg) 1 mKQUJOTEUH313GHz(1deg) 1 mK6 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI maps at 17 GHz smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: intensity 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle: polarization 𝑄 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: polarization 𝑈 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUIJOTE117GHz combined H2+H4(1deg) 5 mk 20QUijOTEQ17GHzcombinedH2+H4(1deg) mkQUjOTEU17GHzcombinedH2+H4(1deg) mkQUIJOTE MFI wide survey 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI maps at 19 GHz smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: intensity 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle: polarization 𝑄 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: polarization 𝑈 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUJOTE119GHz combinedH2+H4(1deg) 5 mk 20QUljOTEQ19GHzcombinedH2+H4(1deg) mkQUljOTEU19GHzcombinedH2+H4(1deg) mk8 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI maps at 11 GHz smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: polarized intensity 𝑃 = √︁ 𝑄2 + 𝑈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle: polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: Polarization angle at 11 GHz, rotated by 90◦ to indicate the direction of the Galactic magnetic field projected on the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colours represent the polarized intensity signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The "drapery" pattern was obtained with the healpy routine line_integral_convolution, and it is smoothed to 2◦ for display purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUUJOTEP11GHz(1deg) 0 mk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3QUJOTEang11GHz(1deg) 90 deg 90MFI 11GHz - LICQUIJOTE MFI wide survey 9 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' List of telescope elevations used for the wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The second column indicates the total observing time (𝑇observed) in hours dedicated to each elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Columns 3 to 6 show the total observing time for the actual subset of observations used for the final intensity (𝑇used,I) and polarization (𝑇used,P) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the later case, different subsets of data are used for each particular horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Observations are separated in periods (column 7), which correspond to specific epochs (column 8) and instrumental configurations (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Elevation (◦) 𝑇observed (h) 𝑇used,I (h) 𝑇used,P,H2 (h) 𝑇used,P,H3 (h) 𝑇used,P,H4 (h) Period Range of Dates 30 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 0.' metadata={'source': 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430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 6 03/2017–04/2017 70 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 6 02/2017–04/2017 TOTAL: 9193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 8476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 5813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 6824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 4720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Definition of the four observing periods during which we carried out wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Last column indicates the instrument configuration and main changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Configuration 1 corresponds to the original MFI design (Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2012), while configuration 2 corresponds to the installation of 90◦-hybrids (Pérez-de-Taoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Period From To Effective year Comments (dd/mm/yyyy) (dd/mm/yyyy) 1 12/11/2012 10/04/2014 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 Configuration 1 for all horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' No extended shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2 11/04/2014 30/11/2015 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 Horn 1 in configuration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Extended shielding installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5 01/05/2016 14/10/2016 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 All horns in configuration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horn 1 not operative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6 15/10/2016 01/11/2018 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 All horns in configuration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horn 1 not operative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ration (Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2012) was used during periods 1 and 2 (see Table 2), but a new configuration was later implemented using 90◦- hybrids (Pérez-de-Taoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this second configuration, all MFI channels are formally correlated, but for historical reasons we maintain the notation of correlated and uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The sum of pairs of channels provides two independent mea- surements of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' for the first MFI configu- ration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' we have 𝑉x + 𝑟u𝑉y = 𝑠x𝑔2𝐼 (1) 𝑉x+y + 𝑟c𝑉x−y = 𝑠x+y𝑔2𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2) while the difference of the pairs of channels provides two measure- ments of the linear polarization 𝑉x − 𝑟u𝑉y = 𝑠x𝑔2� 𝑄 cos(4𝜃pm + 2𝛾p) − 𝑈 sin(4𝜃pm + 2𝛾p) � (3) 𝑉x+y − 𝑟c𝑉x−y = 𝑠x+y𝑔2� 𝑄 sin(4𝜃pm + 2𝛾p) + 𝑈 cos(4𝜃pm + 2𝛾p) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (4) where 𝑉𝑖 represents the output voltage for channels 𝑖 ∈ {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' x − y},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝑠x and 𝑠x+y are the responsivities of those branches in the MFI instrument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝑔 represents the voltage gain of the two MFI Low Noise Amplifiers (here taken to be the same in the two LNAs for simplicity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝑟c and 𝑟u are the so-called r-factors which measure the possible gain and responsivity imbalance in the pair of channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝜃pm is the position angle of the polar modulator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and 𝛾p is the parallactic angle (see details in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When the two channels in the pair have correlated noise, then the difference cancels significantly the 1/ 𝑓 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the MFI pipeline, maps for correlated and uncorrelated channels are produced separately, and combined afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to their noise properties, in polarization we use only those pair of channels with common 1/ 𝑓 properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the "correlated" channels during pe- riods 1 and 2, and both of them ("correlated" and "uncorrelated" channels) for periods 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The MFI data sampling rate is 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the wide survey, all time streams (hereafter Time-Ordered Data or TODs) are binned in 40 ms samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note that this is different from the binning scheme of 60 ms used for raster scan observations in the past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Génova- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017), due to the higher azimuth scan speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The bin- ning process allows us to assign a variance 𝜎2 𝑖 to each binned sam- MNRAS 000, 1–58 (2022) 10 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE-MFI basic peformance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Values for 11 and 13 GHz correspond to horn 3 of MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Values for 17 and 19 GHz have been obtained as the weighted average of horns 2 and 4, using the relative weights described in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Parameter 11 GHz 13 GHz 17 GHz 19 GHz MFI horns contributing to these bands 3 3 2,4 2,4 Centre frequency (nominal), 𝜈0 (GHz) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 Effective frequency for 𝛼 = −1, 𝜈𝑒 (𝛼 = −1) (GHz) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='85 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='85 Bandwidth (GHz) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 Beam FWHM (arcmin) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='84 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='95 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='32 Main beam solid angle, Ωmb (10−4sr) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='748 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='781 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='362 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='428 Beam ellipticity𝑎, 𝑒 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='035 Antenna sensitivity, Γ (𝜇KCMB/Jy) 961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 White-noise level in timelines (𝜇KCMBs1/2) 858 697 773 866 Knee frequency 𝑓k in polarization (mHz) 254 198 223 556 1/ 𝑓 slope in polarization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 Overall calibration uncertainty I (%) 5 5 5 5 Overall calibration uncertainty Q,U (%) 5 5 6 6 𝑎 The ellipticity is defined here as 𝑒 = 1 − FWHMmin/FWHMmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Colour correction coefficients, 𝐶(𝛼, 𝜈0) = 𝑐0 + 𝑐1𝛼 + 𝑐2𝛼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour corrected temperature is obtained as 𝐶(𝛼, 𝜈0)𝑇 , being 𝑇 the uncorrected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Band 𝜈0 𝑐0 𝑐1 𝑐2 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0015 13 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0012 17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0007 19 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0008 ple 𝑖, which we used to define the associated weights (𝑤𝑖 = 1/𝜎2 𝑖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When propagated through the entire pipeline, the resulting weight maps are used for the combination of maps from correlated and uncorrelated channels, and will be used also in the noise character- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 3 contains the summary of basic parameters (central fre- quencies, beams, solid angles) for all MFI horns, extracted from Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also include the calibration uncer- tainties discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5, and representative noise characteristics (knee frequencies and 1/ 𝑓 slopes) that we have obtained from this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 4 also presents the colour corrections for these maps, de- rived from the associated bandpasses as explained in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Colour corrections are presented here in terms of sec- ond order polynomials as a function of the spectral index 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For a sky emission having a flux density law 𝑆𝜈 ∝ 𝜈𝛼, the coefficients 𝐶(𝛼, 𝜈0) provide the multiplicative correction factor to the mea- sured flux density for the MFI frequency map at nominal frequency 𝜈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These corrections are identical for intensity and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Throughout the paper, we use the following notation to refer to specific MFI maps per horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We will use three numbers, the first one refering to the horn number (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2, 3 or 4), and the other two indicating the nominal frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11, 13, 17 or 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, the 19 GHz map for horn 4 will be cited either as 𝑚4,19, 𝑚419, or directly, 419 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that each map will be made, in principle, from the contribution of both the correlated and uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In some case, we use the same notation to refer to channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, the correlated channels of 419 are obtained from the 𝑉x+y and 𝑉x−y outputs of horn 4 at 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the following, we discuss specific additions to the MFI pipeline in the case of the wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we discuss the gain model for wide survey data and the specific data flagging applied in ”nominal mode".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' After this, we present our approach to correct for Radio Frequency Interference (RFI) signals and at- mospheric contamination in the MFI wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For these corrections, the general philosophy adopted in our pipeline follows a two step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We first implement specific methods to de- tect and mitigate the effect of RFI and atmospheric signals both at the TOD (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4) and at the map-level in the post-processing stage (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, a detailed assessment is made later of residual signals in the maps by a variety of techniques (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In practice, the values of uncertainties in calibration and other error budgets are increased appropriately if there is clear evi- dence of residual effects still being present in the maps (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Gain model Gain calibration and the associated relative gain factors (𝑟c and 𝑟u) between pairs of channels are based on Cas A and Tau A observa- tions taken during each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Relative gain variations with respect to the mean gain value 𝐺0 during the full period are traced using the signal of a thermally stabilized calibration diode, located at the centre of the secondary mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Every 30 s, the diode injects a signal during 1 s, which is used to measure the relative gain of each chan- nel, 𝛿𝐺(𝑡) ≡ 𝐺(𝑡) −𝐺0 (see Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Nominal mode observations used for the wide survey usually have a duration of one day for each polarimeter position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Specifically for this nominal mode data, a smooth (interpolated) gain model is obtained by applying a top-hat smoothing kernel on the individual gain measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The width of this kernel is 30 minutes for low frequency channels, and 120 minutes for high frequency ones, due to the different signal-to-noise ratio of the diode signal in the differ- ent channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have checked that the typical MFI gain variations occur on timescales longer than those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These interpolated models are used to correct the instrument gain as 𝐺(𝑡) = 𝐺0 � 1 + 𝛿𝐺(𝑡) 𝐺0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (5) Once these interpolated gain models are generated for the entire survey, they are inspected in order to find residual features (peaks or jumps) in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These features are introduced in flagging MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 11 tables which are later applied during the generation of the calibrated TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Data flagging Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023) describes the basic data flagging that is applied by default to all MFI observations, including flags due to voltage ranges, house-keeping parameters, emission of the Sun and Moon (using a 10◦ exclusion radius), and also the emission of geo-stationary satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, this last flagging produces the empty strip around declination zero degrees that is seen in the 11 and 13 GHz maps (Figures 1 and 2), and also the noise increase in the same region in the 17 and 19 GHz maps (Figures 3 and 4), due to the lower number of independent crossings in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the wide survey, a specific flagging based on the root mean square (rms) of the data in each scan has been implemented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A first version of the wide survey maps is produced with the default pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From here, and separately for each period, we compute the rms of the data minus the reprojected version of that map onto the TOD, in scales of 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This time value corresponds to the length of one azimuth scan at the default scanning speed, and to the length of half azimuth scan for the scanning speed used in part of period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Histograms with the distribution of these rms values are built for each channel and period, and are used to flag those scans with extreme rms values (either above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 times the median rms value in the entire period, or below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 times that median rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The fraction of excluded data using this procedure depends on the channel, but typically is of the order of 10–20 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Once this flagging is applied, no obvious residual spikes or rings are visible in the reconstructed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, for the final wide survey maps we also exclude Jupiter, Venus and Mars, using a 2◦ exclusion radius directly in the TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix A contains detailed tables with the percentage of used (and flagged) data for each MFI channel in every observing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Those fractions of used data apply to the total number of used hours in each case, which were listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On average we are using 61 % of the data after applying all the different flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Out of the flagged 39 %, most of it (approximately three quarters) is excluded in the specific post-processing stage described in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The percentage of used data was slightly lower in period 2 (52 %), and higher in period 6 (68 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 RFI correction Specifically for the wide survey data, residual random spikes as well as possible RFI signals from satellites not identified in our standard pipeline are flagged using a dedicated matched-filter code that is applied to the one-dimensional TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The only assumption is that the object to be detected is unresolved, and thus should match the beam profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The code3 excludes the location of the known bright radio-sources, which are also easily detected in the TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Residual RFI signals appear at fixed azimuth (AZ) locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of QUIJOTE MFI, most of these signals are due to the radio emission of geo-stationary satellites entering through the beam far sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These signals were particularly visible in period 1 and at low frequencies (horns 1 and 3), until the installation of the extended shielding of the first QUIJOTE telescope was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All other periods are much less affected, due to the significant suppression of the far sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Because of this reason, period 1 was used for the intensity maps only, and not for polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to remove 3 https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='com/HerranzD/quijote-satdet these RFI signals, we generate spatial templates in the azimuth direction, by obtaining stacks of the TOD signal as a function of AZ, 𝑓 (AZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These templates are computed for each period and each elevation separately, and thus rely on the assumption that the RFI signal is stable in time during the whole period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The templates are generated both for the sum and difference of MFI channels, and thus, they are applied to the intensity and the polarization TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, a smoothed version of these templates (in scales of 10◦) is subtracted from the TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 6 shows two examples of the global RFI patterns removed using this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These figures are obtained as the difference between the end-to-end MFI maps with and without applying the RFI correction at the TOD level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also note that once the final maps are produced, any residual RFI signals are effectively corrected in the post-postprocessing stage, using a function of the declination as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Some remaining RFI features and glitches are removed after a careful inspection of the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this purpose, separate maps for each elevation and period are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Once a particular RFI feature is identified in these maps, the corresponding location is introduced in specific flagging tables for each period and elevation, which are later applied to the calibrated TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Atmospheric correction Although the observations are done at (nominal) constant elevation, there are still some residual variations due to changes in the atmo- spheric contribution along different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These variations are seen in the data as correlated patterns repeating in azimuth on very large angular scales, and with the amplitude increasing strongly with frequency, as expected for MFI frequencies due to the proximity of the 22 GHz atmospheric water line (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Paine 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It also evolves and changes on the scale of several hours, which is expected due to varying integrated water vapour content along lines of sight as weather systems blow over the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It is possible to try to remove these effects especially at the more troublesome higher frequencies by a Principal Component Analysis (PCA) decomposition to look for these correlated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To model this atmospheric component in the MFI intensity data, only broad scale features are removed by using baselines up to only 5 harmonics over the azimuth scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A mask is used to avoid bright emission from the Galactic plane and strong point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The baseline atmospheric patterns are generated over an hour, as a compromise between good signal to noise and the time evolution of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The PCA decomposition method used is implemented in Python, using the sklearn module (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2011) on all the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The first most significant component found is one that in- creases strongly with frequency, with the spectrum expected for water vapour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A histogram of the ratios between 17 and 19 GHz, the two most strongly affected frequencies, shows a clear broad peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 near the values expected from atmospheric models for the Teide Observatory of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='49 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Paine 2019, and typical PWV conditions of 3–4 mm), although this sits on a smaller but much broader distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Points outside the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 appear to be for dryer conditions, with the implication that the water vapour signal is too weak to be reliably recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It was decided to use this range ratio of 17 to 19 GHz signal as an indicator of a usable atmospheric signal that can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The removal is done by subtracting the PCA template with the coefficient found for each frequency channel at the TOD level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps of this atmospheric emission can be produced running the full pipeline with and without this atmospheric correction, and MNRAS 000, 1–58 (2022) 12 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' RFI patterns removed from the maps of the QUIJOTE MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top row corresponds to the RFI emission at 11 GHz (horn 3, labelled as "311"), while the bottom row corresponds to 17 GHz (horn 4, labelled as "417").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From left to right, we show the residuals for intensity, Stokes Q and Stokes U parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour scale is the same in all six panels, corresponding to the temperature range ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For visualization purposes, all maps are smoothed to one degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' then taking the difference of the two resulting maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The atmo- spheric emission maps for horns 3 and 4 are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The map for horn 2 is similar to the one for horn 4, so it is omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, this atmospheric contribution is more rel- evant at higher MFI frequencies, and affects large angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown below (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5), when doing a spherical harmonic expansion of the maps, this correction is only relevant in the in- tensity maps at multipoles ℓ ≲ 15 for 11 GHz, and ℓ ≲ 25 for 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' No atmospheric correction is needed in polarization for the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When a similar procedure is applied to the polarization data, the results are consistent with essentially unpolarized atmospheric emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Map-making The QUIJOTE MFI wide survey maps are produced using the PI- CASSO code (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021), a destriping algorithm based on the MADAM approach (Keihänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2005, 2010) but specifi- cally implemented and optimised for QUIJOTE MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The destriping technique corrects for a correlated noise component by modelling the 1/ 𝑓 drifts in the TOD with a set of consecutive offsets with a given time length 𝑡b, the so-called baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The PICASSO code has been tested extensively using realistic simulations matching the actual observations of the MFI wide survey and with realistic noise properties (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In these conditions, the reconstructed maps preserve all angular scales with high fidelity, and in particular, we expect a signal error better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 per cent at 20 < ℓ < 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Those realistic simulations were also used to set the reference parameters adopted for the production of the final MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we use a baseline length of 𝑡b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 s for the entire survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps are generated using the HEALPix pixelization scheme with 𝑁side = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The specific priors for the 1/ 𝑓 noise properties (knee frequency 𝑓𝑘, slope 𝛾, and cutoff frequency 𝑓cut) are shown in Table 5, both for the intensity and polarization maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the later case, the parameters are different depending on the noise levels of the corresponding pair of channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' if they are correlated or uncorrelated channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As discussed in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Map-making parameters and related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We consider three different cases of use with the PICASSO code: intensity maps, polarization maps with correlated channels, and polarization maps with uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For each case, we quote the prior values for the knee frequency 𝑓k, the slope of the 1/ 𝑓 noise component 𝛾, and the low cut-off frequency 𝑓cut, as well as the 𝑁side value of the HEALPix map and the baseline length 𝑡b (in seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Case 𝑓k 𝛾 𝑓cut 𝑁side 𝑡b [Hz] [Hz] [s] I 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='033 512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Q,U corr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='033 512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Q,U uncorr 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='033 512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 (2021), those priors are assumed to be stationary parameters for the whole survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Post-processing of MFI wide survey maps 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Combination of maps For each horn and frequency sub-band, maps for the correlated and uncorrelated pairs are produced running the PICASSO code separately for each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These maps are combined at this post-processing stage, using the weight maps which are also pro- duced by the map-making code as the propagation of the individual weights for each sample in the binned TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The combination of correlated (𝑥c) and uncorrelated (𝑥u) maps is done with the usual formula for the weighted arithmetic mean: 𝑚 = 𝑤c𝑥c + 𝑤u𝑥u 𝑤c + 𝑤u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (6) Given that both correlated and uncorrelated channels share the same amplifier, we expect a high level of correlation between the two intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown below in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4, this cor- relation is indeed of the order of 90–95 per cent for the intensity channels, and consistent with zero for the polarization ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Al- though in principle it is possible to construct a minimum variance MNRAS 000, 1–58 (2022) RFlmap (311,I) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2RFl map (311,Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2RF map (311,U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2RFl map (417,I) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2RFl map (417,Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2RFl map (417,U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2QUIJOTE MFI wide survey 13 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Atmospheric pattern removed from the intensity maps of the QUIJOTE MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From top to bottom, we have the atmospheric emission at 11 GHz (horn 3), 13 GHz (horn 3), 17 GHz (horn 4) and 19 GHz (horn 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour scale is the same in the four panels (±3 mK), in order to visualise the increasing contribution of the atmospheric emission at higher MFI frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For visualization purposes, all maps are smoothed to one degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' estimator accounting for these correlations in the intensity pairs, we still use equation 6 for the combination of the intensity (correlated and uncorrelated) maps, in order to have a more robust estimate of the combination (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Schmelling 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From equation 6, we can derive the expression for the weight map of the linear combination as 𝑤 = (𝑤c + 𝑤u)2 𝑤c + 𝑤u + 2𝜌√𝑤c𝑤u , (7) where 𝜌 stands for the correlation fraction between correlated and uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The map-making code also produces an estimate of the co- variance matrix in polarization, 𝑐𝑜𝑣(𝑄,𝑈), as well as the condition number (𝑟cond) map (see equations 44 and 45 in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Before forming the combination of the polarization maps in the wide survey, those pixels with 𝑟cond > 3 are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In practice, this only affects a small number of pixels close to the boundary of the satellite strip, as well as to the north celestial pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, for the 419 map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' horn 4 at 19 GHz) the number of affected pixels is slightly larger in those areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix C contains the 𝑟cond maps for all the MFI wide survey maps, together with other relevant maps, as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Once the combination of the correlated and uncorrelated maps is carried out in polarization, the corresponding weight maps (𝑤𝑄, 𝑤𝑈) and covariance matrices 𝑐𝑜𝑣(𝑄,𝑈) are also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix C also presents images of the 𝑐𝑜𝑣(𝑄,𝑈) maps for all horns and frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These maps show that, as expected, the normalized covariance 𝑐𝑜𝑣(𝑄,𝑈)/(𝜎𝑄𝜎𝑈) is very small (well below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 %), so effectively each pair of 𝑄 and 𝑈 maps are almost independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Residual interference: the FDEC filtering After the map-making step, the resulting maps still present some residual RFI and large-scale patterns, which are corrected during this post-processing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3, residual RFI signals appear at fixed azimuth locations, so during the map-making process these features are projected onto the maps in stripes of con- stant declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This residual RFI is removed using a function of the declination, 𝑓 (𝛿) (hereafter FDEC4), which is extracted directly from the maps as the median of all pixels with the same declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This template function is built using a |𝑏| < 10◦ mask to exclude the Galactic emission, and specific masks in intensity and polarization for each frequency channel excluding the 10 per cent of the brightest pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The procedure is applied both in intensity and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, the maps are first rotated to local (equatorial) coor- dinates in order to extract the correction function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this way, the RFI contamination from static sources in local coordinates appears as a constant signal in a given declination band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 8 shows the correction functions for intensity and po- larization for all MFI maps based on correlated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similar curves are obtained for uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note that in this fig- ure, the panel for Stokes Q parameter corresponds to equatorial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected for RFI signals, these correction functions are larger in the vicinity of the geo-stationary strip (around declina- tion zero) and at low declinations (corresponding to low elevation values of the telescope, where the RFI is expected to be larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also note that they are also larger in intensity than in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Once these correction functions 𝑓 (𝛿) are derived, they are re- projected onto a map in order to produce a RFI template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='com/jarubinomartin/sancho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='git MNRAS 000, 1–58 (2022) Atmosphere (311,D) mk 3 3Atmosphere (313,1) mk 3 3Atmosphere (417,) mK 3 3Atmosphere (419,D) 3 mk 314 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Examples of 𝑓 ( 𝛿) correction functions (FDEC) to remove resid- ual RFI in the MFI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Stokes I FDEC for correlated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: Stokes Q parameter in equatorial (RADEC) coordinates for corre- lated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' templates are subtracted from the data before carrying out the com- bination of correlated and uncorrelated maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 9 illustrates the final FDEC correction applied to the maps of horn 3 at 11 GHz, after combining the correlated and uncorrelated maps in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Monopole and dipole removal Finally, a monopole and a dipole component are subtracted from the correlated and uncorrelated maps before their combination, using the remove_dipole routine of HEALPix with a Galactic mask excluding the region |𝑏| < 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The removed dipole is consistent with the expected CMB dipole, as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Effective transfer function The PICASSO map-making code essentially preserves all angular scales in the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The expected signal error is better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 per cent in the multipole range 20 < ℓ < 200 both for intensity and polarization maps, and stays well within one per cent down to ℓ = 10 (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021, and see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' How- ever, some of the specific procedures applied in the MFI pipeline to correct for RFI signals and atmospheric contributions might have an impact on the effective transfer function of the wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we should consider the impact of the RFI (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Example of the effective correction map based on a function declination (FDEC) for the 311 map (horn 3 at 11 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Stokes I, with a colour scale in the range ±2 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Middle and bottom: Stokes Q and U parameters, with a colour scale in the range ±1 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and atmospheric (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4) corrections at the TOD level, and the RFI correction at the post-processing stage using a function of the declination FDEC (see previous subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In terms of their am- plitudes at the map level, the largest correction corresponds to the third case (subtracting a function of declination), so we discuss the transfer function of this case in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It is important to note that, by construction, after applying this FDEC correction, the zero mode at constant declination will be missing from the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To characterize its impact on the effective transfer function of the wide survey, we follow the methodology described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 of Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we use simulations in the ideal case including CMB and foregrounds, but without a noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The transfer function is then computed in terms of the power spectra of the map with residuals 𝐶res ℓ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' reconstructed map minus input sky) and that of the reconstructed map 𝐶map ℓ , both MNRAS 000, 1–58 (2022) Stokes I, corr 217c 219c 311c 313c 417c 419c ()[mk] 2 20 0 20 40 60 80 Declination [deg]Stokes corr 10 217c 219c 311c 313c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 417c 419c [mk] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 20 0 20 40 60 80 Declination [deg]FDEC (311, I) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKFDEC (311, Q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKFDEC (311, U) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKQUIJOTE MFI wide survey 15 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Transfer function (TF) of the QUIJOTE MFI wide survey map at 11 GHz, after accounting for the post-processing stage of a subtraction of a function of the declination (FDEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The TF for TT is marked with circles connected by red solid lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the EE case with triangles and red dashed lines, and the BB with diamonds and red dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, in green we also include the TF of the PICASSO map-making code (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' computed within the same mask, using this expression: 𝑓ℓ = 1 1 − 𝐶res ℓ /𝐶map ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (8) Figure 10 presents the result obtained for the 311 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, we find that the FDEC correction is affecting low multipoles (ℓ ≲ 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reconstruction of the sky signal is better than one per cent down to ℓ ≈ 10 in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, the correction stays within one per cent down to ℓ ≈ 30, being at ℓ = 10 of the order of 20 % for BB, and 5 % for EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Because of this reason, and although we are able to reconstruct the sky signal to lower multipoles, as a conservative approach the power spectra analyses in this paper will be restricted to ℓ ≥ 30, so no transfer function correction will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix B contains a more detailed discussion on how a given map is affected by the FDEC filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The impact of the RFI and atmospheric corrections at the TOD level is discussed in detail in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4, although we anticipate that their impact is lower than the 𝑓 (𝛿) discussed here (except maybe for 19 GHz, where the atmospheric contribution becomes comparable to the FDEC correction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 Recalibration of the wide survey maps using Tau A Once the MFI wide survey maps are produced using the pipeline described above, we re-evaluate three aspects of the calibration using Tau A: i) the global calibration scale in intensity, ii) the polarization angle calibration, and iii) the polarization efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We discuss them in detail here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Global recalibration in intensity Tau A and Cas A are the two main primary calibrators of QUI- JOTE MFI (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Daily observations of these sources in raster scan mode are used to obtain the overall gain scale in intensity for each MFI channel in every observing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' How- ever, as daily calibrator observations might suffer from 1/ 𝑓 noise and other uncertainties, we recalibrate the MFI wide survey maps in the post-processing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this recalibration, we use Tau A as the reference source, because it is located on a cleaner background than Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this, we first generate wide survey maps for each individual period (four maps in total for each horn and frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These four maps per period are degraded to one degree angular resolution, and then we apply beam fitting photometry (hereafter BF1d) on Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The derived flux densities are compared, accounting for colour corrections, with a spectral emission model that we have specifically obtained for Tau A, using WMAP and Planck data together with some ancillary measurements, and applying the same BF1d methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The new model will be presented and discussed in detail in a separate paper (Génova-Santos & Rubiño-Martín, in preparation), and builds on that presented in Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2011), but including several improvements: i) improved treatment of the colour-corrections and beam effects on WMAP data, ii) inclusion of Planck data, iii) improved variability model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The adopted Tau A model for the recalibration of MFI maps has the shape 𝑆𝜈(Tau A) = 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 � 𝜈 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 GHz �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='297 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (9) This model is evaluated at epoch 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3, which corresponds to the effective central epoch of the wide survey, and we use a secular decrease of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='218 % yr−1 (Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From this compar- ison, we derive global recalibration factors for each MFI frequency map and for each individual period, accounting for the secular de- crease of Tau A and the effective epochs in each period (see values in Column 4 of Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The mean value of these recalibration fac- tors results in an overall 4 per cent recalibration of the wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The accuracy of the MFI wide survey intensity calibration is discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Polar angle recalibration The reference angle for each MFI polarimeter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the reference for 𝜃pm in equations 3 and 4) changes across the spectral band, and thus from band to band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this reason, the reference angle for each frequency map is calibrated separately, despite of the fact that the two frequency bands of the same horn share the same polar modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This procedure is based on daily Tau A observations, and it is described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, the adopted model for the Tau A angle in Galactic coordinates is given by 𝛾Tau A = 𝛾0 + 𝑅𝑀𝜆2, (10) where 𝑅𝑀 = −1406 ± 12 deg m−2 and 𝛾0 = −88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='31◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our daily calibration provides a reference polar angle for Tau A with a statistical error of approximately 1◦ within a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' But similarly to the intensity calibration, daily observations of Tau A might suffer from 1/ 𝑓 noise or other effects, so the polar angles of the final wide survey maps are recalibrated in each period with Tau A again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As for the global recalibration in intensity, we also use BF1d in Tau A to extract the fluxes in Stokes Q and U parameters in the maps per period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From there, recalibration offsets in the reference angles are computed for each channel and each period, and applied in order to generate the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The accuracy of the angle calibration in the MFI wide survey is discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) Transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 EE = 1/(1 - C, res/Ce,map) BB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 FDEC TT FDEC EE FDEC BB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 101 10216 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Polarization efficiency for horns 2, 3 and 4 in period 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars for all measurements are 2 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel 𝜌corr 𝜌uncorr 217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='93 419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Change in the polarization efficiency for horns 2, 3 and 4 in period 6 due to errors in the 𝑟-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel Horn 2 Horn 3 Horn 4 Low freq, corr −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='006 High freq, corr −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='016 Low freq, uncorr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 High freq, uncorr −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='020 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='011 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Polar efficiency Detailed measurements of the polar efficiency of the MFI polarime- ters in horns 2, 3 and 4 were obtained in period 6, once the MFI observations concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The description of the instrumental setup and the final measurements are presented in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), and summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to transfer this polar efficiency information to the other observing periods where we do not have laboratory measurements, we use again BF1d photometry on Tau A, using the MFI wide survey maps per period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The polar efficiency in each period 𝑝 is transferred from period 6 according to the relative value of the Tau A polarized intensity 𝑃TauA(𝑝) in that period and in period 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' using the ratio 𝑃TauA(𝑝)/𝑃TauA(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On average, this photometry method introduces errors of approximately 1 % for horn 3, and 2 % for horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we also account for possible errors in the determination of the 𝑟 factors in equations 3 and 4, using wide survey data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown in Appendix D, an error 𝜖 in the determination of the 𝑟 factors translates into a modification of the polar efficiency, and the appearance of a small leakage term in the TOD polarization timeline which is proportional to the intensity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the PICASSO map-making code to fit for an intensity-to-polarization leakage global component in period 6 data, in a two step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we solve for the intensity map 𝐼 for each case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' horn, frequency and channel), and then we use it to fit for an additional term 𝛼𝐼 when solving for the polarization map in equations 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These values are used to correct for the polar efficiency of each channel in period 6, using the equations derived in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 7 shows the effective correction terms 𝛼 ≡ 𝜖/(2𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We can see that in the case of horn 3, this correction introduces a change of 2–3 per cent in correlated channels, and below 1 per cent for uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horn 4 is almost unaffected, while the largest correction factor appears for the correlated channels in horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The accuracy of the polar efficiency calibration in the MFI wide survey is discussed again in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3 MFI WIDE SURVEY MAPS: INTENSITY AND POLARIZATION Following the methodology described in the previous section, we produced intensity and polarization maps for each MFI horn and fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Images of these individual maps (per horn and frequency) are shown in Appendix C, at their original resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the angular resolution listed in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The resulting maps cover a sky fraction of 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='71 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='73 (equivalent to sky areas of 30 900, 29 300 and 30 100 deg2) for horns 2, 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All MFI maps are produced in CMB thermodynamic units (mKCMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For simplicity, throughout this paper we drop the subindex CMB and use the notation mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Nevertheless, we recall that the correction to Rayleigh-Jeans units is very small at MFI frequencies (at most 1 per cent at 19 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Smoothed maps at 1◦ resolution are gen- erated by convolving those original maps with the corresponding transfer function 𝑇ℓ ≡ 𝑊1 deg ℓ /𝑊MFI ℓ , which converts the spherical harmonic window function for each horn (𝑊MFI ℓ ) into a gaussian beam with FWHM= 1◦ (𝑊1 deg ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps are displayed in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that QUIJOTE-MFI Stokes Q and U param- eter maps and data follow the COSMO convention for polarization angles from HEALPix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Grey regions correspond to the sky areas not observed by QUIJOTE MFI: the southern sky (approximately below 𝛿 = −34◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' a small area around the North Celestial Pole (NCP) for some of the horns (depending on their location in the MFI focal plane);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and the band of geostationary satellites close to declination zero degrees, which mainly emit at 11 and 13 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix C also contains the associated number of hits (𝑁hit) and weight maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Both set of maps are outputs of the PICASSO map-making code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The hit maps (𝑁hit) correspond to the total num- ber of 40 ms samples in each HEALPix pixel of 𝑁side = 512 reso- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The weight maps correspond to the propagation through the map-making process of the errors (weights) associated with each individual 40 ms sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Both sets of maps clearly show the im- print of the scanning strategy of the QUIJOTE MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The ring structures around the North Celestial Pole correspond to the boundaries of the different elevations considered in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to projection effects, the number of hits is significantly larger in those borders (and thus, the noise levels are smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the low declination band of the maps (below the masked area due to geosta- tionary satellites), the number of hits is significantly lower due to the combined effect of a lower number of observations at these low elevations (mainly 30◦, 35◦ and 40◦), and projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that the number of hits in the intensity maps is larger than in polarization due to the fact that some intensity data are not used in polarization (period 1 data are not used for any polarization maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' data from period 2 are not used in polarization for horn 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and data from period 5 are not used in polarization for horn 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' see summary information in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final QUIJOTE MFI wide survey maps at 11 and 13 GHz presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1 and 2 are directly the maps from horn 3, smoothed to 1◦ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final maps at 17 and 19 GHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3 and 4 have been produced as a linear combination of those for horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For simplicity in the computation of effective beams, frequencies and colour corrections, we adopted constant weights for this com- bination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have checked that the resulting maps have comparable noise levels to the maps obtained using spatially-varying weights based on the actual weight maps for each individual map in the combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, the combined maps at 17 GHz can be obtained as 𝑚17 = 𝑤2,17𝑚2,17 + 𝑤4,17𝑚4,17 (11) MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 17 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' List of periods contributing to each final MFI map per horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Column 1 indicates the map per horn with the usual notation: the first number indicates the horn/pixel (column 2), and second and third numbers indicate the nominal frequency (column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Column 4 shows the list of periods contributing to the map based on the correlated channels 𝑉x+y and 𝑉x−y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Column 5 shows the list of periods used for the map based on the uncorrelated channels 𝑉x and 𝑉y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final map is the combination of both correlated and uncorrelated maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Map Horn/Pixel Nominal Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (GHz) Corr Uncorr Intensity 311 3 11 1,2,5,6 1,2,5,6 313 3 13 1,2,5,6 1,2,5,6 217 2 17 1,2,5,6 1,2,5,6 219 2 19 1,2,5,6 1,2,5,6 417 4 17 1,2,5,6 1,2,5,6 419 4 19 1,2,5,6 1,2,5,6 Polarization 311 3 11 2,5,6 5,6 313 3 13 2,5,6 5,6 217 2 17 2,6 6 219 2 19 2,6 6 417 4 17 5,6 5,6 419 4 19 5,6 5,6 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Constant weight factors used to produce the combined 17 and 19 GHz MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We include only the weight factors for horn 4, as those for horn 2 can be obtained as 𝑤2,17 = 1 − 𝑤4,17 and 𝑤2,19 = 1 − 𝑤4,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' I Q U 𝑤4,17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='732 𝑤4,19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='788 for 𝑚 = 𝐼, 𝑄,𝑈, and similarly for 19 GHz, we have 𝑚19 = 𝑤2,19𝑚2,19 + 𝑤4,19𝑚4,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (12) Table 9 contains the final weights used for this linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These values have been derived from the white noise level of the individual frequency maps for each horn, using optimal (inverse variance) weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that horn 2 dominates the linear com- bination in intensity, while horn 4 contributes with a higher weight to the polarization maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The actual values of noise levels for these maps are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final maps in polarization (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1–4) are dominated by the Galactic synchrotron emission (the spectral index of the observed signal is discussed below in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Large scale features such as the Fan region or the North Polar Spur are clearly seen in the four frequency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The MFI instrument is not optimized to measure the intensity signal, and thus the intensity maps present worse noise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, the two highest frequency channels show clear large scale 1/ 𝑓 residuals, particularly at negative declinations, due to the fact that they are observed only with the lower elevations (higher air masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Analysis masks Figure 11 shows the footprint of the different analysis masks which are specific for the QUIJOTE wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' There are three distinct regions that are considered when building these masks: Satellite band ("sat").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The masked region around declination zero is used to block the RFI contamination of geostationary satel- lites mainly affecting 11 and 13 GHz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the MFI pipeline, the emission from each geostationary satellite is flagged at the TOD level using a mask of 5◦ radius around each satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Other satellites or RFI signals are flagged as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' After this pro- cess, the resulting masked area (with zero number of hits) is located approximately between declinations −10◦ to −2◦ (note that geo- stationary satellites are seen at slightly negative declinations from the Teide Observatory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The proposed mask to remove the satellite band (−12◦ < 𝛿 < 6◦) is a conservative choice based on a close inspection of the final maps, extending the unobserved area by two degrees in the negative declination direction, and by eight degrees in the positive direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This choice accounts for low-level RFI residuals in the intensity maps (some of the residual RFI signals corrected during the post-processing stage are located in that area), while keeping a relatively high number of hits per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' North Celestial Pole ("NCP") region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given the latitude of the Teide Observatory (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30◦N) and the minimum elevation observed with QUIJOTE MFI (EL= 30◦), some of the maps present a small area of unobserved pixels around the NCP, depending on the loca- tion of the MFI horns in the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The maximum observed declination is approximately 86◦ for horn 3, and 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦ for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horn 4 covers up to 90◦ in declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In any case, the pixels sur- rounding this NCP area are only accesible with the lowest elevation bands, which usually present the largest levels of atmospheric con- tamination in the intensity maps, particularly at 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this reason, for some of the analysis we mask the region above 𝛿 = 70◦, in order to keep a sky area that is observed practically by all the elevations considered in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Low (negative) declinations ("lowdec").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similarly to the NCP area, this region is only observed when using low elevations (below 40◦), and thus the corresponding intensity maps, specially at the two highest frequencies, are more affected by 1/ 𝑓 residuals from atmospheric emission (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3 and 4, and also the individual maps for horns 2 and 4 in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The proposed mask to exclude this area covers all declinations below 𝛿 = −12◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All different combinations of those three masked regions produce the reference set of specific masks for the MFI wide survey used in this and all accompanying papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, unless other- wise stated, the default analysis mask used in most of the scientific analyses in this paper, and in particular, in all power spectrum computations, corresponds to the superposition of the three re- gions (sat+NCP+lowdec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This mask preserves a sky fraction of 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='418, equivalent to approximately 17 200 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4 DATA VALIDATION In order to characterize the properties of the wide survey maps, we carry out a number of tests and studies in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Most of them rely on different types of null tests, which can be used to detect possible remaining systematic effects in the data, including residual RFI signals, calibration issues, changes in the operational or instrumental conditions, or even unknown effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 18 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Footprint of the wide survey in Galactic coordinates, and pro- posed analysis masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The background image corresponds to the 9-yr WMAP-K band polarized intensity map (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Light colours indicate the observed MFI wide survey regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The band excluded due to satellite contamination corresponds to −12◦ < 𝛿 < 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The default mask adopted for the analyses in this paper preserves the band 6◦ < 𝛿 < 70◦, which is marked as the brightest region in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This mask is labelled as sat+NCP+lowdec (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Null tests A “null test” is defined as the difference between the maps produced from two independent sub-sets of files from the full data base, which are expected to give the same signal under the assumption of a perfect calibration and no systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Null tests have been shown to be a powerful mean to assess the contribution of residual systematic effects in CMB analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014c, 2016d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the characterization of the QUIJOTE MFI wide survey data, we produced the following set of null tests: (i) Half mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The full database is divided in two halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The separation is done according to the calendar date inside each period and each elevation, producing maps labelled as “half1” and “half2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this way, both null test maps contain data from all periods, and have a similar sky coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This is the reference null test used to characterize the overall noise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (ii) Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The MFI wide survey maps are produced using the so- called nominal observing mode, in which the QUIJOTE telescope scans the sky using a circular scanning strategy with a continuous movement in azimuth direction while maintaining a constant ele- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Each azimuth scan is called a "ring".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this null test, the full database is divided in odd (“rings1”) and even (“rings2”) rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' With the nominal azimuth scan speed of 12 deg s−1, each ring is completed in 30 s, so this null test can be used to test for instrumen- tal variations in these short time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As the instrument gain is stable in time scales much longer than one minute, this null test is not expected to reflect gain variations, and will essentially contain white noise plus a 1/ 𝑓 -noise component in scales of 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (iii) Daynight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to evaluate possible residual system- atic effects due to day-night variations of the system gain or cal- ibration factors, this null test is produced by dividing the full database into day observations (“daynight1”) and night observa- tions (“daynight2”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For simplicity, we define here “day” as all observations from 8 AM to 8 PM (UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (iv) PWV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using the information from GPS measurements at the Teide Observatory of the precipitable water vapour (PWV) content of the atmosphere during each individual observation5, we divide 5 The GNSS antenna that provides these PWV measurements is located at the full data base in two sets of low (“pwv1”) and high (“pwv2”) pwv values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As in the case of the half mission null test, the separation is done inside each period and elevation, to guarantee that both splits contain a similar sky coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, the resulting median pwv in these two data splits is 2 mm and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mm, for "pwv1" and "pwv2", respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (v) Halfrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This null test separates the data by dividing each ring in two halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Data taken with telescope azimuth values 0◦ ≤ 𝐴𝑍 ≤ 180◦ correspond to "halfring1", while data with 𝐴𝑍 > 180◦ are part of "halfring2".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although these maps are expected to be noisier than the other null tests due to 1/ 𝑓 contributions (note that in this case we are basically decreasing by a factor of two the number of independent crossings in each pixel when solving the conjugate gradient inside the map-making algorithm), they are still extremely useful to detect residual RFI signals arising from local structures, which usually appear at fixed AZ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, these maps can be also used to test residual pointing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (vi) 𝑇BEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As explained in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), the overall gain of the instrument is strongly correlated with the physical temperature in the electronic boxes containing the Back-End Module (BEM) of the MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a further test to explore possible residual variations after our gain model correction, we use the values of one of the temperature sensors 𝑇BEM, which is monitored every second as part of the house-keeping data, to separate the data in two halves, according to low ("tbem1") and high ("tbem2") values of the BEM temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, the median temperature for these two data splits is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1◦C and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1◦C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As for the half mission and PWV null tests, we do the division in two halves for each period and elevation configuration separately, and then we combine the sub-lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For simplicity, we refer to this case as "tbem null test" in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Two separated lists of calibrated TOD files are produced for each one of those six null tests cases, and the corresponding maps ℎ1 and ℎ2 are produced with fully independent runs of the map- making code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The post-processing of each null test is identical to the procedure applied to the full maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From this point, a "null-test difference map" can be produced for each case, as 𝑛 = ℎ1 − ℎ2 𝑤 , (13) where the normalizing weight is computed as 𝑤 = √︃ (𝑤1 + 𝑤2)(𝑤−1 1 + 𝑤−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (14) Here 𝑤1 and 𝑤2 are the individual weight maps of the null tests ℎ1 and ℎ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They are computed as 𝑤𝑖 = 1/𝜎2 𝑖 , with 𝑖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Defined in this way, equation 13 provides a map with similar noise levels as the residual noise for the weighted-sum of the two halves (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014b, 2016f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Null tests with a common baseline solution For those six cases listed above we have also produced a different set of null test maps, named as "null test with common baselines", as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we run the map-making code for the complete database, and record the baseline solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, each pair of null test maps is generated using that recorded solution, instead of solv- ing for the baselines with half of the data only, as it was the case before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' By construction, this procedure cancels out an important the Izaña Atmospheric Observatory (IZO) just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 km away from QUIJOTE, and virtually at the same altitude (≈ 10 m below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUljOTE MFI masks 6 =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='70 6 12QUIJOTE MFI wide survey 19 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Half-mission null test difference maps for horn 3 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top row shows the Stokes I (left), Q (centre) and U (right) difference maps for the case of "independent baselines".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom row corresponds to the case of "common baselines" (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, all maps are smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour scale corresponds to ±1 mK for the intensity maps, and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 mK for polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' part of the 1/ 𝑓 noise contribution associated with long time-scale variations, partly due to the fact that the baseline solution is better constrained when using the full database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Differences between the two halves ℎ1 and ℎ2 now will be entirely due to the fact that each half uses different input data, and not to the possible uncertainties in the determination of the baseline solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this reason, these null test maps are found to be particularly useful to study those variations in the data which can be (mainly) ascribed to calibration uncertainties, instrument changes or to variability of the sky signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, these maps will be used specifically in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 to assess the internal calibration of the wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For all the remaining analyses, and in particular, for assessing the noise levels in the wide survey maps, we will always use the default set of null tests maps ("with independent baselines").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As illustration, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 12, 13 and 14 present few examples of null test difference maps for horn 3 11 GHz, after smoothing to one degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 12 shows the half mission difference map both for the "independent baselines" and the "common baselines" cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13 contains the ring, halfring and tbem null tests for the case of independent baselines, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 14 shows the same three cases for the "common baselines" solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Other data splits In addition to the null tests described above, other data splits have been considered and generated for the MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particu- lar, we generated the four "maps per period", in correspondence to periods 1, 2, 5 and 6, both for the case of "independent baselines", and also with "common baselines".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although these four maps per period do not have exactly the same sky coverage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' elevation 30 is only used in period 5) or the same format (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' polarization maps are not generated in period 1), they are still very useful for valida- tion purposes (RFI residuals, gain model, calibration), as shown in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, these maps are also used for the study of transients and in particular, to characterise the potential variability of some bright point sources (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Herranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Assessing systematic effects with null tests in power spectra and maps 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Power spectra Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 15 presents the binned raw power spectra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' uncorrected for the beam and pixel window functions) of the six null-test difference maps described in the previous section and computed using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13, compared to the raw power spectra of the final maps for each horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For simplicity, we show only two cases, for horn 3 (11 GHz) and horn 4 (17 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The equivalent figures for other horns and frequencies provide qualitatively similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this section, the 𝐶ℓ’s are computed with the publicly available code Xpol6, which is based on a pseudo-𝐶ℓ estimator, and accounts for incomplete sky coverage (Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The mask adopted for this computation is the default one described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 (NCP+sat+lowdec), using a 5◦ apodization with a cosine function, as implemented in the NaMaster library (Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In all panels, we show as a reference the angular power spectrum of the final map in black, and the spectra of the different "null test difference maps" (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13) in various colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For completeness, these figures also include the power spectra (as dotted lines) of the null-test difference maps for the case of "common baselines".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also include the ideal white noise level for each map, computed from the normalized weights (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All six null test difference maps present a similar behaviour, being asymptotically flat at high multipoles when reaching the white noise level, and increasing at low multipoles (large angular scales) as expected for residual 1/ 𝑓 noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A comparison of these six null test power spectra provides a useful tool to identify and isolate different sources of systematic effects or calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, 6 https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='fr/tristram/Xpol MNRAS 000, 1–58 (2022) half noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKhalf noise map (311, Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalf noise map (3l1, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalf noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKhalf noise map (311, Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalf noise map (3l1, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mK20 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Three examples of null test difference maps for horn 3 11 GHz, for the case of "independent baselines": ring (top), halfring (centre) and tbem (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From left to right, each row shows the Stokes I (left), Q (centre) and U (right) difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, all maps are smoothed to 1 degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour scale corresponds to ±1 mK for the intensity maps, and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 mK for polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' all null test spectra are basically consistent among them, except the ring case, which presents a slightly lower level of 1/ 𝑓 residuals at low multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This behaviour is expected because the ring null test maps probe noise variations in scales of one minute, while the others cases (half, daynight, tbem, pwv) probe longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also note that the halfring null test tends to be slightly above the other noise estimates, but again this is expected as this null test uses basically half of the possible crossings for each pixel, and thus the baseline solution is less constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, this is not the case of halfring null test with common baselines, as in this case the baseline solution was obtained with the complete dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the intensity maps, the qualitative behaviour is similar to polarization, although the scatter among the null tests in the 1/ 𝑓 residuals at low multipolesislarger, particularlyat11 GHz where the RFI contamination due to geostationary satellites was higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this case, the largest 1/ 𝑓 residuals at low multipoles correspond to the tbem, daynight and halfring cases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' By construction, the halfring case amplifies the presence of residual RFI signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of tbem and daynight, this might indicate some low-level RFI residual which becomes visible when splitting the data according to the daily gain variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have confirmed that this is indeed the case, by constructing a new set of maps excluding period 1 in intensity, which was the period most affected by RFI due to the absence of the extended shielding in the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When generating the halfring null test for the case of no period 1, that small excess disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we note that the power spectra for the null test difference maps with "common baselines" present a significantly lower level of 1/ 𝑓 residuals, as anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Maps Visual inspection of the null test difference maps provides comple- mentary information to the one obtained from the power spectra analysis, in terms of identifying localised features due to systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, the halfring null test maps (see the example for horn 3 at 11 GHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13 and 14) can be used to assess the residual systematic effects due to uncertainties in the pointing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), the pointing model solution for each MFI horn provides a reconstruction of the pointing with an overall 1 arcmin accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Any residual pointing error will produce a characteristic feature in the halfring null-test map, as each one of the two sub-maps (halfring1 and halfring2) uses totally different ranges of local coordinates of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Indeed, the morphology and amplitude of the features appearing in the intensity map along the Galactic plane, both around the Galactic centre and the Cygnus area, match the expected residual signals for a shift of 1 arcmin between the halfring1 and halfring2 sub-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Null test difference maps can also be used for assessing the level of residuals in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, a cross correlation analysis of each null test difference map (𝑛) with the corresponding signal map (𝑚) can be used to trace the presence of both errors in the overall gain model or time-dependent RFI residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As usual, MNRAS 000, 1–58 (2022) ring noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKring noise map (311, Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKring noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalfring noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKhalfring noise map (311, Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalfring noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKtbem noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKtbem noise map (311, Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKtbem noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKQUIJOTE MFI wide survey 21 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13, but for the case of "common baselines" difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The colour scale corresponds also to ±1 mK for the intensity maps, and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 mK for polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cross-correlation in real space between the half mission difference maps and the final signal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Columns 2–4 correspond to the case of half mission maps with common baselines, while columns 5–7 show the results for the case of independent baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel 𝛼T 𝛼Q 𝛼U 𝛼T 𝛼Q 𝛼U [%] [%] [%] [%] [%] [%] Common baselines Indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' baselines 217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 311 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 313 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 419 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 a cross-correlation coefficient 𝛼 can be obtained as the minimum variance estimator that minimizes 𝑛 − 𝛼𝑚 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hernández- Monteagudo & Rubiño-Martín 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 10 presents the corre- lation coefficients 𝛼, in percent units, for the case of the half mission null tests both for common and independent baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis is carried out using the standard mask NCP+sat+lowdec defined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These numbers are consistent with the power spectra analyses described in the previous subsection, and lie below the calibration uncertainty of the wide survey (see details in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5 and Table 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, for horn 3, these values are within one per cent, both in intensity (𝐼) and polarization (𝑄, 𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, in polarization all values are below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Noise characterization: 1/ 𝑓 noise and correlations Noise parameters for the MFI instrument have been described in (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023), and are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Those values determine some of the noise properties of the final wide sur- vey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we use the half-mission difference maps (hereafter HMDM), constructed as in equation 13 and for the case of "inde- pendent baselines", to assess the overall noise properties of the MFI wide survey, including white noise levels, 1/ 𝑓 -type components and correlation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analyses are done both in harmonic (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1) and real (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2) space, using the standard mask defined as NCP+sat+lowdec in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, which contains the region in the declination range 6◦ < 𝛿 < 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, and due to the MFI receiver design, there are well-known noise correlations at the TOD level (also called "common mode 1/ 𝑓 noise") between channels of the same horn, which are inherited by the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the cross-spectra of different HMDM to characterize these noise correlations at the map level, both between the two frequen- cies of the same horn (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3) and between the correlated and uncorrelated channels contributing to a given map (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) ring noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKring noise map (311, Q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKring noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalfring noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKhalfring noise map (311, Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKhalfring noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKtbem noise map (311, I) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKtbem noise map (311, Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mKtbem noise map (311, U) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 mK22 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Binned raw power spectra (Δℓ = 11) of the six null test difference maps discussed in the text, for horn 3 at 11 GHz (left) and horn 4 at 17 GHz (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For comparison, we also include as dashed lines the spectra of the null test difference maps for the case of "common baselines".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Black solid lines depict the spectra of the signal maps, while the horizontal dashed lines indicate the ideal white noise level for each map (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Noise properties in harmonic space Our analysis of the noise properties in harmonic space is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 16 and Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The power spectra for the HMDM are com- puted using NaMaster and then fitted with the following empirical model: 𝐶ℓ = 𝐶w � 1 + �ℓk ℓ � 𝛼� , (15) which accounts for a 1/ 𝑓 noise component projected on sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We fit for the three parameters in this equation in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we obtain the white noise level 𝐶w as the average level of the angu- lar power spectrum at high multipoles (ℓ ∈ [700, 800] for TT, and ℓ ∈ [600, 800] for EE and BB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, the knee-multipole ℓk and the slope 𝛼 are obtained analytically after fitting for a linear relation in log10(𝐶ℓ − 𝐶w) 𝑣𝑠 log10(ℓ), in the multipole range ℓ ∈ [20, 100] for both intensity and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To have a better fit in the high multipole range for the EE and BB case of horn 2, we use here the range ℓ ∈ [80, 300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The parameter 𝐶w, which represents the white noise level of the full maps, can be translated into the com- monly used quantity 𝜎1-deg, the equivalent noise level (rms) of the map for a 1-degree beam, with the relation 𝜎1-deg = √︁𝐶w/Ω1-deg, where Ω1-deg is the solid angle of a Gaussian beam with a FWHM of 1-degree, which corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='345 msr= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='133 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These numbers (third column in Table 11) can be directly compared to those obtained with real space statistics in the next subsection7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In summary, for the intensity spectra, horn 3 presents the low- est noise levels both for the 1/ 𝑓 and the white noise components, while horn 4 is the most noisy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, in polarization, horn 4 has a much better performance, yielding the lowest noise levels, while horn 2 is the noisiest in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although the noise levels for horn 3 in polarization are slightly higher than those for horn 4, 7 Note that if we want to quote the map sensitivity in the usual units of 𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='arcmin (or 𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='deg), we can not use directly 𝜎1-deg, as we have to account for the √︁Ω1-deg factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For instance, the white noise level of the MFI 311 map in polarization is 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 𝜇K per 1-degree beam, or equivalently, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='deg = 2695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='arcmin, consistently with the reported 𝐶w value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) Horn 3 11 GHz △lbin=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 half halfring pwv Map ring daynight tbem 10 10-1 [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 10-2 10 10-4 10-5 10-6 10-3 10-4 10 10-6 10-7 10-3 10-4 [mk2] 10- 5 10- 6 101 102Horn417GHzAlbin=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 half halfring pwv Map ring daynight tbem 100 10-1 [mk2] 10-2 10 10-4 10-5 10-6 10-3 10-4 [mk²] 10 5 10- 10-7 10-3 10-4 [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 10 10- 10- 101 102QUIJOTE MFI wide survey 23 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Best-fit solutions to the power spectra of the half-mission differ- ence maps (HMDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 15, we obtain the best-fit models depicted here as dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The corresponding coefficients are listed in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' given that the sky signal is significantly brighter at lower frequencies (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 15), the wide survey polarization maps of horn 3 (11 and 13 GHz) have the better signal-to-noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Regarding the cor- related noise component, we find that the noise spectra in intensity are dominated by the 1/ℓ component down to scales of 1 degree, as a consequence of the large 1/ 𝑓 noise in the intensity TODs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, we find typical knee-multipoles of ℓk = 54–86 for horns 3 and 4, as expected for the significantly lower correlated noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Noise properties in real space First, we normalize the HMDM by dividing each individual pixel by the square root of its covariance as computed from the map weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', 𝜎𝑖 = 𝑤−1/2 𝑖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that those weights are propagated through the pipeline and the map-making code, and were computed Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Noise levels from the fit to the noise power spectra based on the parametric equation 15, computed from the half-mission null tests with independent baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, we show the results of the fit to the EE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Results for BB are fully consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel 𝐶w 𝜎1-deg 𝛼 ℓk [mK2 sr] [𝜇K] Intensity (TT) 217 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 × 10−6 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 × 10−5 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='82 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 311 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='56 × 10−6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='27 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 313 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='29 × 10−6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='60 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 × 10−5 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='45 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 419 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='40 × 10−5 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='82 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 Polarization (EE) 217 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='21 × 10−6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='87 × 10−6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 311 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 × 10−7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 313 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='95 × 10−7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='35 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 417 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 × 10−7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 419 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 × 10−7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='24 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Recalibration factor of the noise standard deviation included in the weight maps, based on null test maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Map H2,17 H2,19 H3,11 H3,13 H4,17 H4,19 Half mission null test I 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='974 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='596 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='424 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='016 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='695 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='108 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='723 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='471 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='372 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='292 U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='723 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='473 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='373 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='292 Ring null test I 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='896 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='449 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='410 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='993 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='641 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='978 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='717 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='994 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='471 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='370 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='286 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='289 U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='716 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='991 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='473 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='370 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='291 from the variance of each individual 40 ms sample in the TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this normalized map, we fit for the standard deviation within the reference mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results are shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, these values are reasonably close to unity for the case of the polar- ization maps, while in intensity these factors are greater than 3 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These deviations from unity are generally consistent with the level of 1/ 𝑓 noise in each case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This set of values could be used to renormalize the weight maps, so they would be representative of the actual noise levels, while preserving the underlying spatial distribution of the hit maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Indeed, these factors are used to estimate the ideal white noise of each map at the power spectrum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 15 are com- puted with these rescaled weight maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, these rescaled weight maps can be used to produce signal-to-noise maps for each frequency (see Appendix C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a second analysis, we repeat the same procedure but now we normalize each difference map according to the square root of the number of hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Taking into account that hits correspond to 40 ms samples, we can obtain from here representative normalization val- ues to describe the noise standard deviation as 𝜎 = 𝜎0 √𝑁hit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (16) MNRAS 000, 1–58 (2022) FitnoiseClAl=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 10-2 h2 17GHz h2 19GHz h3 11GHz 10-3 h3 13GHz [mk2] h4 17GHz h4 19GHz 10-5 10-B 10-4 [mk²] 10- 10-6 10-3 10-4 [mk2] l adb 10- 5 10-6 101 10224 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Characteristic value of the sensitivity for each channel, 𝜎0, in units of mK s1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Based on the half-mission null test maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Map H2,17 H2,19 H3,11 H3,13 H4,17 H4,19 Half mission null test I 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='896 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='445 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='481 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='422 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='939 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='427 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='878 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='280 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='371 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='188 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='059 U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='875 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='273 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='372 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='188 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='064 Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mean noise figures in the final MFI maps, in units of 𝜎1-deg (𝜇K per 1-degree beam), using real-space statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A variance map is estimated based on the half-mission nulltest maps, computing the variance within a circle of 1 degree radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Those values are then converted into 𝜎1-deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Map H2,17 H2,19 H3,11 H3,13 H4,17 H4,19 Half mission null test I 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 Q 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 U 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 Our results are shown in Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The values obtained for the MFI wide survey in polarization are comparable to those obtained for raster scan observations with the MFI in smaller regions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' last column in Table 1 from Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017), and represent the actual sensitivity of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we can also estimate the noise variance directly from the HMDM, using apertures of 1-degree radius across the same mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average values obtained from this analysis are given in Ta- ble 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To facilitate the comparison with the numbers in the previous subsection, these values are re-scaled by the factor √︃ Ωpix/Ω1-deg, so they represent 𝜎1-deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In summary, the final combined maps of the MFI wide survey in polarization present sensitivities within the range 35–40 𝜇K per 1-degree beam for the four frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Noise correlations between frequencies of the same horn Two MFI frequency channels from the same horn have a corre- lated ("common mode") 1/ 𝑓 noise component, due to the fact that they share the same LNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This fact is particularly relevant for the intensity maps, which are strongly dominated by correlated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Because of this reason, our final wide survey maps at 11 and 13 GHz have correlated noise between them, as is the case for the maps at 17 and 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to characterize the actual degree of correlation be- tween two wide-survey maps obtained from the same horn, we use the normalized cross-spectra between the corresponding null-test difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As in the previous section, we use as a reference the HMDM for the case of independent baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following the notation in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, here 𝑛ℎ, 𝑓 represents the half-mission differ- ence map for horn ℎ and frequency 𝑓 (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, for a given horn ℎ(= 2, 3, 4), the normalized correlation between the lowest frequency band 𝑓1 and the highest frequency band 𝑓2, is given by 𝜌ℓ ≡ 𝐶 𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2 ℓ √︃ 𝐶 𝑛ℎ, 𝑓1 ℓ 𝐶 𝑛ℎ, 𝑓2 ℓ , (17) where 𝐶 𝑛ℎ, 𝑓1×𝑛ℎ, 𝑓2 ℓ is the cross-spectrum between the two difference maps, and 𝐶 𝑛ℎ, 𝑓𝑖 ℓ for 𝑖 = 1, 2 represents the auto-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 17 shows this normalized cross-spectrum 𝜌ℓ in the final MFI wide survey maps for horns 2, 3 and 4, both in intensity and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, the resulting noise correlation is of the order of 75–85 per cent for the three horns, being relatively flat in the multipole range 20 ≲ ℓ ≲ 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, the correlation is found to be ∼ 20–60 per cent depending on the horn, with a moderate dependence on the multipole, being slightly lower at higher multipoles (smaller scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to obtain a representative value for this correlation, we compute the average (and standard deviation) of 𝜌ℓ in the multipole range [20, 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For TT, we obtain 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 %, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 % and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 % for horns 2, 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, for EE we obtain 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 %, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 % and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 %, and for BB we have 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 %, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 % and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 %, again for horns 2, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This high degree of correlation has to be taken into account when doing combined analyses of the two frequency maps of the same horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a consistency check, and in order to test that these inter- frequency correlations are entirely due to instrumental (common mode) 1/ 𝑓 noise, and not to external correlated signals produced either by the atmosphere or by RFI, we performed the same analysis but now comparing two frequencies coming from two different horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we evaluated the cross-correlation of horn 2 at 17 GHz with horn 4 at 19 GHz, obtaining −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='64 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='47 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 % and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 % for TT, EE, and BB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, the cross-correlation of horn 4 at 17 GHz with horn 2 at 19 GHz gives −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='32 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='76 %, again for TT, EE and BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In both cases, the results are consistent with zero within the error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Noise correlations between channels As described above, for any given horn and frequency sub-band of MFI, we produce two versions of the intensity and polarization maps, the so-called correlated (𝑥c) and uncorrelated (𝑥u) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to the MFI design, we expect a high degree of correlation between the noise affecting those two versions of the intensity maps, due to the fact that they all share the same LNAs and there is no cancellation of the 1/ 𝑓 noise in any of the sums of channels contributing to 𝑥c and 𝑥u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We can use the same methodology applied in the previous sub-section to characterize this correlation level of the noise between correlated and uncorrelated channels maps for a given horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also use the half-mission null test maps as a reference for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' But now, in the post-processing stage, we generate two independent versions for each individual map, using either the correlated or the uncorrelated information only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' With these maps, and using again eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 13, for a given horn and frequency we can produce 𝑛c and 𝑛u, the half-mission difference maps of the correlated and uncorrelated channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In analogy to equation 17, we now compute 𝜌ℓ ≡ 𝐶𝑛c×𝑛u ℓ √︃ 𝐶𝑛c ℓ 𝐶𝑛u ℓ , (18) where 𝐶𝑛c×𝑛u ℓ is the cross-spectrum between the two difference maps, and 𝐶𝑛c ℓ and 𝐶𝑛u ℓ are the auto-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 18 shows the resulting correlation level between correlated and uncorrelated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, we find a very high degree of correlation (of the order of 90 per cent) in intensity, and a signal consistent with zero in polarization (both for EE and BB spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 25 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cross-correlation spectra of the half-mission difference maps between the two frequencies of the same horn, for TT (left), EE (centre) and BB (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cross-correlation spectra of the half-mission difference maps between the correlated and uncorrelated channels from the same horn and frequency, for TT (left), EE (centre) and BB (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Average inter-channel correlations < 𝜌ℓ > of the half-mission difference maps between the correlated and uncorrelated channels for a given horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The values correspond to the mean and standard deviation of the 𝜌ℓ displayed in Figure 18, computed in the multipole range 20–200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel TT (%) EE (%) BB (%) 217 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='36 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='37 219 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='40 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='60 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='04 311 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 313 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91 417 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='04 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 419 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='61 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 Again, as a representative value for this correlation, we compute the average and standard deviation of 𝜌ℓ in the multipole range [20, 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results are shown in Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These average corre- lation values in intensity are used in the pipeline in order to produce the final combinations of correlated and uncorrelated channels, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Impact of residuals on the power spectra: atmospheric and RFI corrections As described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2, the MFI wide-survey pipeline incorporates several steps tailored to correct for the contribution of atmospheric and RFI signals in the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Atmospheric corrections are applied at the TOD level (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4), and for intensity maps only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When projected on maps, they appear as large scale patterns with an increasing amplitude in frequency (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' RFI signals are corrected both in the intensity and polarization maps, in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, RFI signals at the TOD level are corrected using spatial templates as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When projected on sky, they also appear as large scale patterns with a moderate amplitude (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 mK) and presenting a higher amplitude in intensity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Later, in the post-processing stage (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4), any residual RFI signals emerging after co-adding all data in the map-making process are corrected using a function of the declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In terms of relative amplitude, this is by far the largest correction applied to the MFI wide-survey polarization data, with its amplitude being higher in the 11 and 13 GHz channels due to the emission of geo-stationary satellites entering through the far sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Indeed, the effective transfer function of the MFI wide survey in polarization is mainly determined by this effect (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to quantify the relative importance of these three MNRAS 000, 1–58 (2022) Correlationlow/highfreguencies TT EE BB 100 80 60 % 40 20 h2 17x19 0 h3 11x13 20 h4 17x19 101 102 103101 102 103101 102 103 l l lCorrelation corr/uncorr TT EE BB 100 80 60 % 40 20 217 313 0 219 417 20 311 419 101 102 103101 102 103101 102 103 l l l26 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Raw angular power spectra of the ATMOS (red), RFI (green) and FDEC (blue) patterns removed from the MFI wide survey maps, for horn 3 at 11 GHz (top row) and horn 4 at 19 GHz (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For each case, we represent TT (left), EE (centre) and BB (right) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Solid black lines correspond to the angular power spectra of the corresponding wide survey maps, while dashed lines correspond to the half-mission difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All spectra are computed using the default analysis mask (NCP+sat+lowdec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' corrections, and to evaluate the possible impact of any residual systematic effects due to uncorrected contamination in the final wide survey maps, we have computed the angular power spectra of those patterns that are removed from the maps, and we have compared them with the spectra of the final maps and the half- mission noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 19 shows the resulting power spectra for the two extreme frequency values (11 and 19 GHz) taken here as representative cases, with 11 GHz being the one with highest RFI contamination, and 19 GHz the one with the highest atmospheric contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this plot, we use the notation of ATMOS, RFI and FDEC for "atmospheric", "RFI at TOD level using a function of azimuth", and "RFI at the map level using function of declination" corrections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Regarding the atmospheric contribution (ATMOS) to the in- tensity power spectra, the removed pattern is subdominant at all an- gular scales in the 11 GHz case when compared to the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At 19 GHz, we have a similar behaviour at small angular scales (ℓ >∼ 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, the atmospheric residuals become comparable to the noise levels for multipoles ℓ ≲ 20, as can be anticipated from the visual inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the RFI contribution at the TOD level, the removed patterns both in intensity and polarization are always below the noise levels at all frequencies, although they become comparable to the noise at large angular scales ℓ ≲ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, in this RFI case, as well as for ATMOS, any residual systematic effect with an amplitude being a fraction of the applied correction will have negligible impact at the power spectrum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, for the removed FDEC patterns, the largest amplitude is found at 11 GHz, as anticipated from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At this frequency, the removed pattern in intensity is above the correlated noise level for multipoles ℓ ≲ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, in polarization, the applied correction is found to be critical, in the sense that its amplitude is above the sky signal for multipoles ℓ ≲ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When looking at the 19 GHz FDEC patterns, in intensity the corrected amplitude is always below the noise levels for all multipoles, while in polarization again becomes comparable to the sky signal for ℓ ≲ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this case, although the underlying assumption for modelling residual RFI signals using a function of the declination is very robust and well tested, it is im- portant to keep in mind that residual contributions might have an impact on the polarization maps of the MFI wide survey on large angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, as explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5, the FDEC pro- cedure also affects the same multipole range by introducing a signal error in the reconstructed sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For these reasons, in the following sections involving scientific analyses based on power spectra of the polarization signals in the wide survey, we adopt the conservative choice of restricting the study to multipoles ℓ ≥ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Inter-frequency comparison of the MFI maps As an additional validation test, here we present an inter-frequency comparison of the MFI wide survey maps, together with a com- parison with external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this test, we rely on the assump- tion that the average spectral index of the polarized synchrotron emission in the QUIJOTE maps is 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 (see discussion be- low in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We then rescale the MFI wide survey maps at 1 degree resolution to the central frequency of the WMAP K-band map, 𝜈 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 GHz, accounting for colour correction factors both for MFI and WMAP maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 20 shows the rescaled MFI po- larization maps at 11 and 13 GHz compared to WMAP-K, while Figure 21 shows the differences for pairs of those maps (313−311, 311−WMAP, 313−WMAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A visual inspection shows that there is obvious polarized emission in the Galactic plane which is not consistent with the 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 spectral index, mainly towards the Galactic centre or the Fan region (𝑙 ≈ 135◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the maps we can also identify some residual intensity to polarization leakage in the Cygnus area (around 𝑙 ≈ 80◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, the large scale emission MNRAS 000, 1–58 (2022) TT H3, 11GHZ TT RFI 100 noise FDEC ATMOS 10-2 C,T [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 10-4 10-6 10-8 10-10 101 102EE H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11GHZ 10-2 EE RFI noise FDEC 10-4 CEE [mK?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2] 10-6 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10-10 101 102BB H3, 11GHZ 10-2 BB RFI noise FDEC 10-4 10-6 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10-10 101 102TT H4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 19GHZ TT RFI 100 noise FDEC ATMOS 10-2 C}T [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 10-4, 10-6 10-8, 10-10 101 102EE H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 19GHZ 10-2 EE RFI noise FDEC 10-4 CEE [mK?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2] 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10-10 101 102BB H4, 19GHZ 10-2 BB RFI noise FDEC 10-4 CBB[mK2] 10-6 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10-10 101 102QUIJOTE MFI wide survey 27 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Comparison of the rescaled polarization MFI maps at 11 and 13 GHz with the 9-yr WMAP-K band map (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI maps are rescaled to 23 GHz using an average spectral of 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, and accounting for colour corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps use the same colour scale, saturated at ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' From left to right, we show MFI 11 GHz (rescaled), MFI 13 GHz (rescaled) and WMAP-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top row: Stokes Q maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom row: Stokes U maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, to facilitate the comparison of the different structures near the mask edges, we applied here the QUIJOTE MFI sky mask to the WMAP map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Inter-frequency comparison of the rescaled maps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top (bottom) row shows differences of Stokes Q (U) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First column shows the difference between the rescaled 11 and 13 GHz MFI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Second and third column show the MFI 11 GHz minus WMAP-K, and MFI 13 GHz minus WMAP-K maps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps use the same colour scale as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 20, saturated at ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' far from the Galactic plane is largely suppressed in this difference, showing a good consistency of the MFI and WMAP-K maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The residual emission in the difference map 313-311 is basically consis- tent with the expected noise level for the difference of both maps, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this comparison, we use the EE power spectra for the rescaled maps using the default QUIJOTE mask with the Galactic cut |𝑏| > 10◦, and restricting the comparison to multipoles ℓ ≥ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5 ACCURACY OF THE WIDE SURVEY CALIBRATION In this section we assess the overall calibration uncertainty of the QUIJOTE MFI wide survey maps in intensity and polarization, using the information described in the pipeline paper (Génova- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023) to account for known systematics, and also presenting a set of consistency checks based on the null test maps, in order to evaluate the impact of unknown systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 16 shows the summary of all types of uncertainties considered in this MNRAS 000, 1–58 (2022) QUOTE Q H3 11GHz (1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1QUOTE Q H3 13GHz (1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1WMAP 23GHz Q (1deg) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1QUlOTE U H3 11GHz (1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1QUlOTE U H3 13GHz (1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1WMAP 23GHz U (1deg) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 13GHz - MFI311 (Q, 1deg,rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 11GHz - WMAP (Q, 1deg, rescaled mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 13GHz - WMAP (Q, 1deg, rescaled mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 13GHz - MFI311 (U, 1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 11GHz - WMAP (U, ldeg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1MFl H3 13GHz - WMAP (U, 1deg, rescaled) mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='128 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' EE power spectra of the inter-frequency comparison of the MFI rescaled maps 313−311, shown in the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Black and red solid lines show the EE power spectra of the rescaled MFI 11 and 13 GHz maps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Blue solid line is the power spectrum of the difference map 313−311, while the yellow dashed line shows the expected noise level for that difference map, assuming an average inter-frequency correlation of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 % (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' work, as well as the impact of each of them in the overall calibration error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Statistical uncertainty and known systematics 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Calibration model An important contribution to the global systematic uncertainty bud- get comes from calibration uncertainties, and in particular, the cali- brator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As discussed in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), the two main amplitude calibrators of QUIJOTE MFI are Tau A and Cas A, which are amongst the brightest sources on the sky in this fre- quency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As explained in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6, the wide survey maps have been recalibrated using flux densities extracted on these maps at the position of Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These flux densities are measured with sensitivities better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 % in all frequencies (see also Table 24 and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9) while the internal calibration accuracy of QUIJOTE is better than 1 % as shown below in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Therefore in our case the dominant error component is associated with the cal- ibration models that are used as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As will be discussed in detail in Génova-Santos & Rubiño-Martín in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (see also sub- section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6), using different tests we estimate that the Tau A model has an uncertainty of ≈ 4 % in our frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We believe this value is dominated by calibration errors of the different data that are used to model this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Tau A there is also an important contribution due to the modelling of its secular decrease, which leads to errors when data taken at different epochs are combined to model its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We decide to set a conservative overall calibration uncertainty of 5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reliability of this number is supported by the tests on radiosources and planets presented in Section 9, as well as other calibration tests based on the detection of primary CMB anisotropies shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Colour corrections The overall 5 % calibration uncertainty would strictly apply to any analysis performed in our maps on sources or regions with a power- law spectrum with index 𝛼 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3, as that of Tau A (our primary calibrator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For a different spectrum, uncertainties in the colour corrections must be factored in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These are mainly associated with errors in the measurement or characterisation of the instrument bandpasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI bandpasses were last measured in 2020, for the instrumental configuration corresponding to period 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The statisti- cal uncertainties of these measurements are very low, such that they lead to errors in the global calibrated antenna temperature below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01% for a range of spectral indices 𝛼 ∈ [−3, +3] and for all horns and frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On the other hand, MFI suffered various modifi- cations over its lifetime (see Table 2), which may have introduced modifications in the actual bandpass shapes of periods 1, 2 and 5 with respect to period 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the MFI wide survey, we conservatively assign errors to the colour corrections by comparing the last bandpass measurement from period 6 with a previous one performed in 2013 during period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Through comparing the colour correction coefficients obtained in both cases we find that channel 219 presents the largest differences, and in this case the error scales approximately as 𝜖×|𝛼+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3| %, with 𝜖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note that the error increases as the spectral index of the observation, 𝛼, departs from that of the primary calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For 311 and 313, we obtain 𝜖 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 and 𝜖 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='53 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For 217 we have 𝜖 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51, while for horn 4 we have values between 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We must note that these uncertainties are somewhat conservative, as differences between the two measured bandpasses may not be entirely real, but could also be due to shortcomings in the 2013 measurements, which are deemed much less reliable than those of 2020 due to measurement techniques (see details in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Taking this into account, and the fact that errors in the other channels are smaller, as a conservative choice for this paper, in Table 16 we have assigned an overall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 × |𝛼 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3| % error to colour corrections for 11 and 13 GHz, and 1 × |𝛼 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3| % for 17 and 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note that these errors in the colour correction coefficients should impact the consistency checks presented in Section 9, where we compare with models flux densities of sources with spectral in- dices ranging between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2, and of planets with 𝛼 ≈ 2, or those presented below in this section where we correlate our maps with templates tracing the CMB anisotropies or the CMB dipole that also have 𝛼 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the former case, we find differences of ∼ 5% which we are confident are due to uncertainties in the source calibration models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the CMB anisotropies and CMB dipole, the differences are ∼ 3 % and ∼ 10 % respectively, and are driven by statistical noise (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Beams One of the instrumental aspects that are most carefully charac- terised in CMB experiments are the beams and derived window functions, as they have a direct impact on the amplitude of the derived power spectrum and thence on cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In QUIJOTE MFI this is even more important as its calibration is tied to unresolved point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Comparison between beam radial profiles derived from observations of bright point sources and the numerical optical simulation based on CST software8 described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023) demonstrates an accuracy in the deter- mination of the intensity beam typically below the 2 % level (with respect to the centre of the main beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given that the MFI maps are (re)calibrated using a beam-fitting photometry on point sources, errors in the beams will directly impact the global map temperature 8 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='com/products-services/simulia/products/ cst-studio-suite/ MNRAS 000, 1–58 (2022) Mask: default + Ibl > 10° 10-6 311 rescaled Difference 313-311 313 rescaled Expected noise 313-311 10-7 10-8 102QUIJOTE MFI wide survey 29 Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Accuracy of the calibration in the QUIJOTE MFI wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Second column indicates if the type of uncertainty is applicable to intensity (I) and/or to polarization (P) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Type of uncertainty Applies to 11 GHz 13 GHz 17 GHz 19 GHz Method Reference Calibration model I,P 5 % 5 % 5 % 5 % Model for calibrators Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Colour corrections𝑎 I,P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 % 1 % 1 % Bandpass measurements Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Beam uncertainty I,P 2 % 2 % 2 % 2 % CST beam model, Tau A Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Zero level [mK] I −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 0 0 Plane-parallel model Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 I→P leakage P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='65 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 % Cygnus area Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Polarization efficiency P 3 % 3 % 4 % 4 % Lab measurements, Tau A Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Polarization angle (deg) P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Tau A, WMAP/Planck Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Unknown systematics: Real space (𝜇K/beam) I < 53 < 49 < 118 < 224 Null tests at 𝑁side = 64 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Real space (𝜇K/beam) P < 12 < 15 < 10 < 13 Null tests at 𝑁side = 64 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Harmonic space (30 < ℓ < 200) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 % Null tests Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Harmonic (30 < ℓ < 200) P 3 % 4 % 6 % 6 % Null tests Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Overall calibration error𝑏 I 5 % 5 % 5 % 5 % Overall calibration error𝑏 P 5 % 5 % 6 % 6 % 𝑎 These numbers should be multiplied by |𝛼 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3|, being 𝛼 the spectral index of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝑏 Obtained as the maximum value of the following errors: for intensity, calibration, beam uncertainty and unknown systematics in harmonic space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and for polarization, we add also I→P leakage and polar efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We confidently estimate the error in this temperature scale to be below 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note though that in extracting flux densities of point sources using the same beam-fitting photometry that is used for the main calibration, these errors would be largely suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In Table 16, we adopt a conservative value of 2 per cent, which cor- responds to the maximum error associated with the determination of the brightness of a beam-filling emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, a detailed description of the MFI beams can be found in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), where we use the CST optical simulations and the Mueller matrix formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to the MFI optical design, the cross-polar terms are significantly smaller than the copolar terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, for horn 3, the cross-polar terms are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 % of the copolar beams across the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This implies that the diagonal components of the Mueller matrix (𝑀II and 𝑀QQ) can be considered nearly identical (with that accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, the leakage terms 𝑀IQ and 𝑀QI are also identical in this limit, and are given by one half of the difference of the copolar beams at 0◦ and 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), these terms have a quadrupolar structure with two positive and two negative lobes, with typical peak amplitudes (relative to the copolar peak) of ≲ 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown below in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9, when studying bright compact sources in the MFI wide survey, these patterns are clearly visible around Tau A (in Stokes 𝑈 parameter, because most of the signal appears in 𝑄) and Cas A (in this case, as the source is essentially unpolarized, they are seen both in 𝑄 and 𝑈 maps, rotated by 45◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When integrated on scales larger than the beam, these patterns average to zero, and thus have minimum impact on the photometry analyses (see also Leahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2010, for the case of Planck beams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For example, for the MFI 311 map, the impact on a photometry measurement using either aperture photometry in 1 deg, or beam fitting, is well below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 % across the full frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, we neglect this contribution to the overall calibration error due to beam uncertainties, and in Table 16 we adopt the same calibration uncertainty in polarization as for intensity beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Intensity-to-polarization leakage Despite of the fact that the MFI is a true polarimeter, in the sense that the polarization signal is produced directly for each individual horn and frequency band, there are several known systematic effects that may lead to spurious polarization signals, particularly in bright regions in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the previous subsection we have already dis- cussed, for bright point sources, the intensity-to-polarization leak- age (hereafter IPL) terms due to beam non-idealities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we discuss the IPL terms arising from the bandpass mismatch between the two pairs of channels that contribute to a given polarization timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the MFI instrument, the 𝑟-factors in equations 3 and 4 are determined using Tau A observations (see details in Génova- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When observing a sky region with a bright intensity emission, the effective 𝑟-factor might change depending on the spectral index of the sky emission, particularly if it differs from that of Tau A (𝛼 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using the detailed measurements of the bandpasses, we have estimated that for spectral indices typical of Galactic emission (𝛼 ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5, 0]), the amount of signal leaked into Stokes 𝑄 or 𝑈 due to this effect is typically below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 % of the intensity signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For a CMB spectrum (𝛼 ≈ 2), it is still below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we provide an independent confirmation of the order of magnitude of the IPL in the MFI wide survey maps using the sky emission in the Cygnus region, located at Galactic coordinates (𝑙, 𝑏) = (80◦, 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 23 shows this area in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As the intensity emission in this region is dominated by free-free, it is expected to be almost unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use a cross-correlation analysis (similar to the one used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2) to obtain the corre- lation coefficient 𝛼 that minimizes 𝑄 − 𝛼𝐼 within a region centred at (𝑙, 𝑏) = (80◦, 0◦) with a radius of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The values are always below 1 per cent for all cases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For 311, we find 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='65 % for Stokes Q and U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The largest values are found for 419, where we obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91% and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='41% for Stokes Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This effect in the Cygnus area is clearly seen in the maps of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 21 and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The values reported in Table 16 correspond to the most conservative case (Stokes Q or U) at each frequency, in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 30 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Minimaps of 15◦ × 15◦ around the Cygnus region, located at Galactic coordinates (𝑙, 𝑏) = (80◦, 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show the horn 3 11 GHz (top) and horn 4 19 GHz maps (bottom) at their original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The circle indicates the region where the IPL is computed (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The two bright compact objects in the polarization maps located outside the circle, W63 and Cygnus A, are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Polarization efficiency As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6, the calibration of the polarization ef- ficiency of the MFI wide survey data is done in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we use laboratory measurements taken at the end of period 6 to calibrate the polar efficiency of each individual MFI channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, we use the wide survey data in period 6 to add also the correction factors to these polar efficiencies associated with a pos- sible error in the determination of the 𝑟-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These procedures provide a determination of the polarization efficiency in period 6 with a relative accuracy of 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, in a second step these values from period 6 are transferred to the other two periods that are used in the construction of the MFI wide survey polarization maps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2 and 5), using beam fitting photometry (BF1d) measurements on Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The error budget for these factors is given by the accuracy of the flux density extraction, which is found to be of the order of 1 % for horn 3, and 2 % for horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As they correspond to systematic errors, we adopt the conservative approach of adding them linearly, and we quote an overall 3 % error in the polar efficiency for horn 3, and 4 % for horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the following subsections we evaluate unknown systematic effects in the polarization maps, noting that in those cases, the global errors include the polar efficiency error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9 we also discuss the polarization fraction of Tau A and Cyg A, and the polarized flux in W63, as further consistency tests for this polar efficiency calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Internal calibration of the wide survey and consistency checks: evaluating unknown systematics Following the methodologies outlined in Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2014c) and Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2014d), we use internal consistency checks based on null test maps and other data splits of the wide survey in order to estimate the impact of systematic effects Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Systematic effects in the MFI wide survey maps, evaluated in the maps degraded to 𝑁side = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The excess signal (last column) is computed as the quadratic difference between the values for half and ring null test difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel T,Q,U p-p (half) rms (half) rms (ring) Excess rms [𝜇K] [𝜇K] [𝜇K] [𝜇K] 217 T 1177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 217 Q 410.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 419 U 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 in the overall calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This is particularly useful for assessing the impact of "unknown systematics", i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' those for which we do not have specific measurements or numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the MFI wide survey, and given that we want to focus on the relative calibration of the instrument, we use as a reference the set of null test maps and data splits labelled as "with common baselines" in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Unknown systematics in real space Uncertainties due to (unknown) calibration or systematics effects at the pixel scale have been calculated using the HMDM for common baselines, degraded to 𝑁side = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At this resolution, each pixel roughly corresponds to the beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reference mask for the analysis is the default one (sat+NCP+lowdec) as defined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 17 lists the rms values and peak-to-peak (p-p) variation for the HMDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2014c), the p-p values are computed as the difference between the 99 % and the 1 % quantiles in the pixel value distribution, in order to neglect possible outliers9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A comparison between these numbers for the half-mission null tests and those for the ring null tests is useful for checking residual calibration and/or systematic effects on large angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given that the ring null test maps cancel out possible variations in scales longer than 30 s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the duration of one azimuth scan), they can be used as our best estimate of the noise level, which includes white noise and 1/ 𝑓 on degree scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Any variation on scales longer than one minute, due either to calibration uncertainties in the gain model or systematic effects, will appear as a signal excess in the HMDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As illustration, the top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 14 shows the ring null-test difference maps for the 311 (horn 3 at 11 GHz) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results of this comparison are shown in Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Column 5 presents the rms value for the ring difference maps, and column 6 shows the 9 Note that for a Gaussian distribution, we should have p-p=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='65𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 0 50 100150200 1 0 1 2 3 4 6 4 2 0 2 4 6 4 00 2 0 b 2 4 6 11 GHz Q 11 GHz U 11 GHz 86848280787674 86 84 82 80 78 76 74 86848280787674 1 (deg) 1 (deg) 1 (deg)0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mK CMB 6 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2 0 b 2 4 6 I 19 GHz 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' GHz U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' GHz 86848280787674 86848280787674 86848280787674 1 (deg) 1 (deg) 1 (deg)QUIJOTE MFI wide survey 31 signal excess in the half-mission difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Comparing these values with those in Tables 11 and 14 for the noise levels for the wide survey, we find that in polarization, the rms excess due to unknown systematics is well below the white noise levels, with typical values in the range 5–20𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, we find a similar situation for horn 3 and the 17 GHz frequency maps of horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the two maps at 19 GHz (horns 2 and 4), the residuals are slightly larger than the white noise levels, but still well below the total noise contribution in those channels (column 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, for horn 3, the residuals at beam scales are of the order of ∼ 50𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These numbers are used to complete the main table 16, appearing as "unknown systematics" in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a conservative choice, the values for horns 2 and 4 are combined linearly instead of using a quadratic combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Unknown systematics in harmonic space We use the ratio of cross-power spectra of the null test maps with some external maps, as the reference tool to validate the calibration in harmonic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The use of cross-spectra to external maps min- imises the effects of noise bias on the power spectrum estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In practice, given two maps 1 and 2 that we want to compare, we compute 𝐴1,2 = �� 𝐶1,X ℓ 𝐶2,X ℓ � ℓ � X , (19) where 𝐶𝑖,X ℓ is the cross-spectrum of map 𝑖 (=1, 2) with some other external map X, with X running over all possible uncorrelated ex- ternal maps, and the brackets represent the (unweighted) average in a given multipole range (< .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' >ℓ) or over all external maps (< .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' >X), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For completeness, we also evaluate the uncertainty on this parameter (𝜎𝐴1,2) as the standard deviation of those ratios over the external maps, 𝜎𝐴1,2 = 1 √𝑛X 𝑠𝑡𝑑𝑋 �� 𝐶1,X ℓ 𝐶2,X ℓ � ℓ � , (20) where 𝑛X is the number of external maps involved in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this section, all cross-spectra are obtained using Xpol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reference mask adopted for this computation is the default one (sat+NCP+lowdec), which preserves the declination range 6◦ ≤ 𝛿 ≤ 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This mask is apodized using a 5◦ cosine function, as implemented in the NaMaster library (Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps have been smoothed to a common resolution of one degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For MFI, the ratios are evaluated and averaged within the multipole range ℓ = 30 to ℓ = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The lower value of ℓ = 30 guarantees that the pseudo-𝐶ℓ estimation is not affected by mode coupling due to incomplete sky coverage, and constitutes a conservative choice regarding possible large scale residuals due to RFI and atmosphere, as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As external maps, we decided to use low frequency maps (≤ 70 GHz) from satellites, in order to have similar foreground components to the signal in the QUIJOTE maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we use the 9-year WMAP maps (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013) for bands K, Ka, Q and V, and the PR2 Planck-LFI maps at 30, 44 and 70 GHz corrected from bandpass leakage (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Intra-nulltest calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We first evaluate the relative calibration of the wide survey, using the six null test maps described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, namely half (mission), rings, halfring, daynight, pwv and tbem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For each case, we compare the relative calibration of the Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Intra-nulltest calibration of the MFI widey survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show the consistency of the null test maps, for intensity (TT, top) and polarization (average of EE and BB, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' two maps in each pair ℎ1 and ℎ2, as in equation 19, and we evaluate the error bar using equation 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 24 shows the result both for intensity (TT) and polarization (average of EE and BB) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, we find a good consistency of all the different data splits well within one per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At 11 and 13 GHz, the maximum discrepancy is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average of the six null test cases is consistent with one (perfect relative calibration) within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At 17 and 19 GHz, the maps from horn 4 present a maximum discrepancy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 %, and the scatter of the six measurements stays within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Horn 2, which is known to be the noisiest one, presents the larger discrepancy of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 % for the half-mission null test, and the average of the six values is consistent with one within 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, we find larger values of the scatter, as expected due to the lower signal-to-noise ratios of these maps, although we remind that in this case our analysis also probes possible time vari- ations of the polarization efficiency values on top of the global cal- ibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For horn 3, the maximum discrepancy is associated with the halfring null test, which presents deviations of +7 % for 311, and -8 % for 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, we note that this null test is expected to be noisier than the others, due to the lower number of independent crossings in each half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average of the six measurements is fully consistent with one, and has a scatter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 % and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 % for 11 and 13 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For horn 4, we find a maximum discrepancy of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average of the six measurements is again consistent with one, and the scatter is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 % and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 % for 17 and 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, for horn 2, as in intensity, we find the largest scatter of the MNRAS 000, 1–58 (2022) Intra-nulltest calibration TT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03 half halfrings pwv rings daynight tbem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 217 219 311 313 417 419 Channelntra-nultestcalibrationEE+BB half halfrings pwv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 rings daynight tbem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 217 219 311 313 417 419 Channel32 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The largest discrepancy is found to be 17 % but with a large error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average of the six measurements is slightly biased towards positive values of 𝐴 for 219, but not significantly (two sigmas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The scatter of the measurements is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 % and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 % for 217 and 219, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In summary, the internal calibration scale of the MFI wide survey seems to be consistent within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 per cent in intensity for all horns, reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 % for horn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, we find consistency within 3–4 per cent in for horns 3 and 4, and within 10 % for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To put in context these values, it is useful to compare them with the expected scatter in the 𝐴 values in the case of a perfectly calibrated instrument with the realistic noise levels of the MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this purpose, we have repeated this analysis using simulations including realistic 1/ 𝑓 noise levels as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5 of Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' According to these simulations, the expected scatter of the six null tests in intensity is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 %, while in polarization we expect 2 % for horn 3 and horn 4 at 17 GHz, and we could have up to 5–6 % for horn 2 and horn 4 at 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We stress that these numbers are driven by the 1/ 𝑓 noise in the maps, and therefore they represent the actual sensitivity of this method to detect calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Any calibration uncertainty due to systematic effects in the real data will add to these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When comparing these values from simulations with those found for real sky measurements, we find that they are consistent in intensity, but the real data produce slightly larger scatter in polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This small excess of uncertainty in the polarization values from the real maps can be ascribed to polarization efficiency sys- tematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a conservative approach, we decided to quote as calibration uncertainty in Table 16 the final numbers obtained from this test, thus including also the 1/ 𝑓 noise contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Inter-period calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We now evaluate the time sta- bility of the wide survey calibration, using the four maps per period described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2, again for the case of "common baselines".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also note that period 1 only has observations at high elevations, so in order to have a common sky coverage for this comparison in the four maps, we restrict the analysis in this particular case to a sky mask covering the declination range 8◦ ≤ 𝛿 ≤ 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As usual, this extended mask is apodized using a 5◦ cosine function, as im- plemented in the NaMaster library (Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 25 shows the comparison of the 𝐴 factors for the four maps by period used for the wide survey (periods 1, 2, 5 and 6), when compared to the total final map for each horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, the internal consistency is found to be again better than 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The largest discrepancy in absolute value is found for the map 419 in period 1, at the level of -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The standard deviation of the four 𝐴 values for each horn and frequency is found to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 % for channels in horns 2 and 3, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7–1 % for horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, we recall that some periods are not used for the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, period 1 is not used in polarization, period 2 is not used for horn 4, and period 5 is not used for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The maximum discrepancy with respect to the final map is found in 313 for period 5, at the level of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Taken as a whole, these values suggest that the calibration scale is stable within 1 per percent in intensity, and within 2 per cent in polarization, during the six years of observations covered by the wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Inter-horn calibration for horns 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given that the frequencies of 17 and 19 GHz are observed with horns 2 and 4, we also carry out an inter-horn comparison of the final wide survey maps at these frequencies using the same methodology as above, and where the 𝐴 factor in equation 19 now compares the ratio of the Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Inter-period consistency checks, in intensity (TT, top) and polar- ization (average of EE and BB, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show the 𝐴 factor computed as in equation 19, when comparing the map per period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' using the data of that given period only) to the total final map, for each horn and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' two maps of a given frequency from the two horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this case, we obtain two values, 𝐴217,417 and 𝐴219,419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 26 both for intensity (TT) and polarization (EE and BB, here plotted separately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find that the relative calibration of the wide survey between horns 2 and 4 is consistent within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 per cent in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, this test is not providing very restrictive results due to the high noise levels of horn 2 in comparison to horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Nevertheless, we can conclude that the relative calibration of the two 17 GHz maps is found to be consistent within 2 per cent, while for 19 GHz we find consistency within 4 per cent if we average the values for EE and BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this later case, our simulations show that the separated values for EE or BB alone might differ by more than 4 per cent in the ideal case of a perfect calibration, due to the (white plus 1/ 𝑓 ) noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Summary of the internal calibration tests The overall calibration uncertainty quoted for the QUIJOTE MFI wide survey maps is 5 % in intensity for all frequency maps, 5 % in polarization for 11 and 13 GHz, and 6 % in polarization for the combined 17 and 19 GHz maps (see last two rows in Table 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These values are mainly limited by the physical modelling of the point-sources (Tau A, Cas A) used to calibrate the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, all the tests in this section show that the internal consis- tency of the calibration and gain model, which spans 6 years of measurements, is within the one per cent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, the MNRAS 000, 1–58 (2022) Inter-period calibration TT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='020 p1 p2 p5 p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='980 217 219 311 313 417 419 ChannelInter-periodcalibrationEE+BB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03 p2 p5 p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 217 219 311 313 417 419 ChannelQUIJOTE MFI wide survey 33 Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Inter-horn consistency check between horns 2 and 4, in intensity (top) and polarization (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' internal consistency tests show that the calibration is controlled at the 2–3 per cent level for frequencies 11, 13 and 17 GHz, while for 19 GHz, and particularly for horn 2, this uncertainty could be up to 6 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, we note that in this later case, the quoted uncertainty includes calibration errors, polarization efficiency uncertainties and 1/ 𝑓 noise contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Other calibration tests 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 CMB anisotropies CMB anisotropies in intensity can be measured in the QUIJOTE MFI wide survey maps using a cross-correlation with an external CMB template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We follow the methodology described and validated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 of Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021), and use a template fitting method with two templates: a reference CMB map (mCMB), and a "foreground" map to account for chance alignments between the CMB and the Galactic foregrounds (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The basic assumption is that the QUIJOTE map (mMFI) can be written as a linear combination of these two maps as mMFI = 𝐴mCMB + 𝐵f + n, (21) where 𝐴 and 𝐵 are the parameters of the linear combination, and n represents a noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using the cross spectra of the QUI- JOTE maps with both external templates, 𝐶MFI,CMB ℓ and 𝐶MFI,f ℓ , we can extract both 𝐴 and 𝐵 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021), this method produces unbiased results for the CMB recon- struction (𝐴 = 1), provided that there is a perfect consistency with Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Relative amplitude of the CMB signal in the QUIJOTE MFI maps, using cross-correlations with the Planck SMICA map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are obtained using rotations of the CMB map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For consistency, we show that the average signal of the cross-correlation with rotated CMB maps is consistent with zero, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the calibration of the CMB map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, the method can be used as an additional calibration test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we use as a reference the SMICA 2018 map (Planck Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020d), but we have checked that consistent values are obtained using other versions of the Planck CMB map (NILC, COMMANDER, SEVEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As foreground template, we use the WMAP 9-year K-band map (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013), after subtracting the CMB component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis mask is the same as in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021), which combines the default QUIJOTE analysis mask (NCP+sat+lowdec) with the Planck common confidence mask for temperature analyses (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020d), apodized with a simple 2-degree smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All cross-spectra in this section are computed using Xpol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are obtained using rotations of the CMB map in steps of Δ𝑙 = 18◦, as in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis is carried out in the multipole range [100, 200], but consistent results are obtained in other ranges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' we also tested [30, 200], although the overall significance is lower in this case due to the larger 1/ 𝑓 contribution of lower multipoles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final results are shown in Figure 27 and Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The CMB signal is detected in all channels, with a significance larger than 10-sigma in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These error bars are consistent with the level of 1/ 𝑓 noise in the QUIJOTE maps (see Table 4 in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that, due to the strongly correlated noise in the MFI intensity maps, es- timates from the same horn tend to deviate in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All values are consistent with 𝐴 = 1, providing an independent confirmation of the calibration scale of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we also provide a combined measurement of the CMB signal present in the QUIJOTE MFI maps, using a weighted average combination of all channels and accounting for the noise correlation between frequen- cies of the same horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The overall result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03) provides a 35-sigma detection of the CMB anisotropies in the QUIJOTE MFI intensity maps, and shows a consistent calibration with Planck within three per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 CMB dipole As an additional calibration test, we present here the detection of the CMB dipole in the MFI wide survey maps, using a cross-correlation technique similar to the one used in the previous subsection for the CMB anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analysis, specific MFI wide survey maps are generated excluding the dipole removal and the atmospheric cor- MNRAS 000, 1–58 (2022) nter-hornscalibrationTT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='999 17 19 Frequency[GHz]Inter-horns calibration EE,BB EE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 BB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='92 17 19 Frequency[GHz]CMB Cross-correlations l E[100, 20o] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 217 219 3i1 3i3 417 419 Channels34 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Relative amplitude (𝐴) of the CMB component in the QUIJOTE- MFI wide survey maps with respect to the SMICA Planck map, obtained with cross-correlations in the multipole range 100–200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are obtained using rotations of the CMB map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel A Uncertainty 217 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='068 219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='086 311 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='037 313 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='033 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='086 419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='097 Combined 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='029 Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI wide survey 311 (horn 3 at 11 GHz) map, with the dipole component not removed from the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, the map has been downgraded to resolution 𝑁side = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' rection steps in the post-processing stage of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 28 shows one example of these maps, for the case of horn 3 at 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use a template fitting method in real space with three tem- plates: a reference CMB dipole template map (mdip), a "foreground" map to account for the Galactic component (f), and a constant map accounting for a residual monopole term (𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As in the previous section, we assume that the MFI wide survey maps (mMFI) can be written as a linear combination of those three templates as mMFI = 𝐴mdip + 𝐵f + 𝐶 + n, (22) where 𝐴, 𝐵 and 𝐶 are the three coefficients to be obtained and n represents the noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The dipole template map mdip is prepared following the methodology outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 of Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021), including both the solar and orbital CMB dipole terms with the measured amplitudes by the Planck collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The dipole prediction is generated at the TOD level, and then this is projected into a sky map using the PICASSO map-making algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the Galactic template, we use again the WMAP 9-year K-band map after subtracting the CMB component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analy- sis, all maps are degraded to a common resolution of one degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis mask combines the default QUIJOTE analysis mask (NCP+sat+lowdec), the Planck confidence CMB mask for temper- ature analyses (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020d), and a Galactic mask |𝑏| < 30◦, in order to avoid a possible bias in the dipole determination due to the Galactic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We first validate the methodology using end-to-end simula- tions of the MFI wide survey including the dipole component and realistic 1/ 𝑓 noise levels as in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find that our approach provides unbiased estimates of the dipole amplitude Table 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fitting for the CMB dipole in the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We present the relative amplitude with respect to the expected CMB dipole, and the associated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel Relative amplitude Uncertainty 217 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='04 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝐴 = 1) for all MFI frequency maps, with typical errors of few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have also tested the impact of the three different corrections that are applied to the maps (RFI, FDEC and ATMOS) on the reconstructed dipole amplitude 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In summary, we find that including or not the RFI and FDEC corrections does not bias the recovered 𝐴 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, the ATMOS correction significantly affects the recovered amplitude, especially in the high frequency MFI bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This is expected because the atmospheric templates are built on approximately one hour timescales, and on those scales the CMB dipole component is a very stable signal in the azimuth scans (rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Because of this, the ATMOS correction is not applied for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The measured values in real data are presented in Table 19, for each one of the MFI wide survey maps separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars have been estimated using the following methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We rely on the null test maps for independent baselines as the most representative method to capture large angular scale noise in the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, we repeat the analysis and detect the CMB dipole in the half1/2, pwv1/2, tbem1/2 and daynight1/2 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reported values correspond to the average dipole of the 8 cases, and the error bar is the scatter of the 8 measurements, taken to be a representative error of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have tested that we obtain almost identical results if we carry out the analysis on maps with no FDEC and/or RFI corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we also present the weighted average combination of all channels, accounting for the correlation between frequencies of the same horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The value is 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09, which corresponds to a 10-sigma detection of the CMB dipole, and it is consistent with the Planck calibration within nine per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Bright point sources and planets Bright radio sources and planets have been used extensively as a basic calibration test for MFI wide survey maps in several stages of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Indeed, the maps in each period are recalibrated in order to match the Tau A model in intensity (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Below in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9 we present a detailed study of few bright objects (Tau A, Cas A, Cyg A, 3C274, W63, Jupiter and Venus), which could be seen as a further validation test of the overall calibration scale of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Setting the zero levels The QUIJOTE MFI wide survey intensity maps produced by our default pipeline are insensitive to the true absolute zero level (monopole) of the sky emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A monopole signal is essentially unconstrained for QUIJOTE MFI, as a global constant added to MNRAS 000, 1–58 (2022) H3, 11GHz (with dipole) 10 mK 10QUIJOTE MFI wide survey 35 the full TOD database is not changing the map-making solution af- ter the basic TOD processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Indeed, in the post-processing stage maps are corrected of any residual monopole and dipole signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to estimate the zero levels of these maps in intensity, we follow a methodology similar to the one adopted by WMAP (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2003), and we assume a plane-parallel model for the Galactic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In that case, the zero level of the maps can be estimated by fitting a cosecant model of the form: Δ𝑇 = 𝐴 csc(|𝑏|) + 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (23) For this analysis, we use the smoothed maps at 1◦ angular reso- lution, and degrade them to 𝑁side = 64 in order to have approx- imately independent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We carry out the fit independently in both hemispheres, using the Galactic latitude ranges 15◦ < 𝑏 < 90◦ and −90◦ < 𝑏 < −15◦ for the northern and southern hemispheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We mask the satellite band, and in the case of the northern sky, our analysis also excludes the region in Galactic lon- gitude corresponding to the North Polar Spur (0◦ ≤ 𝑙 ≤ 35◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are computed using the scatter of the results around the mean value, when adding realistic noise simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analysis, we use 100 of the simulations described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reference results adopted here correspond to the northern hemisphere, due to the larger sky fraction covered by the QUIJOTE MFI footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For QUIJOTE MFI 11 GHz (horn 3), we have 𝐵 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 mK, where the error bar includes both the effect of varying sky emission and the noise variance contained in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similarly, for QUIJOTE MFI 13 GHz (horn3) we have 𝐵 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results for the southern hemisphere are consistent with those (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='27 mK and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 mK for 11 and 13 GHz, re- spectively), although they have larger error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the other two frequency bands (17 and 19 GHz), and both for horns 2 and 4, the zero levels are statistically consistent with zero in both hemispheres (with typical error bars of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These values are inserted in Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we note that there are other methods in the literature for deriving the zero levels of radio maps (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Wehus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017), which could be applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, we emphasize that those analyses should be done carefully, due to the special filter- ing of large angular scales (FDEC) applied to the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Polarization angle As described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), the reference angle for each MFI observation is calibrated using daily Tau A observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our calibration scheme provides a reference angle for each period and channel, as this value changes across the spectral band, from horn to horn, and also with the instrument configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As this daily calibration might suffer from 1/ 𝑓 noise uncertainties, the final QUIJOTE MFI wide survey maps are recalibrated again using Tau A in each period (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here, we can evaluate the error budget associated with the polarization angle in the wide survey maps using Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference method, we use aperture pho- tometry in the polarization maps smoothed to 1 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We adopt an integration radius of 𝑟1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦ for the primary aperture, and an outer annulus between 𝑟1 and 𝑟2 = √ 2𝑟1 to correct for the lo- cal background contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The photometry results are described in Table 24 and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 20 presents the error budget in the polarization angle obtained using two methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, col- umn 2 presents the scatter (standard deviation) of the Tau A angle measurements obtained from the null test maps with independent baselines (half1/2, pwv1/2, ring1/2, daynight1/2 and halfring1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On the other hand, column 3 presents the statistical error obtained Table 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error budget for the polarization angle in the wide survey, based on Tau A photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We include the error budget from the scatter of the measurements in the different null tests (column 2) and the statistical error obtained from the photometry method (column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel Error (null tests) Error (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=') (deg) (deg) 217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='91 219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='67 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='64 419 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 17 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='53 Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 19 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51 Table 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Comparison of the reconstructed angles in the QUIJOTE MFI wide survey data to WMAP-K (column 2), LFI30 (column 3) and MFI 311 (column 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Channel WMAP-K LFI30 MFI-311 (deg) (deg) (deg) 217 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 – 313 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 417 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 17 GHz −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 19 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 from the propagation of the errors from the photometry measure- ment in the final maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a conservative approach, we keep the highest value of each pair as representative of the error budget in the angle determination from Tau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We see that the uncertainty changes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦ for 311, to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7◦ for 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a further consistency check for the polarization angle cal- ibration, we compare the measured MFI wide survey polarization angle maps with those from WMAP 9-year K-band map (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013) and Planck PR4 LFI30 data (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 21 presents the results of this comparison, including also an internal comparison to the MFI 311 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis is car- ried out smoothing all maps to 1 degree resolution, and degrading them to 𝑁side = 64, in order to match approximately the beam scale in one pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the standard analysis mask (NCP+sat+lowdec), but in addition, we keep only those high signal-to-noise pixels with a nominal uncertainty in the MFI 311 angle 𝜎𝜙311 ≤ 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to avoid bright regions that might bias the comparison, pixels that have an absolute value in 𝑄 or 𝑈 that is greater than 2 mK in the WMAP K-band after being rescaled to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 GHz using a spectral index of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 are also flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, we also exclude the bright Cygnus area removing all pixels within 5 degrees around the loca- tion (𝑙, 𝑏) = (80◦, 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The resulting analysis area has 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to correct for residual zero level differences between the MFI and the WMAP/Planck maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' due to unresolved point sources), we use a TT plot technique between WMAP-K and each MFI Stokes Q and U map within the analysis mask, and we remove the fitted zero levels from the MFI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that the resulting values are basically consistent with zero (within the error), but of MNRAS 000, 1–58 (2022) 36 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the order of 20 𝜇K for 311 and 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although small, they might introduce measurable differences (at the level of a degree) in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For each MFI map, we compute the weighted mean of the difference between the two angles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝜙MFI − 𝜙WMAP for the first case), using as weights the inverse variance of the angle, which in turn is derived from the 𝑄 and𝑈 weight maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars in Table 21 are generated with a Monte Carlo method using 100 of the noise simulations described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We add each noise simulation to the corresponding MFI map, and repeat the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The error bar corresponds to the standard deviation of the 100 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In general, all the measured differences are statistically consistent with zero given the noise uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MFI 311 (horn 3 at 11 GHz) is consistent with both WMAP-K and LFI30 within the quoted uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The situation is similar for the 19 GHz maps (both horns 2 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, we note that there is a moderate tension with the MFI 313, which deviates in the case of LFI30 up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 sigmas, and the 17 GHz cases, which deviates 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 sigmas for the combined map of horn 2 and horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to investigate this possible discrepancy, we have repeated the analysis but using all the different null test maps with independent baselines (half1/2, pwv1/2, ring1/2, daynight1/2, halfring1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The error is now computed as the standard deviation of all those values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The result for the MFI 313 comparison with LFI30 now gives −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9, showing that maybe the error in this case is slightly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' While we are still finding a discrepancy, the significance is now reduced to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 sigmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Another point that we have studied is the possible impact of Faraday rotation in this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using the Galactic Faraday depth maps from Hutschenreuter & Enßlin (2020), we estimate that in our analysis region the mean rotation measure is −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This would introduce differences of the order of approximately −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4◦ between MFI311/MFI313 and LFI30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although this value is not enough to explain the discrepancy, it helps to further decrease the tension below the 2 sigma level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The final results in Table 16 contain the worst case value based on the three values reported in this section (two values for Tau A in Table 20, and the standard deviation of the comparison with WMAP/Planck in Table 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6 SIMULATIONS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Sky signal Some of the analyses in this paper make use of sky simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our reference sky simulations were developed within the context of the RADIOFOREGROUNDS project10, and are described in detail in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 of Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They contain different foreground components from the Planck FFP10 sky model (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020b,c), a CMB realization, and the CMB dipole contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For some applications, these sky simulations are projected into the MFI wide survey TODs, and the PICASSO map-making code is used to generate synthetic maps with the same flagging and number of hits as in the real wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These simulated data can also include a noise contribution, injected at the TOD level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As explained in this paper, this approach has been extensively used to validate some aspects of the pipeline (map- making, transfer function, null tests, determination of the CMB dipole, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These sky simulations are also used below to evaluate the statistical errors associated with the power spectra (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='radioforegrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='eu 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Simulated noise maps In addition to the end-to-end noise simulations that have been pro- duced as explained in the previous subsection, we also construct noise simulations for the different channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', pair frequency- horn) maps, starting from the HMDM of totally independent splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The simulations aim to account for the measured anisotropic be- haviour, spatial correlations and the correlations between the two frequency channels of the same horn in the wide survey maps (∼ 60– 80% in intensity, and ∼ 20% in polarization, as seen in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The anisotropic behaviour follows the properties of the cor- responding Local Variance (LV) maps per channel and per Stokes parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These LV maps are estimated from the HMDM maps, by assigning, at each pixel at resolution 𝑁side = 512, the variance com- puted from the surrounding pixels at a given distance (39 arcmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' this value has been chosen as a compromise to have enough pixels to provide an accurate estimation and, at the same time, to preserve as much information at small scale as possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The estimation of the variance takes into account only those pixels which are within the observed sky at each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Each one of the HMDM are normal- ized by dividing them by the square root of the corresponding LV maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The non-observed pixels of the normalized HMDM are filled with a Gaussian random realization with unit dispersion, building in this manner extended-normalized HMDM maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We now compute the noise spatial correlation by computing the TT, TE and BB angular power spectra (APS) of the extended- normalized HMDM maps, and from there, we derive a model of these APS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This is done by estimating a smoothed version of the observed APS of the extended-normalized HMDM maps for each channel, using a polynomial fit (of order 4), and by defining the max- imum multipole that provides a variance (at the map level) as close as possible to the one of the corresponding extended-normalized HMDM maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following this process, we end-up with a model for the noise correlations that provides the right power level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A noise simulation is now generated by drawing a Gaussian random map in harmonic space, following the corresponding mod- els of the noise APS for each frequency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The maps (T, Q and U) are further multiplied by the square root of the corresponding LV map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the correlation coefficient between frequency maps of the same horn to further modify the simulated map of the second member of the pair (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', the 13 GHz frequency channel in the case of horn 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we construct the final version of the simu- lated map of the second member of the pair as a linear combination of the first member of the pair and the initial version of the second map, taking into account the correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this way, all the pixel-based statistics are maintained for the two members of the pair, as well as the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The APS of the first member are also maintained, but, eventually, we modify the APS properties of the second member and the cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The correlation at pixel level is imposed for T, Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Notice that for the Stokes parameters this is done as if they were scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Nevertheless, the properties of the polarization intensity are preserved, although we are not able to reproduce the observed cross-correlation in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find that this approximation is the most adequate for our further analyses, since most of them are addressed in the pixel domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As illustration, Figure 29 shows the power spectra for a subset of 100 noise simulations for horn 3 at 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7 POWER SPECTRA OF THE WIDE SURVEY MAPS In this section we study the main properties of the auto- and cross- spectra of the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We consider three masks, MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 37 Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Power spectra (TT, EE, and BB) for 100 noise simulations of the 311 map (horn 3 at 11 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The red line shows the reference noise power spectrum for the half mission difference maps (labelled as HD) which was used to generate the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The individual power spectrum for each simulation is shown in light grey, and the average of those 100 simulations in light blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' corresponding to different Galactic latitude cuts (|𝑏| > 5◦, 10◦ and 20◦), which are always combined with the default QUIJOTE analysis mask (NCP+sat+lowdec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As usual, each of these three masks is apodized with a five degree apodization kernel and the cosine function implemented in Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All spectra have been computed with the NaMaster code, enabling for the option of "purification" of E and B modes, which allows a better reconstruction of the E and B mixing matrix for cut-sky spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In Appendix E we discuss the validity of the use of this pseudo-Cℓ approach for the wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Throughout this section, all power spectra have been corrected by the MFI beam window functions, as well as the pixel window function (which in this case corresponds to a HEALPix map with 𝑁side = 512).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Noise levels (𝑁ℓ) are estimated from the half-mission difference maps (with independent baselines), and then subtracted from the corresponding power spectra of the maps, in order to obtain the spectrum of the sky signal, 𝐶sky ℓ = 𝐶map ℓ − 𝑁ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have tested that using another estimate of the noise power spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the average of several null test difference maps) produces consistent results to those presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All spectra are binned using Δℓ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In all figures in this section, we represent band power values 𝐷ℓ = ℓ(ℓ + 1)𝐶𝑋𝑌 ℓ /2𝜋, where 𝑋,𝑌 ∈ {𝑇, 𝐸, 𝐵}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Uncertainties in the power spectra of the maps 𝜎(𝐶map ℓ ) are estimated using 100 simulations including sky signal (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1) and realistic noise simulations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The same noise simulations Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' TT, EE and BB spectra for |𝑏| > 5◦, and for all frequencies (11, 13, 17, 19 GHz), represented as solid circles with their corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, dashed lines depict the noise spectra 𝑁ℓ for each case, using the same colour scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' are also used to estimate the uncertainties in the noise level, 𝜎(𝑁ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The quoted uncertainties in 𝜎(𝐶sky ℓ ) are obtained as the quadratic sum of both 𝜎(𝐶map ℓ ) and 𝜎(𝑁ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 30 shows the (auto) power spectra (TT, EE, BB) of the wide survey maps, for the particular case of the Galactic mask with |𝑏| > 5◦, combined with the default QUIJOTE analysis mask (NCP+sat+lowdec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have a high significance detection of TT, particularly for the two lowest frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At these frequencies, MNRAS 000, 1–58 (2022) Noise simulations 11GHz, H3 Sims 10-3 [mk2] HD 10-4 10-5 10-4 [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 10-5 10-6 CBB [mK2] 10-5 10-6 101 102TT, Ibl>5° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='100 [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 11 GHz 13 GHz 17 GHz 19GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001 50 100 150 200 Multipole lEE, lbl>5° 11 GHz 13 GHz 17 GHz 19 GHz mk²] 10 D 10 10- 50 100 150 200 Multipole lBB, Ibl>5° 11( GHz 13 GHz 17 GHz 19 GHz D 10 50 100 150 200 Multipole l38 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Best fit results obtained after fitting the model in equation 24 to the wide survey EE and BB power spectra at 11 GHz, in the multipole range 30 < ℓ < 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' No colour corrections were applied when fitting the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mask |𝑏| > 5◦ |𝑏| > 10◦ |𝑏| > 20◦ 𝑓sky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='27 EE and BB fitted separately 𝐴EE [𝜇K2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 𝐴BB [𝜇K2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 𝛼EE −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='16 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='36 𝛼BB −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='87 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03 𝑐EE [𝜇K2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 𝑐BB [𝜇K2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 𝐴BB/𝐴EE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 Joint EE and BB analysis 𝐴EE [𝜇K2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14 𝛼EE (= 𝛼BB) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='21 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='29 𝑐EE (= 𝑐BB) [𝜇K2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 𝐴BB/𝐴EE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 the polarized emission is dominated by Galactic synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The EE synchrotron signal is clearly detected at large angular scales (ℓ ≲ 100) for 11 and 13 GHz, and the BB signal is also significantly detected in that range for 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the next three subsections we discuss the angular and frequency dependence of these spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A multi-frequency analysis of the power spectra of the MFI wide survey maps, in combination with WMAP and Planck data, will be presented in a separate paper (Vansyngel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Fitting the EE and BB auto-spectra at 11 GHz Figure 31 shows the TT, EE and BB auto-spectra at 11 GHz, for three masks with Galactic latitude cuts |𝑏| > 5◦, 10◦ and 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We focus here on the polarization spectra, EE and BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2018), we fit for these spectra in the multipole range 30 < ℓ < 300, using the following parameterization 𝐶XX ℓ = 𝐴XX � ℓ 80 � 𝛼XX + 𝑐XX, (24) where 𝑋 ∈ {𝐸, 𝐵}, 𝐴XX is the amplitude of the spectrum at the pivot multipole ℓ = 80, 𝛼XX is the slope of the multipole dependence, and 𝑐XX is a global constant which represents the contribution of unresolved (Poisson distributed) radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The power spectra are fitted using the EMCEE ensemble sam- pler (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013), and using a standard Gaus- sian likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our best-fit results, obtained from the marginalised posterior distributions for each parameter, are given in Table 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we fit for the EE and BB power spectra sep- arately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In all three cases, the global constants 𝑐EE and 𝑐BB are statistically consistent with zero, as expected given the noise lev- els of the wide survey maps, and the expected contribution from radio sources at these frequencies, estimated to be ≲ 30 𝜇K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='deg at 11 GHz (Puglisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Herranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Both the EE and BB spectra present similar values of the slope, and no dependence on the Galactic latitude cut is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' When combining the ratios of the EE and BB signals, we find that 𝐴BB/𝐴EE is of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 for the two higher Galactic cuts (|𝑏| > 10◦ and |𝑏| > 20◦), and we obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 for the lowest cut (|𝑏| > 5◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to increase the significance of this measurement, and based on these Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' TT, EE and BB spectra for QUIJOTE MFI 11GHz, as a function of the Galactic cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Dashed lines represent the corresponding noise spectra 𝑁ℓ for each case, using the same colour scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' results, we repeat the analysis now assuming that both EE and BB spectra have the same slope (𝛼EE = 𝛼BB) and Poissonian terms contributions (𝑐𝐸𝐸 = 𝑐𝐵𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this case, we can fit simultaneously for the EE and BB spectra using four parameters (𝐴EE, 𝛼EE, 𝑐EE and 𝐴BB/𝐴EE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results for the amplitudes and slopes are con- sistent with the values obtained in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Regarding the ratio of the amplitudes, we have now a higher significance, with 𝐴BB/𝐴EE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='08 for the |𝑏| > 20◦ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In summary, the MFI wide survey data at 11 GHz show more power in the EE spectra MNRAS 000, 1–58 (2022) TT 11GHZ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 [mk²] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 D Ibl> 5° Ibl>10° Ib1>20° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 50 100 150 200 250 Multipole lEE 1 1GHz Ibl> lbl>10° lb/>20° mk²] 10 D 10- 50 100 150 200 Multipole lBB 3 11GHz Ibl> 5° lb/>10° lb/>20° mk²] 10° D 10 50 100 150 200 Multipole lQUIJOTE MFI wide survey 39 Table 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Best fit results obtained after fitting a constant model to the wide survey EB and TB power spectra at 11 GHz, in the multipole range 30 < ℓ < 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' No colour corrections are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mask |𝑏| > 5◦ |𝑏| > 10◦ |𝑏| > 20◦ 𝐴EB [𝜇K2] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='043 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='041 𝐴EB/𝐴EE (ℓ = 80) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='059 𝐴TB [𝜇K2] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 𝐴TB/𝐴EE (ℓ = 80) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='28 than BB, with a typical BB/EE ratio of a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This value is approximately half of the equivalent BB/EE ratio for thermal dust emission, as derived from Planck observations at 353 GHz (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016a, 2020e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our numbers for the synchrotron emission at 11 GHz can be compared with others in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2020d) found 𝛼EE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05, 𝛼BB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 and 𝐴BB/𝐴EE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 for the synchrotron map at 30 GHz obtained with Commander (Eriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2008), and analysing a sky area of 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='78 and a multipole range ℓ = 4-140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following a similar methodology to the one used here, Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2022) carried out a combined analysis of WMAP-K band and Planck LFI30 data, finding very stable values for the slopes and BB/EE ratios as a function of the sky mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the case of a mask preserving 50 % of the sky, they obtain 𝛼EE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05, 𝛼BB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15, and 𝐴BB/𝐴EE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In both cases, the values are consistent with our results at 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On the other hand, using S-PASS data at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 GHz, Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2018) find significantly larger values of the BB/EE ratio for similar Galactic cuts in the southern sky, with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 for |𝑏| > 20◦, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='03 for |𝑏| > 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 TE, TB and EB spectra at 11 GHz Figure 32 shows the TE, EB and TB power spectra for the 11 GHz map, evaluated in the same sky masks as in the previous subsection (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given that the power spectra of the HMDM is statistically consistent with zero in all three cases (TE, EB and TB), we do not apply the 𝑁ℓ correction in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars are computed using the same methodology described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, for the EB spectra, we also add in quadrature the uncertainty on the power spectrum due to the polarization angle (Table 16), using equation 5 in Minami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2019), and assuming that the underlying EB power spectrum is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We detect a positive cross-correlation between the total inten- sity T and the E-mode polarization (TE> 0) at large angular scales for the three considered Galactic cuts (up to ℓ ≲ 80 for |𝑏| > 5◦, and ℓ ≲ 50 for |𝑏| > 10◦ and |𝑏| > 20◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Beyond ℓ >∼ 150, this TE cross spectrum becomes very noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also find a null corre- lation in TB and EB in the range 30 ≲ ℓ ≲ 150, as expected for a parity-invariant emission process and an accurate calibration of the polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Beyond this multipole range, the error bars increase significantly, in particular for the TB case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We provide a quantitative measurement of the TB/EE and EB/EE ratios by fitting these spectra to a constant value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝐶TB ℓ = 𝐴TB, and 𝐶EB ℓ = 𝐴EB), in the range 30 ≲ ℓ ≲ 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results are presented in Table 23, where we have used the EE fits from Table 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the synchrotron emission, the MFI 311 maps provide upper limits on the EB signal at the level of 4 per cent of the EE component at ℓ = 80 for the |𝑏| > 10◦ cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These results are consistent with Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' TE, EB and TB spectra for QUIJOTE MFI 11GHz, as a function of the Galactic cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' those found in Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2022) for WMAP/Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similarly, for the TB component we provide upper limits at the level of 20 % of the EE component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that for the thermal dust emission, the Planck satellite found a positive TE signal at large scales, a weakly positive TB, and a EB statistically consistent with zero (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2016a, 2020e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Frequency dependence of the EE and BB signal We carry out a simultaneous fit of all the power spectra shown in Figure 30, using the parameterization from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 24, but assuming that the amplitudes are related via a power law dependence in frequency with a temperature spectral index 𝛽s,EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In practice, the amplitude MNRAS 000, 1–58 (2022) TB 3 11GHZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 [mk²] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 lbl> 5° /b/>10° Ibl>20° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 50 100 150 200 Multipole lTE 11GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' [mk"] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 lbl> 5° /b/>10° Ibl>20° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 50 100 150 200 Multipole lEB 11GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='006 Ibl> 5° Ibl>10° Ibl>20° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 [mk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='006 50 100 150 200 Multipole l40 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' at a given frequency channel 𝜈 is computed as: 𝐴EE(𝜈) = 𝐴EE � 𝜈 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 GHz �2𝛽s,EE (25) where 𝐴EE represents the EE amplitude in the MFI 311 map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' There- fore, for this fit, we have seven parameters, namely 𝐴EE and 𝛼EE for the amplitude and angular dependence of the synchrotron signal at 11 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the spectral index 𝛽s,EE describing the frequency depen- dence, and four constant coefficients 𝑐11 EE, 𝑐13 EE, 𝑐17 EE and 𝑐19 EE, ac- counting for the unresolved source contributions at each frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analysis, we also introduce the colour correction term based on the fitted spectral index, using values reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the |𝑏| > 5◦ mask, we obtain 𝐴EE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 𝜇K2, 𝛼EE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13 and 𝛽s,EE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similarly, we repeat the analysis for the BB power spectra, finding 𝐴BB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 𝜇K2, 𝛼BB = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='33, and 𝛽s,BB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In both cases, the first two parameters are in agreement with the values reported in Table 22, taking into account that the colour correction term for the 11 GHz map and for a spectral index of 𝛽 ≈ −3 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The power spectrum of the synchrotron emission detected in the MFI wide survey maps scales with an average index of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 for EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The BB analysis is consistent with this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The weighted average of the two values is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13, consistent with the result of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09 for the 50 per cent mask obtained in Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2022) for the combination of WMAP-K and LFI30 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our value also agrees with the study carried out in the next section for a real space analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A more detailed analysis on the reconstruction of the synchrotron spectral index with QUIJOTE MFI wide survey data is presented in two accompanying papers (de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Vansyngel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8 BASIC PROPERTIES OF THE WIDE SURVEY MAPS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Spectral index of the MFI sky emission 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Intensity We first investigate the spectral dependence of the intensity emission in the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use as a reference the MFI 11 GHz map, which presents the largest signal-to-noise, and we evaluate the spectral index of the sky emission when comparing it to the Haslam 408 MHz (Haslam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1982) and WMAP-K 9-year maps (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The version of the Haslam map used here corresponds to the destriped map from Remazeilles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this spectral analysis in real space, all external maps are filtered using the FDEC procedure, degraded to 2◦ angular resolution, and then downgraded to 𝑁side = 64 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Zero levels of all maps are corrected as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Colour corrections for MFI-311 and WMAP-K are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The analysis region is restricted to the sky area covered by MFI 11 GHz, but excluding the satellite band (satband) as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For each 𝑁side = 64 pixel 𝑝 within the allowed mask, we solve for the spectral index 𝛽(𝑝) using a standard gaussian likelihood function L, which for the case of Haslam and MFI 11 GHz reads −2 ln L(𝑝) = � 𝐼408(𝑝) � 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='408 �𝛽( 𝑝) − 𝑐𝑐11(𝛽(𝑝))𝐼11(𝑝) �2 𝜎(𝑝)2 , (26) where 𝑐𝑐11 is the colour correction for MFI 11 GHz, and the noise term 𝜎 is evaluated using 1000 noise simulations for MFI (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2), and accounting for a 10 per cent calibration error in the Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Spectral index of the intensity emission in the QUIJOTE 11 GHz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Spectral index of 𝛽408MHz−11GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average index is 𝛽 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, the Galactic plane regions have a flatter index, while the regions off the plane have steeper values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: Spectral index of 𝛽11GHz−23GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average spectral index in this case is 𝛽 ≈ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this colour scale, dark red corresponds to AME dominated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Haslam map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Similarly, for the spectral index in intensity between MFI 11 GHz and WMAP-K, we use the same approach, accounting for the WMAP noise levels in the evaluation of the noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 33 shows the results for the case of 𝛽408MHz−11GHz (top panel) and 𝛽11GHz−23GHz (bottom panel), both for the inten- sity emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 35 shows a histogram with the distribution of spectral indices in both maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The median intensity spectral index 𝛽408MHz−11GHz in the full analysis mask is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='90, with a standard deviation of the values across the map of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This value is con- sistent with the expectation for the average synchrotron emission at these frequencies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Platania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de Oliveira-Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fernández-Cerezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, the spatial de- pendence confirms the well-known steepening of the spectral index at high Galactic latitudes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the 408 MHz–23 GHz spectral index map in Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The 𝛽11GHz−23GHz intensity spectral index presents a much broader distribution of values, due to the presence of multiple spec- tral components (AME, free-free and synchrotron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to avoid extreme values for low signal-to-noise (high Galactic latitude) pix- els, in this case we also add a broad gaussian prior 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 to the likelihood in equation 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have checked that this has a mini- mal impact in the final histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The median spectral index in this case is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59, and the standard deviation of the values is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Some of the bright AME dominated regions (Perseus, Lambda Orionis and rho Ophiucus) are clearly visible in dark red colour, while free-free dominated regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cygnus area) appear as light red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A more detailed study of the spectral properties of the sky emission in in- tensity along the Galactic plane (|𝑏| ≤ 10◦) in the MFI wide survey MNRAS 000, 1–58 (2022) Spectral index in intensity (Haslam to MFll1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β408MHz - 11GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2Spectral index in intensity (MFIl1 to WMAP-K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHzQUIJOTE MFI wide survey 41 Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Spectral index map of the polarized emission between QUIJOTE 11 GHz and WMAP 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: the associated error map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' maps is carried out in an accompanying paper (Fernandez-Torreiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also present a component separation analysis of the full MFI maps in de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Polarization In polarization, the 𝛽11GHz−23GHz spectral index presents a cleaner interpretation in this case, as we are dominated by synchrotron emission only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 34 presents the recovered polarization spec- tral index map, following the same methodology as for the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The fit is carried out simultaneously in Stokes Q and U parameters, and in order to obtain a stable solution for high Galactic latitude pixels, we add a Gaussian prior 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 to the likelihood in equation 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The bottom panel in that figure shows the asso- ciated error map, derived from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 35 includes also the histogram of these polarization 𝛽11GHz−23GHz values, showing that the median value is −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09, and the standard deviation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For comparison, we also include in this figure the histogram of spectral index values for the PySM synchrotron model 1 (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017), which in turn corresponds to "Model 4" of Miville-Deschênes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2008) calculated from a combi- nation of Haslam and WMAP 23 GHz polarization data using a model of the Galactic magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find that in the same sky mask, the PySM spectral index map peaks at a higher value and presents a much narrower distribution (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a further consistency check, Appendix F presents the results for the same analysis carried out in this section, but using the MFI 13 GHz map as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We can see that both the mean values and widths of the distributions discussed here are consistently reproduced in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These values for 𝛽11GHz−23GHz in polarization are consistent with those measured in the range 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8–100 GHz (𝛽s ∼ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1) by Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Histogram of spectral index values obtained from Figures 33 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show in dashed lines the mean of the prior adopted in the determina- tion of the spectral index in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For comparison, we also include the histogram of spectral index values from the PySM synchrotron model 1 (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that in the intensity case for 𝛽11GHz−23GHz (blue line), the 11 GHz map contains free-free and AME in addition to syn- chrotron, and thus the histogram presents a different shape with a broader distribution (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' other authors (Dunkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A more detailed study of the spectral properties of the sky emission in polarization using the MFI wide survey maps in combination with WMAP and Planck, including a discussion on synchrotron spectral curvature, is carried out in an accompanying paper (de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 E- and B-mode maps As a complementary view of the relative power distribution in the E- and B-mode components for the synchrotron emission traced by the QUIJOTE MFI wide survey map, we have obtained in this section E- and B-mode maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the full QUIJOTE observed area, but we mask the satellite band (satband) as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to minimize the impact of E/B mixing (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2001), we apodize this analysis mask using a Gaussian kernel of 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' E- and B-mode maps are then generated using the standard HEALPix routines anafast and synfast, as 𝐸( ˆ𝑛) = ∞ ∑︁ ℓ=2 ℓ ∑︁ 𝑚=−ℓ 𝑎E ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛) 𝐵( ˆ𝑛) = ∞ ∑︁ ℓ=2 ℓ ∑︁ 𝑚=−ℓ 𝑎B ℓ,𝑚𝑌ℓ,𝑚( ˆ𝑛), (27) where 𝑎E ℓ,𝑚 and 𝑎B ℓ,𝑚 are the corresponding harmonic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 36 shows the derived maps for MFI 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As ex- pected from the power spectrum analysis in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7, there is sig- nificantly more power in the E-mode than in the B-mode map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Moreover, most of the brightest synchrotron features in the polar- ized intensity map (North Polar Spur, Fan region, Galactic centre) appear mostly in the E-mode, as expected due to the underlying magnetic field structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Strongly polarized radio sources (Tau A, Cyg A) appear in these E- and B-mode maps with the characteristic quadrupole patterns with two positive and two negative lobes, and with the B-mode profile rotated by 45◦ with respect to the E-mode map (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Diego-Palazuelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7Error in the spectral index in polarization (MFll1 to WMAP-K) 0 (β11GHz - 23GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Intensity（408MHz-11GHz） Intensity (11GHz-23GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 Polarization （11GHz-23GHz Prior β=-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 (normalized) PySM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 ixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 4 3 2 Spectral index β42 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' E and B-mode maps at 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Most of the brightest features in the QUIJOTE map (North Polar Spur, Fan region, Galactic plane) appear in the E-mode map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Bright structures in the polarized intensity maps The MFI wide survey polarized intensity maps are dominated by several bright and extended structures (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We discuss some of them in four accompanying papers: the Fan region (Ruiz- Granados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023), the Haze and Galactic center (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023), the North Polar Spur (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023), and other syn- chrotron loops and spurs (Peel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 AME in the MFI wide survey maps The MFI wide survey maps can be used to characterize the spec- tral properties of the AME, both in intensity and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, in Fernandez-Torreiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023) we present a study of the diffuse AME emission in intensity along the Galactic plane (|𝑏| ≤ 10◦), while Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023) characterizes the SED in intensity for 52 compact sources with AME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, two additional papers update the constraints in intensity and polarization of the AME in several Galactic regions (Tramonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Lopez- Caraballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9 BRIGHT COMPACT SOURCES AND PLANETS IN THE WIDE SURVEY Despite its coarse angular resolution a high number of point sources are detected to high significance in the QUIJOTE MFI wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In a companion paper, where we discuss radio source de- tectability in these maps and derived statistical properties (Herranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023), we show that we detect 235 point sources at S/N> 3 at 11 GHz, while 85 are detected at S/N> 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a further consis- tency check of the global amplitude calibration, in this section we compare with models the recovered flux densities on four of the brightest sources having well characterised spectra (Tau A, Cas A, Cyg A and 3C274), and in two planets (Jupiter and Venus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also calculate polarization flux densities in three bright polarized sources (Tau A, Cyg A and W63) to assess the accuracy of the polarization calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Compact sources in intensity Tau A (also known as the Crab nebula), Cas A and Cyg A are amongst the brightest compact sources in the microwave range, and hence they have traditionally been used to calibrate experiments op- erating in this frequency range, including CMB experiments (Baars et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using WMAP data, Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2011) presented updated spectrum models in the range ∼ 1–300 GHz of these three sources and of 3C274 (also known as Virgo A or M87) and 3C58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Here we will focus on Tau A, Cas A, Cyg A and 3C274, while 3C58 will be discussed in detail in Ruiz-Granados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 37 shows the MFI wide survey maps on the positions of these four sources (Tau A, Cas A, Cyg A and 3C274) at 11 GHz and 19 GHz, smoothed to a common angular resolution of 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that Tau A and Cas A are the two main calibrators of QUIJOTE MFI, and thus we have much more sensitive data on these two sources obtained in raster mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However we focus here on the wide survey maps only, in order to provide another consistency check for the calibration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We extracted total-intensity flux densities on these maps us- ing a beam-fitting photometry (BF1d), consisting in fitting a 1◦- FWHM Gaussian beam superimposed on a flat background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We applied colour corrections following the methodology described in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), and using for each source a spectral index derived from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We compare these flux densities with spectral emission models that we have specifically derived for these sources, and which will be presented in a separate paper (Génova- Santos & Rubiño-Martín, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' While in that paper we discuss models extracted with different photometry techniques, here we compare with models derived from WMAP and Planck maps convolved to a common resolution of 1◦ and using the same BF1d technique that we applied to QUIJOTE MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, the Tau A model was used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 to recalibrate the wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As it will be discussed in depth in Génova-Santos & Rubiño-Martín (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ), the uncertainties of these models are of the order of 3–5 %, and are driven not by the statistical noise of the individual obser- vations which is well below this value, but by systematic effects and calibration uncertainties of the fitted data, which lead to higher model-fitting residuals than would be expected in the presence of just statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the cases of Tau A and Cas A, modelling of their secular decrease also introduces significant uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Final QUIJOTE MFI flux densities, for each horn and fre- quency, and relative deviation with respect to the fitted intensity models, are quoted in Table 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All values are referred to date 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 (1 April, 2016), which roughly corresponds to the middle of the wide survey observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It can be seen that in most cases the measured flux densities deviate less than 3–5 % with respect to the models, while in the case of Tau A, which is the main am- plitude calibrator, the deviations are within 1 % (the difference is not exactly zero due to the way the different periods are calibrated and combined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The level of these deviations is expected given the typical model uncertainties, and therefore these results give full confidence to our global calibration strategy and the MNRAS 000, 1–58 (2022) MFI 11GHz - E modes mKMFI 11GHz - B modes mKQUIJOTE MFI wide survey 43 Figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Minimaps of 5◦ × 5◦ size around four bright radiosources: Tau A (first row), Cas-A (second row), Cygnus-A (third row), and 3C274 (bottom row) at 11 GHz (first three columns are I, Q, and U), and 19 GHz (columns 4 to 6 are I, Q, U, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, we use the MFI maps degraded to a common angular resolution of one degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' quoted uncertainty (see Table 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A detailed discussion on the vari- ability of these four sources (and others in the wide survey maps) can be found in Herranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Planets in intensity Venus and Jupiter are also detected to high significance in the QUI- JOTE MFI wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Owing to its orbital motion Venus declination varies roughly between ±27◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Given Tenerife’s latitude (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3◦ N), when its declination is close to 27◦ it is always visible in any of the elevations considered in the wide-survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On the contrary, when it reaches its minimum declination of −27◦ it culminates at elevation 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦, and therefore it is only picked up in observations at elevations 30 or 35◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The distance between this planet and the Earth changes between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='27 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='74 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', meaning that there is a factor ≈ 42 variation between its minimum and maximum brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At 19 GHz its flux density is expected to vary between 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 and 445 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, during its inferior conjunction it is amongst the brightest sources on the sky at the QUIJOTE MFI frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Jupiter, being an external planet, this variation is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Its distance to Earth varies between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', producing a variation of its flux density at 19 GHz between 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Between 2012 and 2016 its declination was always positive, reach- ing 23◦, meaning that it was picked up in most of the wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Between 2016 and 2018 its declination dropped below zero, reaching −22◦, and therefore during this period it was only visible on the wide survey observations performed at low elevations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' While further details will be given in a future paper where we will discuss planets and other bright astronomical sources, here we briefly describe the procedure we have developed to estimate planets’ brightness temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We implemented a specific map- making in which we rotate the coordinates of QUIJOTE MFI wide survey data to planet-centred coordinates, to produce planet-centred maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use the same final calibrated data that were used to pro- MNRAS 000, 1–58 (2022) 50100150200250300 20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mKcMB mK CMB 4 (deg) 5 6 b 7 8 Tau A I 11 GHz Tau A Q 11 GHz Tau A U 11 GHz 187186185184183 187186185184183 187186185184183 1 (deg) 1 (deg) 1 (deg)0 20 40 60 80 100 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 mKcMB 4 (deg) 5 6 b 7 8 Tau A I 19 GHz Tau A Q 19 GHz Tau A U 19 GHz 187186185184183 187186185184183 187186 185 184183 1 (deg) 1 (deg) 1 (deg)50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 mK CMB mK CMB 0 1 (deg) 2- b 3 4 Cas A I 11 GHz Cas A Q 11 GHz Cas A U 11 GHz 114 113 112 111110 114 113 112 111 110 114 113 112 111 110 1 (deg) 1 (deg) 1 (deg)10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='15 mK, CMB mK, CMB 0 1 (deg 2- b 3 4 Cas A I 19 GHz Cas A Q 19 GHz Cas A U 19 GHz 114 113 112 111 110 114113112111110 114 113 112111110 1 (deg) 1 (deg) 1 (deg)20 40 60 80 100 0 1 2 3 4 3 2 1 0 mKcMB 8 7 00 (deg 6 b 5 4 Cyg A I 11 GHz Cyg A Q 11 GHz Cyg A U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11 GHz 78 76 75 74 78 76 75 74 78 76 75 74 1 (deg) 1 (deg) 1 (deg)0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8 7 80 (deg 6 b 5 4 Cyg A I 19 GHz Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='. Q 19 GHz Cyg A U 19 GHz 78 76 75 74 78 76 75 74 78 76 75 74 1 (deg) 1 (deg) 1 (deg)0 5 10 15 20 25-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 mKcMB 76 80 75 (deg b 74 73 3C274 I 11 GHz 3C274 11 GH: C274 U 11 GHz 72 286 285 284 283 282 286 285 284 283 282 286 285 284 283 282 1 (deg) 1 (deg) 1 (deg)0 1 2 3 4 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 mK CMB mK, CMB 76 a0 75 (deg b 74 73 3C274 I 19 GHz 3C274 Q 19 GHz 3C274 U 19 GHz 72 286285284283282 286285284283282 286 285 284 283 282 1 (deg) 1 (deg) 1 (deg)44 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Flux densities (Jy), in intensity and in polarization, extracted from the QUIJOTE MFI wide survey maps at one degree resolution on Tau A, Cas A, Cyg A and 3C274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Intensity measurements are based on BF1d photometry, while the polarization measurements used AP1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the intensity measurements, inside parentheses we quote the percent deviation of flux densities with respect to predictions from spectral models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Tau A and Cas A values are referred to an effective date corresponding to 1 April 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All flux densities include colour corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Source Stokes 311 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 GHz) 313 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 GHz) 217 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 GHz) 417 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 GHz) 219 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 GHz) 419 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 GHz) Tau A I 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8) 427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7) 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5) 393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) 378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2) Q −29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51 −31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='43 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='36 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='52 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='66 U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 Cas A I 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1) 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5) 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4) 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0) 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8) 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0) Q −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='32 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='72 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8) Q 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='93 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='48 U −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='56 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='45 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='47 duce the final maps that are presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to ac- count for the 1/𝑑2 effect we define distance bins (3 bins for Jupiter and 6 for Venus), and produce individual maps for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We have verified in the final map that the (symmetrized) beam shape is well preserved, this being a health check both for the tailored map-making that we use here as well as for the pointing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' On these maps we apply a beam-fitting photometry to derive flux den- sities for each distance bin and for each horn/frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These flux densities are then colour-corrected using a Rayleigh-Jeans spectrum (spectral index 𝛼 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition to data maps for each redshift bin we produce maps of 1/𝑑2 using the same noise weights and flags that are applied to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These maps are later used to calculate an effective distance at the position of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using this infor- mation we fit the flux densities measured in each bin to a 1/𝑑2 law in order to derive the final brightness temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Our Venus and Jupiter brightness temperatures derived from QUIJOTE MFI are listed in Table 25 and plotted in Figure 38, in comparison with other data at similar frequencies, as well as with various models giving the spectral dependency of the brightness temperatures of these planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In both cases we have corrected for the planet absorption of the CMB monopole, and therefore the quoted values represent the intrinsic brightness temperature of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Venus, it is seen that the ancillary measurements seem a bit high with respect to the Bellotti (2015) and Fahd (1992) models and therefore we performed a power-law fit to the data in the range 7–100 GHz (dashed line in the figure) and use this fit as a reference to compare with the QUIJOTE MFI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Jupiter we use as reference the model of Karim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2018), which seems to trace better the ancillary data, and in particular the VLA data from de Pater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2019) below the ammonia absorption at 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As can be seen in Table 25, both for Venus and Jupiter the QUIJOTE MFI measurements deviate always less than 5 % from the models (note that in some cases the statistical error bar is larger than this value), which bestows confidence to our calibration strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 Polarized sources Figure 37 also shows wide-survey polarization maps of Tau A, Cas A, Cyg A and 3C274, projected in Galactic coordinates and convolved to an angular resolution of one degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Clear polarized emission is seen in Tau A, mainly concentrated in the 𝑄 map, as expected due to its polarization angle (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The 𝑈 map shows the typical cloverleaf pattern (with the expected peak-to-peak amplitude of ∼ 1 % with respect to the total intensity) arising from the differences between the two co-polar beams (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This pattern is also visible in the 𝑄 and 𝑈 maps of Cas A, more notably at 11 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to it being a very young shell-type supernova remnant (SNR), the magnetic field of Cas A is expected to be radial (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Being ∼ 5′ across, this source is unresolved by the QUIJOTE MFI beam and therefore we expect zero integrated polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Clear polarized emission is also seen in Cyg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A rotation of the polarization angle is apparent between 11 and 19 GHz, which is due to the two jets of this radio galaxy having different rotation measures, the so-called Laing-Garrington effect (Laing 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of 3C274, we only have a marginal polarization detection in the 𝑈 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This is expected given our noise levels (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5–1 Jy), and the fact that the measured polarization fraction at 23 GHz is approximately 4 % (Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to minimize systematic effects introduced by differ- ences between the two co-polar beams, we extract flux densities in polarization through an aperture photometry technique on maps smoothed to one-degree angular resolution (AP1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The circular aperture radius (𝑟1) is taken to be 𝑟1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5◦ for Tau A and 3C274, and 𝑟1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3◦ for Cas A and Cyg A, due to the larger foreground contamination in the surroundings of the latter two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Cas A, we also mask the region centred at Galactic co- ordinates (𝑙, 𝑏) = (111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='11◦, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='53◦), using an exclusion radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The background emission in all four cases is corrected using the mean of the signal in the annulus between 𝑟1 and 𝑟2 = √ 2𝑟1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 24 shows the Stokes 𝑄 and 𝑈 flux densities measured on Tau A, Cas A, Cyg A and 3C274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We now discuss the first three cases in detail, as well as the bright polarized emission in W63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this discussion, we also ap- ply the same methodology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AP1d for polarization and BF1d for intensity) to derive the photometry values for these sources using WMAP 9-year data (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2013) and Planck 2018 maps (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020a) at the common one-degree res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Specially for the cases of Tau A and Cas A, and for Planck LFI, we correct for the intensity-to-polarization leakage due to band- MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 45 Table 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Brightness temperatures (in Kelvin) of Jupiter and Venus extracted from the QUIJOTE MFI wide survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Inside parentheses we quote the percent deviation with respect to predictions from spectral models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planet 311 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 GHz) 313 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 GHz) 217 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 GHz) 417 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 GHz) 219 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 GHz) 419 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 GHz) Jupiter 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8) 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0) 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9) 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) Venus 578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) 568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5) 546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4) 533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4) 518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6) Figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Venus (top) and Jupiter (bottom) brightness temperatures derived from the QUIJOTE MFI wide survey data (red and yellow) in comparison with ancillary data, and with various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Venus data have been obtained from Bellotti (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hafez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Dahal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2021), while the plotted models are from Bellotti (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fahd (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also show a power-law fit to the observed data in the range 7–100 GHz that we use to compare with the QUIJOTE MFI measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Jupiter data come from Hafez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' de Pater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Karim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2016e, 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We plot the model by Karim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2018) and the ESA1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' pass mismatch following the methodology described in Appendix C of Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2016h), using the maps of projection factors described in Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2016c), and the spectral index of each source derived from the intensity SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 Tau A and Cyg A Figure 39 shows our results for the polarization fractions in Tau A and Cyg A at one degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We include also our WMAP (for both sources) and Planck (only for Tau A) measurements, as well as ancillary measurements both for Tau A (Kuz’min & Udal’Tsov 1959;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mayer & Sloanaker 1959;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Davies & Ver- schuur 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hollinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Morris & Berge 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Boland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Gardner & Whiteoak 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hobbs & Haddock 1967a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Sastry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Satoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hobbs 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hollinger & Hobbs 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mayer & Hollinger 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Seielstad & Weiler 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Johnston & Hobbs 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Dmitrenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Wright 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hafez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Aumont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2010) and for Cyg A (Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hollinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Sobol- eva 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Boland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mezger & Schraml 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hobbs & Haddock 1967b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For Tau A, we show in solid grey lines Monte Carlo realiza- tions of a simple model for the spectral dependency of its polar- ization fraction that accounts for the Faraday depolarization (Burn 1966), and that is consistent with both the ancillary measurements at low frequencies (𝜈 ≲ 10 GHz) and existing measurements of the Faraday dispersion in the region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bietenholz & Kronberg 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This model will be described in detail in a separate paper (Génova-Santos & Rubiño-Martín, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that in our recalibration strategy of the MFI wide survey maps, we use Tau A for fixing the intensity calibration scale and the polarization angle, while the polarization amplitude is essentially given by inde- pendent polarization efficiency measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, this analysis provides a consistency test on the MFI polarization calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cyg A data points clearly show the effect of the Faraday de- polarization produced by the Laing-Garrington effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It is evident from this plot that the maximum alignment between the polarization directions of the two jets occurs at ≈ 10 GHz, and then the measured polarization fractions decrease in both sides of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Mod- elling this effect is complicated and beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The QUIJOTE MFI measurements are in good agreement with the other measurements, again providing confidence in our calibration strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Cas A Figure 40 shows the polarization fraction measured in Cas A with QUIJOTE MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also include our photometry results for WMAP and Planck, and ancillary data from the literature (Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hollinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Hobbs & Haddock 1967a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Sastry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Seielstad & Weiler 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Vinyaikin 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All values are noise- debiased using the PMAS estimator (Plaszczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The intensity-to-polarization leakage due to the co-polar beam asym- metry is almost cancelled in the integrated flux densities thanks to the positive and negative structure of this pattern, leading to inte- grated polarization fractions in QUIJOTE of around ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Note that similar levels are detected in WMAP, and could also be due to beam effects as discussed in Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At face value, these numbers can be considered as a conservative upper limit on the overall intensity-to-polarization leakage in the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 46 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Consistency checks on polarized sources detected on the QUI- JOTE MFI the wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show polarization fractions measured in Tau A and in Cyg A, in comparison with our WMAP and Planck results obtained using the same methodology, and with ancillary measurements (see the complete list of references in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the case of Tau A we overplot in grey models for the polarization fraction that account for Faraday rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The Cyg A data show the Laing-Garrington effect arising from different rotation measures in the two lobes of this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 W63 region As an additional test of the polarization calibration of the MFI, we also investigate the polarized intensities of W63, another SNR which appears as a very bright extended structure in the polarization maps at these frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 41 shows the MFI 11 GHz Stokes I, Q and U maps for this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The total-intensity emission of W63 is practically embedded inside the emission of the Cygnus X star-forming complex, so it is difficult to extract re- liable total-intensity flux density estimates in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' However, the polarization signal is reasonably isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Thus, we only discuss its polarized flux density here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In order to capture all the flux in the region, we use an aperture radius of 𝑟1 = 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As in the pre- vious cases, we carry out this analysis in the smoothed maps at one degree resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AP1d photometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The bottom panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 41 shows the SED in polarized intensity 𝑃 = √︁ 𝑄2 + 𝑈2 derived from our photometry measurements, including also our results for WMAP and Planck applying the same methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All values are noise-debiased using the PMAS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Error bars account for the photometry error plus the corresponding calibration uncertain- ties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For MFI, we use the values reported in Figure 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Polarization fractions measured on Cas A in QUIJOTE MFI wide survey data, in comparison with other measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' WMAP and Planck results are obtained using the same methodology as for the MFI maps values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The complete list of ancillary measurements is given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Upper limits are represented with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At degree-beam scales, the polarized emission of Cas A is expected to be zero, so these measurements serve as a consistency check for the overall intensity-to-polarization leakage of the MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table 16, while for WMAP and Planck data we adopt the conser- vative value of 3 %, as done for similar analyses (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Cepeda-Arroita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The WMAP and Planck polarized intensity flux in W63 can be fitted to a power-law 𝑃 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='97(𝜈/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8GHz)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='68 Jy, that is depicted by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The QUIJOTE MFI data are con- sistent within 1-sigma with the fitted model, which gives additional confidence to our calibration strategy in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10 DATA RELEASE AND DESCRIPTION OF THE DATA PRODUCTS Together with this paper, there is a series of further publications containing scientific results derived from the QUIJOTE-MFI wide survey maps presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The titles of all the papers in the series begin with "QUIJOTE scientific results", and comprise: IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A northern sky survey at 10–20 GHz with the Multi- Frequency Instrument (this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The microwave intensity and polarization spectra of the Galactic regions W49, W51 and IC443 (Tramonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The Haze as seen by QUIJOTE (Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Galactic AME sources in the QUIJOTE-MFI North Hemi- sphere Wide-Survey (Poidevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Diffuse polarized foregrounds from component separation with QUIJOTE-MFI (de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Radio sources in the QUIJOTE-MFI wide survey maps (Her- ranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AME variability along the Galactic Plane in the QUIJOTE- MFI wide survey (Fernandez-Torreiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Polarized synchrotron loops and spurs in the QUIJOTE-MFI wide survey (Peel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Analysis of the polarized synchrotron emission at the power spectrum level in the MFI wide survey (Vansyngel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 47 Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Minimaps of 6◦ × 6◦ size around W63 for the MFI 11 GHz I, Q, and U maps at one degree angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A circle with radius of 2◦ indicates the integration area for the photometry analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: Polarized intensity measurements on W63 with the QUIJOTE MFI wide survey data, in comparison with WMAP and Planck measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We overplot with a dashed line a power-law fit representing the spectrum of the synchrotron emission fitted to the WMAP and Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' SNRs in the QUIJOTE-MFI wide survey (Lopez-Caraballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The FAN region as seen by QUIJOTE-MFI (Ruiz- Granados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The North Polar Spur as seen by QUIJOTE-MFI (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' XVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Diffuse intensity foregrounds from component separation with QUIJOTE-MFI (de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023b, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, we have a dedicated paper describing the MFI data processing pipeline (Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The distribution of released data products associated with the QUIJOTE-MFI wide survey papers contain the following items: Four frequency maps (11, 13, 17, 19 GHz) in intensity and polarization, both at native and one degree resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps at 11 and 13 GHz correspond to those produced from MFI horn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps at 17 and 19 GHz correspond to the weighted average of horns 2 and 4, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The associated weight and hit maps for each frequency map at native resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' One set of null tests maps (half1/2 for independent baselines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Instrument Model (IMO), containing central frequencies, beams properties, beam profiles and window functions for each MFI horn, bandpasses and colour corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The default analysis mask (sat+NCP+lowdec), as well as the satellite mask (sat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The later is applied to all the released maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11 CONCLUSIONS This paper presents and characterizes the properties of the QUI- JOTE wide survey maps of the northern sky carried out with the MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They result from approximately 9 000 h of obser- vations spread over six years between 2013 and 2018, and include four frequency maps at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 GHz, with angu- lar resolutions between 55 and 39 arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The maps cover around 29 000 deg2 with sensitivities in linear polarization (Stokes Q and U parameters) within 35–40 𝜇K per 1-degree beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although the MFI instrument is not optimized for intensity measurements, we also present the corresponding intensity maps at those four fre- quencies, with sensitivities in the range 65–200 𝜇K per 1-degree beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Together with the description of the specific aspects of the MFI pipeline related to the production of the wide survey maps, we have presented a detailed validation of the maps, a characterization of residual systematic effects (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 4), and an extensive study of their calibration accuracy (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 5 and Table 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The overall calibration uncertainty of the polarization maps is 5 % for the two lowest fre- quency channels, and 6 % for the highest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These final maps and other derived data products are part of a public data release associated with this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Although a full description of the science results obtained from these maps are given in the accompanying papers listed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10, this paper presents some global properties of the Galactic foregrounds at these frequencies, and in particular, the polarized synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average synchrotron spectral index in polarization between 11 GHz and the WMAP 23 GHz is found to be 𝛽 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='16, showing a much broader distribution (by a factor ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7) than the one adopted in current synchrotron sky models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Miville-Deschênes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Most of the large- scale polarized synchrotron features in the MFI maps appear in the E-mode map, which shows significantly more power than the B-mode at these frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Based on the analysis of the angular power spectra of the measured polarized signal, we find that the BB/EE ratio at multipole scales of ℓ = 80 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='07 for a Galactic cut |𝑏| > 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This value is consistent with that found for WMAP/Planck low frequency maps (Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022), but it is significantly different from the values obtained for the S-PASS polarized signal at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 GHz (Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2018), suggesting that probably there is some contribution of Faraday rotation and/or depolarization at lower frequencies than those probed by QUIJOTE MFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also find a positive correlation in the TE spectrum for 11 GHz at large angular scales (ℓ ≲ 80), while the EB and TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150, as expected for the synchrotron emission, as its polarization orientation is dictated by the Galactic magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The MFI instrument was decommissioned in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' At this moment, QT2 is operating with a combination of the TGI and FGI instruments in a single cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition to QUIJOTE, there are two other CMB polarization experiments at the Teide Observa- tory and providing a similar sky coverage: GroundBird and LSPE- STRIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' GroundBird (Honda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020) is a MKIDs array with two bands centered at 145 and 220 GHz, installed back in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' STRIP (Addamo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021) is part of LSPE, a combined programme of ground-based and balloon-borne polarization observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' STRIP will operate in the 42 and 90 GHz bands, and will be installed at the Teide Observatory in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The QUIJOTE collaboration is devel- oping a new instrument at these frequencies, called MFI2, with an expected sensitivity three times better than the former MFI (Hoyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The new MFI2 is now in the final integration phase, and MNRAS 000, 1–58 (2022) 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 0 1 2 3 mK, CMB mK CMB 8 7 00 6 (deg 5 b 4 3 V63 I 11 GHz W63Q Q11GHz W63 U 11 GHz 85 84 83 82 81 80 85 84 83 82 81 80 85 84 83 82 81 80 1 (deg) 1 (deg) 1 (deg)48 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' it is using a digital back-end based on Field Programmable Gate Ar- rays (FPGAs), that will allow us to identify and filter the RFI signals from geostationary satellites directly in the data processing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A new wide survey at these frequencies (10–20 GHz) will be carried out with MFI2 at the first QUIJOTE telescope (QT1) starting 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' DATA AVAILABILITY All data products described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10 can be freely downloaded from the QUIJOTE web page11, as well as from the RADIOFORE- GROUNDS platform12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They include also an Explanatory Supple- ment describing the data formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps will be submitted also to the Planck Legacy Archive (PLA) interface and the LAMBDA site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Any other derived data products described in this paper (null test maps, simulations, etc) are available upon request to the QUIJOTE collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the staff of the Teide Observatory for invaluable assistance in the commissioning and operation of QUIJOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The QUIJOTE experiment is being developed by the Instituto de Astrofisica de Canarias (IAC), the Instituto de Fisica de Cantabria (IFCA), and the Universities of Cantabria, Manch- ester and Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Partial financial support was provided by the Spanish Ministry of Science and Innovation under the projects AYA2007-68058-C03-01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2007-68058-C03-02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2010-21766-C03-01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2010-21766-C03-02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2014- 60438-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ESP2015-70646-C2-1-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2017-84185-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' ESP2017- 83921-C2-1-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' AYA2017-90675-REDC (co-funded with EU FEDER funds),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' PGC2018-101814-B-I00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' PID2019-110610RB- C21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' PID2020-120514GB-I00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' IACA13-3E-2336,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' IACA15-BE- 3707,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' EQC2018-004918-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the Severo Ochoa Programs SEV- 2015-0548 and CEX2019-000920-S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the Maria de Maeztu Pro- gram MDM-2017-0765,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and by the Consolider-Ingenio project CSD2010-00064 (EPI: Exploring the Physics of Inflation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We acknowledge support from the ACIISI, Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference ProID2020010108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This project has received funding from the Eu- ropean Union’s Horizon 2020 research and innovation program un- der grant agreement number 687312 (RADIOFOREGROUNDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This research made use of computing time available on the high-performance computing systems at the IAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We thankfully acknowledge the technical expertise and assistance provided by the Spanish Supercomputing Network (Red Española de Supercom- putación), as well as the computer resources used: the Deimos/Diva Supercomputer, located at the IAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The PWV data used in the tests presented in Section 4 comes from the Izaña Atmospheric Observatory (IZO), and have been made available to us by the Izaña Atmospheric Research Center (AEMET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' SEH and CD acknowl- edge support from the STFC Consolidated Grant (ST/P000649/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' FP acknowledges support from the Spanish State Research Agency 11 http://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='es/proyecto/quijote 12 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='radioforegrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='eu/ (AEI) under grant number PID2019-105552RB-C43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' DT acknowl- edges the support from the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) with Grant N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2020PM0042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Some of the presented results are based on observa- tions obtained with Planck (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We acknowl- edge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Support for LAMBDA is provided by the NASA Office of Space Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Some of the results in this pa- per have been derived using the HEALPix (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2005) and healpy (Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019) packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We also use Numpy (Harris et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' D, 55, 1830 MNRAS 000, 1–58 (2022) 50 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Zonca A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Singer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Lenz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Reinecke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Rosset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Hivon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', Gorski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=', 2019, The Journal of Open Source Software, 4, 1298 APPENDIX A: DATA FLAGGING IN THE MFI WIDE SURVEY Tables A1, A2, A3 and A4 show the percentage of data used (and flagged) for each period, elevation and horn in the MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' APPENDIX B: IMPACT OF FDEC FILTERING ON POLARIZATION MAPS In this appendix, we investigate the impact of the FDEC filtering on some of the scientific analyses carried out in this paper and in other papers of the associated release (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In particular, we consider here a photometry method (aperture photometry) and correlation method (the so called TT plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this study, we use the sky signal simulations presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure B1 shows the simulated (noiseless) sky maps in polarization at 11 GHz used as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We apply the FDEC filter to these maps, and show in the same figure the resulting filtered maps, as well as the residual maps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' difference between the original and the filtered map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5, the FDEC filtering effectively corresponds to a high-pass filter, which removes the zero mode for any line of constant declination on a map in local (equatorial) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The image illustrates again that the effective transfer function of the FDEC filter leaves unaltered all scales with ℓ >∼ 30, because the residual maps only contain large scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B1 Impact of FDEC on photometry methods: aperture photometry From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B1, one would expect that all analyses in real space involving "local" analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the photometry extraction of a com- pact source with a local determination of the background), should be unaffected by the FDEC filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To test this hypothesis, we take as a reference one of the photometry methods used in this pa- per: the aperture photometry method (AP1d) described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, we apply AP1d to all possible pixels in the simulated maps within the MFI wide survey sky mask, both to the original and to the filtered maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analysis, we use a reference aperture of 𝑟1 = 1◦, and the background is estimated in the annulus between 𝑟1 and 𝑟2 = √ 2𝑟1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We find that the maximum difference between the photometry on both Stokes Q and U parameters obtained in the original map and the filtered one is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 Jy, while the standard de- viation of the difference of the two photometry methods is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='007 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Both values are significantly smaller than the typical error in the photometry (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' the results presented in Table 24, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 9), thus confirming that we can safely neglect any impact on the pho- tometry due to the FDEC filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For completeness, we repeat the analysis for a larger aperture of 𝑟1 = 2◦, and find that in this case the maximum difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 Jy, with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='037 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B2 Impact of FDEC on correlation analyses: recovery of the spectral index We now evaluate the impact of the FDEC filtering on the recovery of spectral index of the sky emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To this end, we use the same simulation set described above, taking as a reference the simulated maps at 11 and 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We now apply two different methods to reconstruct the spectral index 𝛽 of the sky emission between 11 and 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' First, we carry out a direct evaluation of the spectral index at the pixel level (𝑁side = 512) in the original (unfiltered) maps, and also in the filtered ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This methodology is similar to the one used in Sect 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' It is important to emphasize that both maps (the simulated MFI 11 GHz and the simulated WMAP 23 GHz) have to be filtered with the FDEC, in order to have consistent scales between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The reconstructed spectral index is fully consistent with the input one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' If we restrict the comparison to pixels with high emission (polarized intensity at 11 GHz greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 mK), and we compare the reconstructed maps after degrading to 2◦ (to be consistent with Sect 8), we find that the median difference Δ𝛽 between the reconstructed and original spectral index is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0005, while the standard deviation of the difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Second, we use a correlation analysis method (also called TT plot) to recover the spectral index of the emission between 11 and 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this analysis, we degrade the simulated maps at 1 degree resolution to 𝑁side = 64, in order to have approximately independent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Then, we divide the observed sky in patches of ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3◦, using as a reference the pixels of a 𝑁side = 8 HEALPix map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Within each patch, we carry out a TT plot analysis assuming a typical error in each map corresponding to 3 per cent of the sky signal, and accounting for errors in both axes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B3 shows the obtained results from the original maps (top panel), and the FDEC filtered maps (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As expected, the spatial distribution of the reconstructed index has a good correspondence with the maps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' A numerical comparison of both maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' B3 gives that median difference Δ𝛽 between the two maps is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0007, and the standard deviation of the difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Summarising, we can obtain an unbiased reconstruction of the spectral index of the sky signal, provided that both maps are filtered in the same way using FDEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In practice, this means that when doing these type of analyses using QUIJOTE MFI wide survey maps and external ancillary data, we must filter first the external maps using the same procedure as for the MFI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' If the FDEC filtering is not applied to the external ancillary data, we find that for these simulations the standard deviation of the reconstructed spectral index can be as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This issue is further discussed in other papers in the series (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Appendix C in de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2023a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' APPENDIX C: QUIJOTE MFI WIDE SURVEY MAPS PER HORN AT ORIGINAL RESOLUTION Figures C1, C2 and C3 show the final MFI wide survey maps at their original resolution (quoted as beam FWHM in Table 3), obtained for horns 2, 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The intensity maps of horns 2 and 4 show some large angular-scale residual patterns, particularly visible in the highest frequency map (19 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' These are due to a combination of residual instrumental and atmospheric 1/ 𝑓 noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figures C4, C5 and C6 show the corresponding weight maps at the original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figures C7, C8 and C9 show the maps with the number of individual TOD samples in each pixel (the so called "hit maps", 𝑁hit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They correspond to the total number of 40 ms samples in each HEALPix pixel of 𝑁side = 512 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The ring structures correspond to lines of constant declination, and indicate the edges of the declination limits of observations performed at different elevations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Due to projection effects, the number of hits MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 51 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fraction of data used in period 1 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Column 1 indicates the elevation, columns 2 and 3 show the horn and frequency (0 for low and 1 for high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Columns 4 and 5 show the percentage of used data in correlated and uncorrelated channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Columns 6 and 7 show the percentage of flagged data during the post-processing stage, and columns 8 and 9 show the percentage of flagged data due to Sun, Moon and planets (Mars, Venus, Jupiter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Last column indicates the range of dates when each elevation was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Elevation Horn Freq Used c Used u Flag1 c Flag1 u Flag2 c Flag2 u Range of Dates (deg) (%) (%) (%) (%) (%) (%) 60 2 0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': 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65 3 1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 5/2013–3/2014 65 4 0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 5/2013–3/2014 65 4 1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 5/2013–3/2014 Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fraction of data used in period 2 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same format as in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Elevation Horn Freq Used c Used u Flag1 c Flag1 u Flag2 c Flag2 u Range of Dates (deg) (%) (%) (%) (%) (%) (%) 30 2 0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8/2014–3/2015 30 2 1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8/2014–3/2015 30 3 0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 8/2014–3/2015 30 3 1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 8/2014–3/2015 30 4 0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 8/2014–3/2015 30 4 1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 8/2014–3/2015 40 2 0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 8/2014–1/2015 40 2 1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 8/2014–1/2015 40 3 0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 8/2014–1/2015 40 3 1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 8/2014–1/2015 40 4 0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8/2014–10/2015 50 2 1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8/2014–10/2015 50 3 0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 7.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 8/2014–10/2014 is significantly larger in those boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In the low declination band of the maps, particularly for negative declinations, the number of hits is significantly lower due to the combined effect of smaller number of observations at low elevations (mainly 30◦, 35◦ and 40◦) and projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We recall that the number of hits in intensity is larger than in polarization, due to the fact that some intensity data are not used in polarization, as shown in Table 1 (period 1 is not used for any polarization maps, data from period 2 are not used in polarization for horn 4, and data from period 5 are not used in polarization for horn 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C10 shows the 𝑟cond maps in polarization, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C11 shows the normalized covariance 𝑐𝑜𝑣(𝑄,𝑈), both at original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) 52 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fraction of data used in period 5 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same format as in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Elevation Horn Freq Used c Used u Flag1 c Flag1 u Flag2 c Flag2 u Range of Dates (deg) (%) (%) (%) (%) (%) (%) 40 2 0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 33.' metadata={'source': 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+page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 8/2016–9/2016 60 4 0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 8/2016–9/2016 60 4 1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 8/2016–9/2016 Table A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fraction of data used in period 6 after applying the flags for the wide survey observations with the QUIJOTE MFI instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same format as in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Elevation Horn Freq Used c Used u Flag1 c Flag1 u Flag2 c Flag2 u Range of Dates (deg) (%) (%) (%) (%) (%) (%) 35 2 0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 12/2017–6/2018 35 2 1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 45.' 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70 4 0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2/2017–4/2017 70 4 1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 2/2017–4/2017 MNRAS 000, 1–58 (2022) QUIJOTE MFI wide survey 53 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Example of application of FDEC filtering in simulations of the polarized MFI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top (bottom) row corresponds to Stokes Q (U) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Left column shows the simulated MFI 11 GHz map at 1 deg resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' middle column corresponds to the same map, after applying the FDEC filtering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and last column shows the difference of the previous two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All maps use the same colour scale, saturated at ±1 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Impact of the application of FDEC filtering in the reconstruction of the spectral index in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We use simulations of the polarized sky signal in MFI 11 GHz and WMAP 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top panel shows the (true) underlying spectral index of the simulated signal between 11 and 23 GHz, within the MFI observing mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom panel shows the reconstructed spectral index after applying the FDEC filtering to both simulated maps (11 and 23 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Impact of the application of FDEC filtering in the reconstruction of the spectral index using correlation analysis (TTplot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We carry out the correlation analysis in regions defined by HEALPix pixels of 𝑁side = 8, and extract the spectral index of the polarized sky signal between MFI 11 GHz and WMAP 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top panel shows the (true) underlying spectral index of the simulated signal within the MFI observing mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom panel shows the reconstructed spectral index after applying the FDEC filtering to both simulated maps (11 and 23 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) Simulated MFl 11GHz Q (1deg) - original mkSimulated MFI 11GHz Q (1deg) - no FDEC mkSimulated MFl 11GHz Q (1deg) - FDEC mk 一Simulated MFl 11GHz U (ldeg) - original mkSimulated MFI 11GHz U (ldeg) - no FDEC mkSimulated MFl 11GHz U (1deg) - FDEC mkSimulated spectral index in polarization (MFil1 to WMAP-K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7Simulated spectral index in polarization (MFil1 to WMAP-K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7Recovered spectral index in polarization (MFll1 to WMAP-K) - after FDEC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β11GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='754 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Original resolution QUIJOTE MFI wide survey maps for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Maps are shown in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' All figures use the same linear colour scale, saturated at 20 mKCMB for intensity (first column) and 2 mKCMB in polarization for Stokes Q (second column) and Stokes U (third column) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, maps are downgraded to 𝑁side = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C1, but for QUIJOTE MFI wide survey maps for horn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C1 Signal-to-noise of the QUIJOTE MFI maps From the maps at original resolution shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C1–C3, and the noise variance maps estimated from the inverse of the weights presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C4–C6 and rescaled by the factors reported in Table 12, we can produce signal-to-noise maps for the MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' To this end, we downgrade these maps to a HEALPix reso- lution of 𝑁side = 64, which roughly corresponds to the beam size of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table C1 presents some basic statistics about the fraction of 𝑁side = 64 pixels in the maps observed about a certain signal-to- noise significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, the 11 GHz polarized intensity map has 52 % of its pixels with a signal-to-noise ratio larger than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' APPENDIX D: IMPACT IN THE POLARIZATION TOD OF AN ERROR IN THE DETERMINATION OF THE 𝑟-FACTOR We illustrate this effect using the particular case of uncorrelated channels in the first MFI configuration, but the result is equivalent for correlated channels and for all MFI configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We follow the notation introduced in Génova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2023), and used in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Following the notation of Jarosik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (2003), the MFI response for the two uncorrelated channels, 𝑥 and 𝑦, in the first MFI MNRAS 000, 1–58 (2022) QUIJOTE 1 H2 17GHZ 5 mK 20QUIJOTE Q H2 17GHz mK 2QUIJOTE U H2 17GHZ 2 mKQUIJOTE 1 H2 19GHZ 5 mK 20QUIJOTE Q H2 19GHz mK 2QUIJOTE U H2 19GHzZ mK 2QUIJOTE I H3 11GHZ 5 mK 20QUIJOTE Q H3 11GHz mK 2QUIJOTE U H3 11GHzZ 2 mKQUIJOTE I H3 13GHZ 5 mK 20QUIJOTE Q H3 13GHZ 2 mKQUIJOTE U H3 13GHZ 2 mK 2QUIJOTE MFI wide survey 55 Figure C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' C1, but for QUIJOTE MFI wide survey maps for horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey weight maps for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top row is 17 GHz, and bottom row is 19 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Each row shows, from left to right, the weight maps for Stokes I, Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' configuration is given by 𝑉x = 𝑠x𝑔2 1 2 � 𝐼 + 𝜌x(𝑄 cos 𝜃 − 𝑈 sin 𝜃) � (D1) 𝑉y = 𝑠y𝑔2 2 2 � 𝐼 + 𝜌y(−𝑄 cos 𝜃 + 𝑈 sin 𝜃) � (D2) where 𝜃 stands for the argument of the cosine and sine in MFI receivers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝜃 = 4𝜃pm + 2𝛾p, as in equations 3 and 4), 𝑠x and 𝑠y represent the responsivities of the detectors in the two branches, 𝑔1 and 𝑔2 represent the voltage gains of the two amplifiers in each MFI polarimeter, and 𝜌x and 𝜌y are the polar efficiencies in each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The 𝑟-factor is defined as 𝑟u ≡ 𝑠x𝑔2 1 𝑠y𝑔2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (D3) If we are using an incorrect 𝑟-factor 𝑟′u = 𝑟u + 𝜖, where 𝑟u is the correct underlying value, then we have 𝑉x − 𝑟′ u𝑉y = 𝑠x𝑔2 1 �� 𝜌x + 𝜌y 2 + 𝜖 2𝑟u 𝜌y � (𝑄 cos 𝜃 − 𝑈 sin 𝜃) − 𝜖 2𝑟u 𝐼 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (D4) We find that this error on the 𝑟-factor translates into an effective modification of the polar efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and the appearance of a constant MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1–58 (2022) QUIJOTE 1 H4 17GHZ 5 mK 20QUIJOTE Q H4 17GHZ mK 2QUIJOTE U H4 17GHzZ 2 mKQUIJOTE 1H4 19GHZ 5 mK 20QUIJOTE Q H4 19GHz 2 mKQUIJOTE U H4 19GHzZ 2 mK 2QUIJOTE WEIGHTS 1 H2 17GHZ 0 mk-2 30QUIJOTE WEIGHTS Q H2 17GHz 0 mk-2 30QUIJOTE WEIGHTS U H2 17GHZ 0 mk-2 30QUIJOTE WEIGHTS 1 H2 19GHz 0 mk-2 30QUIJOTE WEIGHTS Q H2 19GHz 0 mk-2 30QUIJOTE WEIGHTS U H2 19GHZ 0 mk-2 3056 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey weight maps for horn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey weight maps for horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' offset factor in the polarization timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the particular case of 𝜌x = 𝜌y, then the effective polar efficiency is rescaled by the factor 𝜌x → 𝜌x � 1 + 𝜖 2𝑟u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' (D5) Finally, we note that for the intensity timeline, the same effect gener- ates an overall calibration shift, and a small polarization-to-intensity leakage term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The first term is absorbed once we carry our a re- calibration of the instrument, while the second one can be safely ignored, as the polarization fraction of the sky emission is already small (typically well below 10 per cent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' APPENDIX E: POWER SPECTRUM ESTIMATORS FOR MFI WIDE SURVEY MAPS Throughout this paper, we have been using two power spectrum esti- mation codes, both based on a pseudo-Cℓ approach: Xpol (Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2005) and NaMaster (Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this appendix, we show that both methods produce consistent results for the typical sky masks adopted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For this comparison, we take as a reference case the MFI 11 GHz wide survey map and the default QUIJOTE mask (NCP+sat+lowdec) combined with the Galactic cut |𝑏| > 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In addition, we justify the use of the pseudo-spectra approach by comparing these results with those from an optimal estimator based on a fast implementation of a quadratic maximum- likelihood (QML) estimator (ECLIPSE, Bilbao-Ahedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1–58 (2022) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='QUIJOTE WEIGHTS I H3 11GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS Q H3 11GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS U H3 11GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS 1 H3 13GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS Q H3 13GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS U H3 13GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS I H4 17GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS Q H4 17GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS U H4 17GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS 1 H4 19GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS Q H4 19GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE WEIGHTS U H4 19GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='mk-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='30QUIJOTE MFI wide survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='Figure C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey hit maps for horn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' They show the total number of 40 ms samples in each HEALPix pixel of 𝑁side = 512 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey hit maps for horn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Running this QML code is computationally very expensive, so the comparison is limited to this case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure E1 shows the (binned) low multipoles points of the an- gular power spectra and cross-spectra (30 ≤ ℓ ≤ 80) computed with those three codes using the same mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the case of NaMaster we use the "purification" option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The conclusion is that, within the multipole range used in this paper (ℓ ≥ 30), all methods provide consistent results, so it is justified to use the pseudo-Cℓ approach for our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this work, we use equally Xpol or NaMaster for TT, EE and BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For the cross-spectrum analysis in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 7, we use the NaMaster code, as it provides slightly closer results to the (optimum) QML solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' APPENDIX F: SPECTRAL INDEX OF THE MFI 13 GHZ SKY EMISSION In this appendix we repeat the same analysis carried out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, but using now as a reference the MFI 13 GHz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure F1 shows the result for the intensity spectral index in 𝛽408MHz−13GHz (top panel) and 𝛽13GHz−23GHz (bottom panel), while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' F2 presents the polarization spectral index map 𝛽13GHz−23GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Again, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' F3 we show the histogram with the distribution of spectral indices in both maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In general, all results are consistent with those obtained using MFI 11 GHz as the reference map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In intensity, the median spectral index 𝛽408MHz−13GHz in the full analysis mask is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83, with a standard deviation of the values across the map of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' and the MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1–58 (2022) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='QUIJOTE NHITS I H2 17GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='nhits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1000QUIJOTE NHITS Q H2 17GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='nhits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='500QUIJOTE NHITS U H2 17GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='nhits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='500QUIJOTE NHITS I H2 19GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' 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+page_content='1000QUIJOTE NHITS Q H3 13GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='nhits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='500QUIJOTE NHITS U H3 13GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='nhits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='50058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey hit maps for horn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey 𝑟cond maps for all four horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' During the post-processing stage, all pixels with 𝑟cond > 3 are removed from the final wide survey polarization maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 𝛽13GHz−23GHz spectral index has a median of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='83 and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We note that in this latter case, there is a peak around −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1, which is due to the adopted prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In polarization, the 𝛽13GHz−23GHz spectral index presents a median value −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='09, and the standard deviation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 1–58 (2022) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3QUIJOTE RCOND H4 19GHZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='rcond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3QUIJOTE MFI wide survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='Figure C11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' QUIJOTE MFI wide survey normalized covariance (𝑐𝑜𝑣 (𝑄, 𝑈)) maps for all four horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Comparison of the angular power spectrum estimators used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Using the default QUIJOTE analysis mask together with the Galactic cut |𝑏| > 10◦, we evaluate the TT, EE, BB auto-spectra and the TE, EB and TB cross-spectra of the MFI 11 GHz map, using Xpol and NaMaster (both based on pseudo-Cℓ formalism), and ECLIPSE (based on a QML approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' As a reference, in the first three panels we also show the corresponding noise power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For display purposes, the different data points have been shifted by Δℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) QUIJOTE COV(Q,U) H2 17GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(QQ Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001QUIJOTE COV(Q,U) H2 19GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(QQ Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001QUIJOTE COV(Q,U) H3 11GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(Qo Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001QUIJOTE COV(Q,U) H3 13GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(QQ Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001QUIJOTE COV(Q,U) H4 17GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(QQ Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001QUIJOTE COV(Q,U) H4 19GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001 cov(Q,U)/(QQ Qu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0001TT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Ibl>10° EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='Ibl>10° BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='Ibl>10° 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 10- 10 TT Nmt pure · EE Nmt pure · BB Nmt pure · TT QML O EE QML O BB QML 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 TT Xpol · EE Xpol · BB Xpol ● 10-2E 10-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content="000 [mk'] [mk'] a a 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='100 10~4 10-* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='001L 10-5 10-5 30 40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80 Multipole t Multipole t Multipole l EB TE TB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010 EB Nmt pure · TE Nmt pure · TB Nmt pure · EB QML O TE QML TB QML EB Xpol TE Xpol · TB Xpol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content="005 [mk'] [mk'] [eyw] 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='000 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0010LI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='010L 30 40 50 60 70 80 30 40 50 60 70 80 30 40 50 60 70 80 Multipole t Multipole t Multipole t60 Rubiño-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Fraction of 𝑁side = 64 pixels with signal-to-noise ratio (SNR) above a certain threshold in the four QUIJOTE-MFI frequency maps (horns 2 and 4 have been combined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We report the SNR both for the intensity (I) and the (noise debiased) polarized intensity (𝑃 = √︁ 𝑄2 + 𝑈2) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 11 GHz 13 GHz 17 GHz 19 GHz Intensity (I) SNR> 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='82 SNR> 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='64 SNR> 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='49 SNR> 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='36 SNR> 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='26 Polarized intensity (P) SNR> 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='70 SNR> 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 SNR> 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='16 SNR> 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 SNR> 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='02 Figure F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Spectral index of the intensity emission in the QUIJOTE 13 GHz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Spectral index of 𝛽408MHz−13GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average index is approxi- mately −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: Spectral index of 𝛽13GHz−23GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' The average spectral index is also 𝛽 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' In this colour scale, dark red corresponds to AME dominated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Top: Spectral index map of the polarized emission between QUIJOTE 13 GHz and WMAP 23 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Bottom: error map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Figure F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' Histogram of spectral index values obtained from Figures F1 and F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' We show in dashed lines the mean of the prior adopted in the determination of the spectral index in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' For comparison, we also include the histogram of spectral index values from the PySM synchrotron model 1 (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content=' MNRAS 000, 1–58 (2022) Spectral index in intensity (Haslam to MFl13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β408MHz - 13GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2Spectral index in intensity (MFI13 to WMAP-K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β13GHz - 23GHz-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='4 β13GHz - 23GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='7Error in the spectral index in polarization (MFl13 to WMAP-K) 0 (β13GHz - 23GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='2 Intensity（408MHz-13GHz Intensity（13GHz-23GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='0 Polarization（13GHz-23GHz） Prior β=-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfgA3B/content/2301.05113v1.pdf'} +page_content='3 (normalized) PySM 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index 0000000000000000000000000000000000000000..00d6cbc82a4249a3b2c188b47efd4251b1e24d29 --- /dev/null +++ b/GNFIT4oBgHgl3EQfWStC/content/tmp_files/2301.11239v1.pdf.txt @@ -0,0 +1,1404 @@ +Collective Vortical Motion and Vorticity Reversals +of Self-Propelled Particles on Circularly Patterned Substrates∗ +Haosheng Wen,1, 2 Yu Zhu,1 Chenhui Peng,1, 3 P.B. Sunil Kumar,4, 5 and Mohamed Laradji1, † +1Department of Physics and Materials Science, The University of Memphis, Memphis, TN 38152, USA +2Biophysics Graduate Program, The Ohio State University, Columbus, OH 43210, USA +3Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China +4Department of Physics, Indian Institute of Technology Palakkad, Palakkad 668557, Kerala, India +5Department of Physics, Indian Institute of Technology Madras, Chennai 600036, Tamil Nadu, India +The collective behavior of self-propelled particles (SPPs) under the combined effects of a circu- +larly patterned substrate and circular confinement is investigated through coarse-grained molecular +dynamics simulations of polarized and disjoint ring polymers. +The study is performed over a wide +range of values of the SPPs packing fraction ¯φ, motility force FD, and area fraction of the patterned +region. At low packing fractions, the SPPs are excluded from the system’s center and exhibit a +vortical motion that is dominated by the substrate at intermediate values of FD. +This exclusion +zone is due to the coupling between the driving force and torque induced by the substrate, which +induces an outward spiral motion of the SPPs. +For high values of FD, the SPPs exclusion from the +center is dominated by the confining boundary. At high values of ¯φ, the substrate pattern leads to +reversals in the vorticity, which become quasi-periodic with increasing ¯φ. +We also found that the +substrate pattern is able to separate SPPs based on their motilities. +I. +INTRODUCTION +Active matter systems, which are collections of individ- +ual self-driven units that consume energy from the envi- +ronment to move, have been the subject of a significant +amount of research over the last few decades [1–3]. Active +matter systems range widely from macroscopic systems, +including schools of fish [4], flocks of birds [5], and granu- +lar media [6–8], to microscopic systems including colonies +of bacteria, eukaryotic cells [9–12], actin filaments and +microtubules that are propelled by their respective mo- +tor proteins [13], and active colloids [14]. Active matter +systems often exhibit intriguing collective behavior char- +acterized by clustering of the units and large-scale col- +lective motion [15]. This collective behavior is used, for +example, in bacteria colonies to reduce competition for +nutrients, accelerate growth of the colony, or to increase +resilience in hostile environments [16]. Likewise, the col- +lective behavior of assemblies of eukaryotic cells, such as +epithelial monolayers and cancer cells, has physiological +and pathological implications. These include embryoge- +nesis, wound healing and tumor metastasis [17–19]. +Many studies have shown that clustering and collec- +tive motion of self-propelled particles (SPPs) are influ- +enced by various physical factors, including the packing +fraction of the SPPs, nature of the coupling between +neighboring SPPs, and the type of motion of a single +SPP [20, 21]. Other physical factors include environmen- +tal constraints [3] such as anisotropy of the embedding +fluid [22, 23], geometric confinement [24–36] and obsta- +cles [37, 38]. An interesting effect of circular confinement, +for example, is an induced vortical motion of the SPPs +∗ Physical Review E (in press) +† Corresponding author. mlaradji@memphis.edu +that is concentric with the boundary [26–28]. +While in most studies of SPPs’ collective behavior, the +substrate is non-patterned, the effect of patterned sub- +strates on SPPs collective behavior has recently been in- +vestigated in a few studies. For example, it was shown +that patterning the substrate, into periodic linear fur- +rows, aligns Pseudomonas aeruginosa along the furrows +while greatly supresses their migration across them [39]. +Likewise, collective migration of epithelial cells is sub- +stantially promoted by linear grooves of patterned sub- +strates [40, 41]. However, computational investigations +of the effect of patterned substrates on SPPs collective +behavior are lacking. In this article, we address the effect +of substrates, which are partially circularly patterned, on +the collective behavior of soft SPPs with the ability to +switch their polarity. In particular, we investigate how +a circular confinement that is concentric with the sub- +strate’s pattern further influences their collective motion. +II. +MODEL AND METHOD +We consider a total number of P SPPs in two dimen- +sions, each modeled as a semi-flexible ring polymer com- +posed of N beads in a good solvent. +This model was +recently introduced by us to investigate SPPs collective +behavior on a non-patterned substrate [42] and is a gen- +eralization of an earlier model for strongly adsorbed dis- +joint ring polymers [43]. +The potential energy of the +arXiv:2301.11239v1 [cond-mat.soft] 26 Jan 2023 + +2 +SPPs is given by +Unet= +P +� +l=1 +� N +� +i=1 +Ubond +� +r(l) +i,i+1 +� ++ +N +� +i=1 +Ubend(r(l) +i−1, r(l) +i , r(l) +i+1) ++ +N +� +i=1 +Uwall +� +r(l) +i +� ++ Uarea +� +{r(l) +i } +� ++ Usub +� +r(l) +p1 , r(l) +p2 +� � ++ +� +l1,l2 +� +i,j +Urep +� +|r(l1) +i +− r(l2) +j +| +� +, +(1) +where r(l) +i +is the coordinate of bead i belonging to SPP +l and r(l) +i += |r(l) +i |. The lth SPP has two symmetrically +positioned poles with indices +p1 = 1 and p2 = N/2 + +1. Ubond is a harmonic potential ensuring connectivity +between consecutive beads within an SPP and is given +by +Ubond +� +r(l) +i,i+1 +� += 1 +2k +� +r(l) +i,i+1 − rb +�2 +, +(2) +where k is the spring constant, r(l) +i,i+1 = |r(l) +i+1−r(l) +i | and rb +is the preferred bond length. +In Eq. (2), r(l) +N,N+1 = r(l) +N,1. +The semi-flexibility of an SPP’s boundary is maintained +through a three-body interaction +Ubend(r(l) +i−1, r(l) +i , r(l) +i+1) = κ +� +1 − cos θ(l) +i +� +, +(3) +where κ is the bending stiffness of the polymers and +cos θ(l) +i += r(l) +i−1,i·r(l) +i+1,i/r(l) +i−1,ir(l) +i+1,i. +In Eq. (3), r(l) +0 += r(l) +N +and r(l) +N+1 = r(l) +1 . Eq. (3) implies that the preferred bend- +ing angle of a triplet is 180◦. To account for the polariza- +tion of the SPPs, triplets of beads centered at the pole +beads with indices p1 and p2 have a preferred bending +angle θp ≤ 180◦. Since Eq. (3) does not allow for pre- +ferred angles different from 180◦, beads p1 and p2 are +assigned the following slightly different three-body inter- +action, which allows for any arbitrary splay angle θs, +Ubend(r(l) +p−1, r(l) +p , r(l) +p+1) = 1 +2κ′ � +cos θ(l) +p − cos θs +�2 +, +(4) +where κ′ is the bending stiffness at the poles. Due to +the softness of the potential given by Eq. (4), we found +that achieving the same persistence length of the polymer +with this potential as with that given by Eq. (3) requires +κ′ ≈ 10κ. +The disjointness of the ring polymers is maintained +by the following fully repulsive two-body interaction be- +tween any two non-bonded beads +Urep (r) = +� +1 +2ζ +� +1 − r +rc +�2 +if r ≤ rc, +0 +if r > rc, +(5) +where ζ and rc are the strength and range of the repulsive +interaction, respectively. Finally, the area constraint of +each SPP is maintained by the effective potential energy +Uarea +� +{r(l) +i } +� += 1 +2χA0 +� +�1 − +A +� +{r(l) +i } +� +A0 +� +� +2 +, +(6) +Pl +!" +!# +$%&%' +(%& + (%' /2 +,- +, +Non-patterned +substrate +Patterned +substrate +Confining +wall +FIG. 1. Schematic of the system. Solid black circle of radius R +corresponds the confining wall of the system. The yellow disk +of radius Rp corresponds the region of the substrate that is +patterned. The green annulus corresponds to the region of the +substrate that is non-patterned. The mid-point of the polarity +vector Pl of an arbitrary SPP l is at a distance (rl1 + rl2)/2, +where rl1 and rl2 are the coordinates of the two poles (The +origin of the coordinate system is at the center of the system). +The effect of the patterned substrate is to reorient the SPP’s +polarity through a torque, whose forces are indicated by the +green vectors). In this schematic, the size of the SPP is not +to scale with the system size and the size of the patterned +region. +where χ is the area-stretch modulus, A0 is the SPP’s +preferred area, and +A +� +{r(l) +i } +� +is the area enclosed by +the SPP’s boundary and depends on the coordinates of +the beads belonging to the SPP through the shoelace +formula, +A +� +{r(l) +i } +� += 1 +2 +���� +N +� +i=1 +� +x(l) +i y(l) +i+1 − x(l) +i+1y(l) +i +� ����, +(7) +with x(l) +N+1 = x(l) +1 +and y(l) +N+1 = y(l) +i . +Finally, the SPPs are confined within a circle of radius +R by the interaction potential +Uwall (r) = +� +εwall (r − R + a)n /an if R − a ≤ r < R, +0 +if +r < R − a, +(8) +where εwall and a are the strength and range of this +interaction, respectively. +We choose n = 4 since this +value is large enough to prevent the SPPs from crossing +the circular confining wall. The main difference between +this model and prior models for the collective behavior +of elongated self-propelled particles is that the present +model accounts for the elongation of the self-propelled +particles and their flexibility. +This is in contrast with +previous studies wherein particles are either rigid [44–46] +or deformable with high aspect ratio and with practically +no account for the enclosed volume of the particles [47]. +We consider the case where a region of the substrate +is circularly patterned. Experimentally, this would cor- + +3 +respond, for example, to a substrate that is circularly +grooved [40, 41]. +The effect of the substrate’s pattern +on an SPP is to align it along the local direction of the +pattern. This is achieved by a simple effective potential +energy between the SPP’s poles that produces a torque +on the SPP, +Usub +� +r(l) +p1 , r(l) +p2 +� += ks +2 sin2 ϕl, +(9) +where ks is the strength of the interaction and ϕl is the +angle between the polarity Pl = r(l) +p2 − r(l) +p1 and the local +tangent to a circle of radius (r(l) +p1 +r(l) +p2 )/2 centered at the +origin. This torque tends to align an SPP’s polarity with +the local tangent of a circle centered at the origin and +passing by the mid-point of the two poles, as schemati- +cally shown by Fig. 1. We focus on the case where the +substrate is patterned only within the region (r ≤ Rp). +Otherwise, the substrate is uniform (non-patterned) for +Rp < r ≤ R. +Each SPP is propelled by a motility force of magnitude +FD, along its polarity, that is given by +fl(t) = FD (Pl(t)/Pl(t)) g (¯vl(t), Pl(t)) , +(10) +where g(A, B) = +1 or -1 if A·B > 0 or < 0, respectively, +and where ¯vl(t) is the SPP’s average velocity over the +time interval [t − τm, t], i.e. +¯vl(t) = 1 +τm +� t +t−τm +vl(t′)dt′, +(11) +with vl(t) = (1/N) �N +i=1 v(l) +i (t). +In Eq. (11), we take +τm = τ where τ = rb +� +µ/ε, rb is the preferred bond +length, ε is the energy scale and µ is the bead’s mass. +Beads are moved according to a molecular dynamics +scheme, +˙r(l) +i (t) = v(l) +i (t), and +µ ˙v(l) +i (t) = −∇(l) +i Unet + fl(t) +N +− Γv(l) +i (t) ++Γ +√ +2DΞ(l) +i (t), +(12) +where ∇(l) +i += (∂x(l) +i , ∂y(l) +i , ∂z(l) +i ) and v(l) +i +is the instanta- +neous velocity of bead i belonging to SPP l. In Eq. (12), +Γ is the friction coefficient, D is the diffusion coefficient +of the beads in the ideal limit (i.e. in the absence of inter- +actions and beads connectivity), and Ξ(l) +i (t) is a random +vector that has zero-mean and is δ-correlated for the same +particle and same component, i.e. Ξ(l) +i (t) satisfies +⟨Ξ(l) +i (t)⟩ = 0, +⟨Ξ(l1) +i,α (t) Ξ(l2) +j,β (t′)⟩ = δl1l2δijδαβδ (t − t′) , +(13) +where α, β = x or y, δnm is the Kronecker delta, and +δ(t) is the Dirac delta-function. +The equations of motion are integrated using the +velocity-Verlet algorithm with a time step ∆t = 0.01τ. +The numerical value of a component of the random force +is given by +Ξ(l) +i,α = +� 3 +∆t +�1/2 +λ(l) +i,α, +(14) +where λ(l) +i,α is a random number generated from a uni- +form distribution in the interval [−1, 1]. +Each SPP is +composed of N = 40 beads. The values of the param- +eters of the model SPPs, +which are kept fixed in the +present study, are given by +k = 100ε/r2 +b, κ = 100ε, κ′ = 1000ε, θs = 120◦, ζ = 50ε, +rc = rb, χ = 1ε/r2 +b, A0 = 100r2 +b, τm = τ, +D = 1.0r2 +b/τ, and Γ = 1.0µ/τ. +(15) +III. +RESULTS +A. +Effects of Patterned Substrate and Motility +Force on SPPs’ Collective Behavior +We first focus on the combined effect of the patterned +substrate and circular confining wall on the SPPs col- +lective behavior at an average packing fraction ¯φ = +PA0/πR2 = 0.398 with R = 200rb. This corresponds +to P = 500. +Steady-state snapshot (a) in Fig. 2(A) +and Movie 1, at FD = 20ε/rb and non-patterned sub- +strate (ks = 0), indicate a small amount of clustering +and a weak collective motion, in agreement with prior +results [42]. Fig. 2(C) shows that at these conditions, the +radial distribution of the SPPs packing fraction, φ(r), is +almost uniform. As FD is increased to 24ε/rb at ks = 0, +the motility force drives many SPPs to the boundary +leading to their accumulation as shown by snapshot (b) +in Fig. 2(A) and collective unidirectional vortical motion +(see Movie 2). This is also demonstrated by the time de- +pendence of the average tangential velocity of the SPPs +in an annulus of thickness 10rb near the boundary (red +graph in Fig. 2(B) at ks = 0 and FD = 24ε/rb). In con- +trast, the SPPs motion in an annulus close to the center +is fairly turbulent (blue graph in Fig. 2(B) at ks = 0 and +FD = 24ε/rb). SPPs accumulation at the boundary is +due to the asymmetry between the effect of the motil- +ity force, which drives the SPPs toward the boundary, +and thermal effects, which drive the SPPs away from the +boundary, and has been observed in earlier studies [3]. +In contrast, although the SPPs that are away from the +boundary move collectively in clusters, they do not ex- +hibit a net vortical motion, as demonstrated by the fluc- +tuations around 0 of the average tangential velocity of +the SPPs in the annulus close to the center (blue graph +in Fig. 2(B) at ks = 0 and FD = 24ε/rb). +Interaction between the SPPs and the patterned sub- +strate leads to a much richer dynamical behavior. Snap- +shots (c) and (d) in Fig. 2(A) and their corresponding +tangential velocities vs. +time in Fig. 2(B) show that, +at ks = 100ε and FD = 18 or 20ε/rb, the patterned + +4 +0 +50 +100 +150 +200 +0.00 +0.25 +0.50 +0.75 +1.00 +(A) +!" = 0, &'= 20)/+, +!" = 100), &'= 20)/+, +!" = 100), &'= 24)/+, ++ [+,] +!" = 0, &'= 24)/+, +!" = 100), &'= 22)/+, +(C) +!" = 100), &'= 18)/+, +_. +2[3] +45 +,/3 +b +!" = 0, &' = 24)/+, +c +!"= 100), &' = 18 )/+, +d +!" = 100), &' = 20 )/+, +(B) +e +!" = 100), &' = 22)/+, +a +!" = 0, &' = 20)/+, +; + +&' = 24)/+, +f !" = 100) +(a) +(b) +(f) +(d) +(e) +(c) +FIG. 2. +Panel (A): Steady-state snapshots at (a) FD = 20ε/rb and ks = 0, (b) FD = 24ε/rb and ks = 0, (c) FD = 18ε/rb and +ks = 100ε, (d) FD = 20ε/rb and ks = 100ε, (e) FD = 22ε/rb and ks = 100ε, and (f) FD = 24ε/rb and ks = 100ε. Panel B: Time +dependence of the average tangential velocity for different values of ks and FD corresponding to those in Panel (A). The blue +(red) graphs correspond to SPPs in the blue (red) annulus, shown in snapshot (A). Shaded yellow (green) region corresponds to +the regime where the vortices in the patterned and non-patterned regions are in same (opposite) directions. Panel (C): Radial +profiles of the packing fraction, ¯φ at values of FD and ks corresponding to those in Panel (A). All data shown in this figure are +at ¯φ = 0.398, Rp = 100rb and R = 200rb. +substrate and the driving force collectively lead to (1) +a tangential alignment of the SPPs in the patterned re- +gion, (2) their accumulation at the periphery of the pat- +terned region, and (3) their exclusion from the center. At +ks = 100ε and FD = 20ε/rb, Fig. 2(B) and Movie 3 show +that the SPPs move as a vortex, in the patterned region of +the substrate, with very few reversals in its direction. In +contrast, the SPPs outside the patterned region exhibit +a weak collective behavior, as demonstrated by the fact +that the SPPs’ average tangential velocity in this region +fluctuates around 0 (red graph in Fig. 2(B) at ks = 100ε +and FD = 20ε/rb). +As FD is further increased to FD = 22 or 24ε/rb, at +ks = 100ε, the corresponding snapshots (d) or (e), re- +spectively, shown in Fig. 2(A), show that more SPPs are +driven to the confining wall. This is also demonstrated by +increased packing fraction next to the boundary at these +values of FD in Fig. 2(C). Fig. 2(B) shows that, at these +values of FD, the SPPs exhibit collective vortical motion +in both patterned and non-patterned regions. These vor- +tices can move either in the same direction (shaded yellow +regions in Fig. 2(B) and Movie 4) or opposite directions +(shaded green regions and Movie 5) with frequent rever- +sals. Inspection of the vorticity reversals indicates that +they are due to collectively moving clusters in the non- +patterned region, which collide with the vortices in the +patterned region or in the boundary layer. +The SPPs collectivity is quantified through the vortical +order parameter defined as +Sv = ⟨| +P +� +l=1 +σl|⟩/P, +(16) +where σl = +1 (-1) if the direction of the tangential ve- +locity of SPP l is clockwise (counter-clockwise). Fig. 3, +which depicts Sv vs. FD at ¯φ = 0.398, shows that the +substrate pattern shifts the onset of vortical collective +motion to smaller values of FD. Four distinct regimes +in the case of ks = 100ε are identified. +In regime I +(FD ≲ 16ε/rb), there is no collective motion. In regime +II (16ε/rb ≲ FD ≲ 21ε/rb), the collective behavior is +dominated by the patterned region, and is characterized +by an almost unidirectional vortical motion. Fig. 2(C) +shows that regime II is also characterized by an in- +crease in the maximum of the SPPs packing fraction +in the patterned region with increasing FD. In regime +III (21ε/rb ≲ FD ≲ 25ε/rb), both patterned substrate +and confining wall independently promote SPPs collec- +10 +15 +20 +25 +30 +35 +0 +0.2 +0.4 +0.6 +0.8 +1 +!" +#$ = 0 +#$ = 100( (opposite vorticities) +#$ = 100( (same vorticities) +I +II +III +IV +67 (/9: +0 +50 +100 150 +0 +0.1 +0.2 +0.3 +0.4 +!" +#$ ( +67 = 20(/9: +FIG. 3. +SV vs. FD at ¯φ = 0.398, Rp = 100rb and R = 200rb +for ks = 0 (red circles) and ks = 100ε (blue circles). +Full +(open) blue circles correspond to Sv at ks = 100ε where the +vortices in the patterned and non-patterned regions have same +(opposite) directions.(Inset) Sv vs. ks at FD = 20ε/rb. The +solid lines are simply guides to the eye. + +0.4 +0 +-0.4 +0.4 +0 +-0.4 +0.4 +0 +-0.4 +0.4 +0 +-0.4 +0.4 +0 +-0.4 +0.4 +0 +-0.4 +20000 +25000 +30000 +35000 +40000800 +Q05 +0 +50 +100 +150 +200 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +50 +100 +150 +200 +0.00 +0.25 +0.50 +0.75 +1.00 +10 +15 +20 +25 +30 +35 +0 +0.2 +0.4 +0.6 +0.8 +1 + + + + + +1 pt + + 20 + 32 + + + +(A) +Circular boundary +Periodic boundary +conditions +! " +" ["$] +(C) +0 +50 +100 +150 +200 +0.00 +0.02 +0.04 +0.06 +0.08 +" ["$] +(B) +&' " +" ["$] +(D) +125"$ +150"$ +175"$ +25"$ +50"$ +75"$ +100"$ +FIG. 4. (A) trajectories of a single SPP starting from a po- +sition near the center, for differemt values of FD and ks. (B) +Radial profile of the radial velocity of the SPPs for the case +of a circular confining wall. (B) Radial profile of the packing +fraction for the case of a circular confining wall (solid line) +and PBC (dashed line). Data shown in (B) and (C) are in +the case of FD = 24ε/rb, ks = 100ε, ¯φ = 0.398, Rp = 100rb +and R = 200rb. (D) Radial profiles of the packing fraction for +different values of the radius of the patterned region, Rp, indi- +cated in the legend. These data correspond to FD = 22ε/rb, +ks = 100ε, R = 200rb and ¯φ = 0.398. The vertical dashed +lines in (B-D) indicate the location of the boundary between +the patterned (left) and non-patterned (right) regions of the +substrate. +tive motion, and lead to vortical motion in both regions +with same or opposite directions. This results in a bi- +furcation of Sv into two branches: one branch with high +values of Sv (solid blue circles in Fig. 3) where the two +vortices have same direction, and a second branch with +low values of Sv (open blue circles in Fig. 3) where the +two vortices have opposite directions. Regime III marks +the beginning of the decrease in the value of the maxi- +mum of the SPPs’ packing fraction in the patterned re- +gions. Finally, in regime IV (FD ≳ 24ε/rb), the major- +ity of the SPPs are accumulated near the confining wall, +where they move as a unidirectional vortex. +Interestingly, snapshots (c) to (f) of Fig. 2(A) and +Fig. 2(C) demonstrate that the patterned substrate in- +duces an exclusion zone in the center with a diameter +that increases with FD. This is contrasted with the case +of a non-patterned substrate, in which the radial profile +of the packing fraction is almost uniform, except at the +boundary. The source of this exclusion zone, is inferred +from simulations of a single SPP (dilute regime) at finite +values of ks and FD, starting from a location near the +center. Fig. 4(A) (see Movie 6 as well) shows that the +SPP’s trajectory is an outward spiral, with a number of +turns that increases with increasing ks or decreasing FD. +Hence, the motility force and the substrate’s pattern co- +operatively drive the SPPs away from the patterned re- +gion with a rate that increases with FD and decreases +with ks, leading to an exclusion zone in the center. +In addition to the exclusion zone in the center, +Fig. 2(C) shows that the radial profile of the packing +fraction exhibits a broad peak within the patterned re- +gion, and close to the boundary between the patterned +and non-patterned regions. The emergence of this peak +is understood as follows. The motion of the SPPs within +the patterned region is mainly tangential, while in the +non-patterned region (but away from the confining wall), +the motion is more turbulent. +As a result vp +⊥ < vn +⊥, +where vp +⊥ and vn +⊥ are the averages of the magnitudes of +the radial components of the SPPs velocities in the pat- +terned and non-patterned regions, respectively, as shown +in Fig. 4(B). Steady state requires that the outflux of +the SPPs from the patterned must be equal to the influx +of the SPPs from the non-patterned regions to the pat- +terned region, i.e. φpvp +⊥,out = φnvn +⊥,in, where φp (φn) is +the packing fractions of the SPPs in the patterned (non- +patterned) region, close the boundary between the pat- +terned and non-patterned regions. +vp +⊥,out is the average +of the radial component of the velocity of the SPPs out- +going from the patterned region at the boundary between +the patterned and non-patterned regions. Likewise, vn +⊥,in +is the average of the radial component of the velocity of +the SPPs incoming from the non-patterned region at the +boundary between the patterned and non-patterned re- +gions. Therefore, mass balance between the outflux and +influx of the SPPs across this boundary, at steady state, +imposes φp > φn. Combined with the fact that the inter- +play between the motility force and the torque induced +by the patterned substrate, which leads to SPPs exclu- +sion from the center, the argument above implies that the +radial packing fraction profile must exhibit a peak within +the patterned region, and close to the boundary between +the patterned and non-patterned regions, as shown by +Fig. 2(C). FD enhances the SPPs outflux from the pat- +terned region, i.e. it increases vp +⊥, while it decreases the +influx from the non-patterned region, due to increased +accumulation of the SPPs near the confining wall. As +a result, the size of the exclusion zone increases with +FD (see Fig. 2(C)). Elimination of SPPs accumulation +at the boundary, through imposing periodic boundary +conditions (PBC), enhances SPPs influx from the non- +patterned region to the patterned region. This leads to +a decrease in the size of the exclusion zone, as demon- +strated by Fig. 4(C). +The results thus far presented correspond to the case +of a radius of the patterned region of the substrate, +Rp = 100rb. To infer the effect of the size of the patterned +region, we performed a series of simulations in the case +of ¯φ = 0.398, FD = 22ε/rb, ks = 100ε, and R = 200rb. +Fig. 4(D) shows the radial profile of the packing fraction +of these systems with Rp varying between 25rb and 175rb. +This figure demonstrates that the diameter of the deple- + +/r;ks = 100 +/rb; ks = 160c +ε/rb; ks = 100c32206 +tion zone increases with Rp, which implies that the size +depletion of the SPPs from the middle is also affected +by the behavior of the SPPs in the non-patterned region +of the substrate, in line with the arguments presented in +the previous paragraph. +B. +Effect of SPPs’ Packing Fraction on their +Collective Behavior on a Patterned Substrate +We now turn to the effect of SPPs packing fraction +on their collective motion. We consider the case where +FD = 24ε/rb and ks = 100ε. The packing fraction is var- +ied by changing the number of SPPs from P = 59 to 540, +while the radius of the system is kept fixed at R = 138rb. +Corresponding Sv vs. ¯φ, shown in Fig. 5, reveals three +main regimes. For ¯φ ≲ 0.3, most SPPs accumulate at +the boundary where they move as a unidirectional vor- +tex (see Movie 7). For 0.3 ≲ ¯φ ≲ 0.8, the amount of SPPs +is increased in the patterned region, where they move as +a vortex with same direction as that in the boundary +layer (see Movie 8). Fig. 5 shows that for ¯φ ≲ 0.8, Sv in- +creases monotonically with ¯φ. Surprisingly, however, Sv +decreases with ¯φ for ¯φ ≳ 0.8. This decrease is interest- +ingly correlated with the disappearance of the exclusion +zone in the center as demonstrated by the profiles of the +packing fraction in the inset of Fig. 5. In fact, the inset +of Fig. 5 shows that an excess of SPPs at the center is +induced at ¯φ ≳ 0.8. +Inspection of movies at ¯φ ≳ 0.8 reveals an emergence +of reversals in the vorticity (demonstrated by SPPs ve- +locities snapshots in Fig. 6(A) and by Movie 9). These +reversals are quantified by the time dependence of vT (t), +defined as the average of the tangential velocity of the +SPPs in an annulus of thickness 10rb near the system’s +boundary. +Fig. 6(B) shows that vT is essentially con- +stant in the case of a non-patterned substrate (ks = 0) +at FD = 24ε/rb, indicating a unidirectional vortical mo- +tion. At ks = 40ε and same FD, Fig. 6(B) shows that +vT exhibits a single reversal during the time interval +[20 000τ, 40 000τ]. +In stark contrast, however, vT ex- +hibits many reversals at ks = 100ε and same FD dur- +ing the same time interval. Therefore, at high packing +fractions, the rate of vorticity reversals (i.e., number of +reversals per unit of time), κ, increases with increasing ks +beyond some threshold value. Likewise, Fig. 6(C) shows +that κ increases with ¯φ for ¯φ ≳ 0.8. The decrease in Sv +at ¯φ ≳ 0.8, shown in Fig. 5(B), is simply due to coexis- +tence of two vortices with opposite directions during the +reversal events, as demonstrated by a series of snapshots +in Fig. S1 in Supplemental Information [48]. +Correlations between reversal events are inferred from +the power spectrum F(ν), defined as the Fourier trans- +form of the velocity autocorrelation f(t) = ⟨vT (t0 + +t)vT (t0)⟩, where ν is frequency. Fig. 6(D) shows that, +at ¯φ = 0.836, F(ν) is peaked at ν ≈ 0. +This indi- +cates that reversal events are weakly correlated at pack- +ing fractions around this value of ¯φ. +Fig. 6(D) shows +0 +0.2 +0.4 +0.6 +0.8 +1 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +!" +#$ +0 +25 +50 +75 +100 125 +0.6 +0.7 +0.8 +0.9 +" % +% [%'] +" = 0.887 +" = 0.861 +" = 0.836 +" = 0.803 +" = 0.769 +" = 0.736 +FIG. 5. +Vortical order parameter vs. packing fraction at +ks = 100ε, FD = 24ε/rb, Rp = 100rb and R = 138rb. Vor- +tical motion is dominated by the circular confining wall at +low ¯φ (green region). Both circular confining wall and pat- +terned substrate contribute to vortical motion at intermediate +¯φ (blue region). At high ¯φ, vortical motion exhibits reversals +(red region). Inset shows radial packing fraction profiles at +different values of ¯φ. Steady state snapshots at different pack- +ing fractions are shown at the top of the figure. The dashed +circles in these snapshots indicate the boundary of the pat- +terned region of the substrate. +that F(ν) exhibits a well-defined peak at ¯φ = 0.887. +Therefore, reversal events of the vorticity become inter- +estingly quasi-periodic with increasing ¯φ. The emergence +of quasi-periodic reversals at high densities is also demon- +strated by the time dependence of the tangential velocity +in Fig. 7. +Inspection of Movie 9 shows that vorticity reversals +always originate from the center of the system. +This +concurs with the fact that vorticity reversals are absent +at low packing fractions, i.e. when the exclusion zone is +present. To demonstrate that the geometry of the confin- +ing wall has a weak effect on vorticity reversals, we per- +formed a simulation on a system with a square boundary, +of linear size Lx = 400rb, and same circular pattern with +ks = 100ε, FD = 24ϵ/rb, ¯φ = 0.887 and Rp = 100rb, +and found reversals in the vorticity similar to the case +with circular boundary and with about same value of κ, +as demonstrated by Fig. S2 [48]. Likewise, Fig. S3 [48] +shows that systems with periodic boundary conditions, +at same values of FD, ks, ¯φ, Rp and Lx, also exhibit vor- +ticity reversals, albeit not as correlated as in the case of +circular or square boundary. This is due to the fact the +periodic boundary conditions induce more turbulent flow +of the SPPs in the non-patterned region. +As stated above, reversals in the vorticity are associ- +ated with an increase in SPPs packing fraction at the + +7 +0 +0.01 +0.02 +0.03 +0.04 +0 +2 +4 +6 +8 +0.8 +0.85 +0.9 +0 +1 +2 +3 +4 +5 +6 +20000 +25000 +30000 +35000 +40000 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 + ks=0 + ks=40 + ks=100 +v (t)[rb/ ] +t [ ] +! = 37000& +38000& +39000& +40000& + B + B + B + B +(A) +(B) +(D) +(C) +*+ +, &-. (×10-2) +4(5) +5 &-. ++ = 0.836 ++ = 0.887 +FIG. 6. (A) Time-sequence of velocity snapshots showing vorticity reversals at FD = 24ε/rb, ¯φ = 0.836, Rp = 100rb, R = 138rb +and ks = 100ε. (B) Tangential velocity vT (t) vs. time at FD = 24ε/rb and ¯φ = 0.836. (C) Rate of vorticity reversals vs. ¯φ at +ks = 100ε. (G) The Fourier transform, F(ν), of the velocity autocorrelation function f(t) = ⟨vT (t0 + t)vT (t0)⟩, vs. frequency +at ks = 100ε at two high values of the packing fraction. +center. This is found to also be associated with an in- +crease in the misalignment between the SPPs polarities +and velocities, as shown by Fig. S4 (A) [48]. This re- +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +35000 +37500 +40000 +42500 +45000 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +̅"# $ [&'/)] +$[)] +FIG. 7. +Time dependence of the tangential velocity of an +annulus of thickness 10rb near the system’s boundary for the +case of FD = 24ε/rb, Rp = 100rb, R = 200rb and ks = 100ε. +Top and bottom graphs correspond to ¯φ = 0.836 and 0.887, +respectively. +sults in a high degree of fluctuations in the average of +the tangential velocity of the SPPs in the center as op- +posed to those away from the center, as shown by Fig. S4 +(B) [48]. These increased fluctuations at the center leads +some SPPs to move in a direction opposite to that of +the vortex, and in some cases these SPPs force neighbor- +ing SPPs to follow, leading to the observed intermittent +vorticity reversals. +C. +Patterned-substrates induced segregation +between fast and slow SPPs +Our simulations show that at low and intermediate val- +ues of the packing fraction, the SPPs spatial distribution +depends on their motility force. One would therefore ex- +pect that patterning the substrate may be used as a tool +to spatially separate SPPs, based on their motility force. +To verify this hypothesis, we performed a simulation of a +binary system, at an average packing fraction ¯φ = 0.6, in +which half of the SPPs are slow (with Fd = 20ε/rb) and +the other half are fast (with FD = 24ε/rb). The two types +of SPPs are otherwise identical. +The packing fraction +profiles of the two components and a steady-state snap- +shot, depicted in Figs. 8(A) and (B), respectively, show +that the fast and slow SPPs mostly segregate such that +the fast SPPs are highly concentrated in the patterned +region and the slow SPPs are more concentrated in the +non-patterned region. In comparison, the two types of +SPPs are mixed in the case where the substrate is fully + +8 +0 +25 +50 +75 +100 125 150 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +25 +50 +75 +100 125 150 +0.00 +0.25 +0.50 +0.75 +1.00 +! [!#] +(A) +(B) +% ! +&' = 24+/!# +&' = 20+/!# +Overall packing +fraction +% ! +&' = 24+/!# +&' = 20+/!# +Overall packing +fraction +(C) +(D) +FIG. 8. (A) Radial profile of the packing fraction in the case +of a binary system of fast SPPs, with FD = 24ε/rb (blue) +and slow SPPs, with FD = 20ε/rb (red), in the case where +the average packing fraction is 0.6, ks = 100ε, R = 162rb +and Rp = 100rb. +(B) A snapshot of the binary system at +steady state. Blue and red SPPs correspond to fast and slow +SPPs, respectively. The dashed vertical line and circle in (A) +and (B), respectively, indicate the boundary of the patterned +region. (C) and (D) same as in (A) and (B), respectively, but +in the case of a non-patterned substrate (ks = 0). +uniform, as shown by Figs. 8(C) and (D), except that the +fast SPPs are more concentrated at the confining wall +than the slow SPPs. +The separation between the fast and slow SPPs shown +in Figs. 8 (A) and (B) is counterintuitive in that the +coupling between the pattern of the substrate and the +motility force tend to expel the SPPs from the patterned +region. Therefore, one would expect that the fast SPPs +are more concentrated in the non-patterned region and +that the slow SPPs are more present in the patterned +region, as discussed earlier in Section III.A, which is op- +posite to what is observed from Figs. 8 (A) and (B). +The fact that the patterned substrate is able to segre- +gate the SPPs based on their motilities is very interesting +and potentially very useful. However, an explanation of +this phenomenon is lacking at the moment and requires +further systematic simulations. This segregation could +be understood from a balance of the normal stresses ex- +erted by the SPPs at the interface between the patterned +and non-patterned regions, using for example the Irving- +Kirkwood formalism [49]. This study is planned to be +performed by the authors in the near future. Separation +between SPPs may also be induced through differences in +their interaction strength with the substrate and possibly +the degree of their flexibility. +IV. +SUMMARY AND CONCLUSIONS +We showed in this article that a complex collective be- +havior is exhibited by SPPs that are confined in a circular +geometry and that interact with a circularly patterned +substrate, which tends to orient the SPPs polarities with +the local tangent of the pattern. This collective behav- +ior is characterized by SPPs vortical motion, accumula- +tion in the outer portion of the patterned region and/or +the system boundary, and SPPs exclusion from the cen- +ter. This collective behavior is enhanced with increasing +SPPs driving force. The size of the exclusion zone is de- +termined by an interplay between, on one hand, the com- +bined effects of the driving force and the patterned sub- +strate, which tends to drive the SPPs outward, and, on +the other hand, motion of the SPPs in the non-patterned +region of the substrate which drives the SPPs into the +patterned region. Interestingly, the vortices in the pat- +terned and non-patterned regions, at intermediate values +of the SPPs packing fraction, may have same or opposite +directions. +Another interesting feature of this system is that at +intermediate packing fractions and intermediate values +of the motility force, the radial profile of the packing +fraction is non-monotonic, with a peak in the patterned +region close to its boundary with the non-patterned re- +gion. A simulation of a binary system, composed of slow +and fast SPPs (i.e., SPPs with a low and motility forces, +respectively) show that they can be segregated such that +the fast SPPs are mostly trapped in the patterned region, +while the fast SPPs are mainly in the non-patterned re- +gion. This implies that SPPs can be segregated based on +their motility. +With increasing packing fraction, the exclusion zone in +the center disappears. High misalignment between the +SPPs polarities and tangential velocities, in the center of +the system, leads to an increased degree of fluctuations +in their tangential velocities and reversals in the vorticity +that originate from the center. Interestingly, these rever- +sals become quasi-periodic at high packing fractions. It +is worth noting that while the system exhibits vorticity +reversal at both intermediate and high packing fractions, +the mechanisms leading to the two types of reversals are +different. The results of the present work implies that +circular patterning of the substrate can be used as a tool +to guide the motion of SPPs into a collective vortical mo- +tion, and that at high packing fractions, can be used to +create quasi periodic reversals in their vortical motion. +We also showed that the patterned substrate is able to +segregate a binary mixture of slow and fast SPPs. We +expect that SPPs can likewise be segregated based on +their degrees of adhesion to the substrate. This segrega- +tion can be enhanced by further increasing the adhesion +strength of the fast SPPs to the substrate. +We note that the present model of SPPs accounts for +details often not accounted for in other models. These +include elongation of the self-propelled particles, their +flexibility, and enclosed area of the SPPs. It would of + +9 +course be very desirable to determine the effects of each +of these ingredients on the details of the results. There is +of course a close connection between the SPP dynamics +described here with that of swimming bacteria. +How- +ever, it is important to note that the estimated value of +the Reynolds number based on the parameters used in +this study (Eq. (15)) is about 1, which is much larger +than that of swimming bacteria. Using the present ap- +proach to investigate the collective motion of cells such as +bacteria requires a much smaller Reynolds number which +can be achieved by increasing the value of the drag coef- +ficient Γ in our model. We plan to investigate the effects +of these parameters on the observed phenomena in the +present study in the near future. +V. +ACKNOWLEDGEMENTS +All simulations were performed on computers of the +High Performance Computing Facility of the University +of Memphis. This work was funded by the University of +Memphis. +[1] S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, +323 (2010). +[2] S. Ramaswamy, J. Stat. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5 and Mohamed Laradji1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' † 1Department of Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The University of Memphis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Memphis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' TN 38152,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' USA 2Biophysics Graduate Program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Columbus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' OH 43210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' China 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Indian Institute of Technology Palakkad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Palakkad 668557,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Kerala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' India 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Indian Institute of Technology Madras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Chennai 600036,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Tamil Nadu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' India The collective behavior of self-propelled particles (SPPs) under the combined effects of a circu- larly patterned substrate and circular confinement is investigated through coarse-grained molecular dynamics simulations of polarized and disjoint ring polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The study is performed over a wide range of values of the SPPs packing fraction ¯φ, motility force FD, and area fraction of the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' At low packing fractions, the SPPs are excluded from the system’s center and exhibit a vortical motion that is dominated by the substrate at intermediate values of FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This exclusion zone is due to the coupling between the driving force and torque induced by the substrate, which induces an outward spiral motion of the SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' For high values of FD, the SPPs exclusion from the center is dominated by the confining boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' At high values of ¯φ, the substrate pattern leads to reversals in the vorticity, which become quasi-periodic with increasing ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We also found that the substrate pattern is able to separate SPPs based on their motilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' INTRODUCTION Active matter systems, which are collections of individ- ual self-driven units that consume energy from the envi- ronment to move, have been the subject of a significant amount of research over the last few decades [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Active matter systems range widely from macroscopic systems, including schools of fish [4], flocks of birds [5], and granu- lar media [6–8], to microscopic systems including colonies of bacteria, eukaryotic cells [9–12], actin filaments and microtubules that are propelled by their respective mo- tor proteins [13], and active colloids [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Active matter systems often exhibit intriguing collective behavior char- acterized by clustering of the units and large-scale col- lective motion [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This collective behavior is used, for example, in bacteria colonies to reduce competition for nutrients, accelerate growth of the colony, or to increase resilience in hostile environments [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Likewise, the col- lective behavior of assemblies of eukaryotic cells, such as epithelial monolayers and cancer cells, has physiological and pathological implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These include embryoge- nesis, wound healing and tumor metastasis [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Many studies have shown that clustering and collec- tive motion of self-propelled particles (SPPs) are influ- enced by various physical factors, including the packing fraction of the SPPs, nature of the coupling between neighboring SPPs, and the type of motion of a single SPP [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Other physical factors include environmen- tal constraints [3] such as anisotropy of the embedding fluid [22, 23], geometric confinement [24–36] and obsta- cles [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' An interesting effect of circular confinement, for example, is an induced vortical motion of the SPPs ∗ Physical Review E (in press) † Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' mlaradji@memphis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='edu that is concentric with the boundary [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' While in most studies of SPPs’ collective behavior, the substrate is non-patterned, the effect of patterned sub- strates on SPPs collective behavior has recently been in- vestigated in a few studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' For example, it was shown that patterning the substrate, into periodic linear fur- rows, aligns Pseudomonas aeruginosa along the furrows while greatly supresses their migration across them [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Likewise, collective migration of epithelial cells is sub- stantially promoted by linear grooves of patterned sub- strates [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' However, computational investigations of the effect of patterned substrates on SPPs collective behavior are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In this article, we address the effect of substrates, which are partially circularly patterned, on the collective behavior of soft SPPs with the ability to switch their polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In particular, we investigate how a circular confinement that is concentric with the sub- strate’s pattern further influences their collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' MODEL AND METHOD We consider a total number of P SPPs in two dimen- sions, each modeled as a semi-flexible ring polymer com- posed of N beads in a good solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This model was recently introduced by us to investigate SPPs collective behavior on a non-patterned substrate [42] and is a gen- eralization of an earlier model for strongly adsorbed dis- joint ring polymers [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The potential energy of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='11239v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='soft] 26 Jan 2023 2 SPPs is given by Unet= P � l=1 � N � i=1 Ubond � r(l) i,i+1 � + N � i=1 Ubend(r(l) i−1, r(l) i , r(l) i+1) + N � i=1 Uwall � r(l) i � + Uarea � {r(l) i } � + Usub � r(l) p1 , r(l) p2 � � + � l1,l2 � i,j Urep � |r(l1) i − r(l2) j | � , (1) where r(l) i is the coordinate of bead i belonging to SPP l and r(l) i = |r(l) i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The lth SPP has two symmetrically positioned poles with indices p1 = 1 and p2 = N/2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Ubond is a harmonic potential ensuring connectivity between consecutive beads within an SPP and is given by Ubond � r(l) i,i+1 � = 1 2k � r(l) i,i+1 − rb �2 , (2) where k is the spring constant, r(l) i,i+1 = |r(l) i+1−r(l) i | and rb is the preferred bond length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (2), r(l) N,N+1 = r(l) N,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The semi-flexibility of an SPP’s boundary is maintained through a three-body interaction Ubend(r(l) i−1, r(l) i , r(l) i+1) = κ � 1 − cos θ(l) i � , (3) where κ is the bending stiffness of the polymers and cos θ(l) i = r(l) i−1,i·r(l) i+1,i/r(l) i−1,ir(l) i+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (3), r(l) 0 = r(l) N and r(l) N+1 = r(l) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (3) implies that the preferred bend- ing angle of a triplet is 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' To account for the polariza- tion of the SPPs, triplets of beads centered at the pole beads with indices p1 and p2 have a preferred bending angle θp ≤ 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (3) does not allow for pre- ferred angles different from 180◦, beads p1 and p2 are assigned the following slightly different three-body inter- action, which allows for any arbitrary splay angle θs, Ubend(r(l) p−1, r(l) p , r(l) p+1) = 1 2κ′ � cos θ(l) p − cos θs �2 , (4) where κ′ is the bending stiffness at the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Due to the softness of the potential given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (4), we found that achieving the same persistence length of the polymer with this potential as with that given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (3) requires κ′ ≈ 10κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The disjointness of the ring polymers is maintained by the following fully repulsive two-body interaction be- tween any two non-bonded beads Urep (r) = � 1 2ζ � 1 − r rc �2 if r ≤ rc, 0 if r > rc, (5) where ζ and rc are the strength and range of the repulsive interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Finally, the area constraint of each SPP is maintained by the effective potential energy Uarea � {r(l) i } � = 1 2χA0 � �1 − A � {r(l) i } � A0 � � 2 , (6) Pl !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content="# $%&%' (%& + (%' /2 ,- , Non-patterned substrate Patterned substrate Confining wall FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Schematic of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Solid black circle of radius R corresponds the confining wall of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The yellow disk of radius Rp corresponds the region of the substrate that is patterned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The green annulus corresponds to the region of the substrate that is non-patterned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The mid-point of the polarity vector Pl of an arbitrary SPP l is at a distance (rl1 + rl2)/2, where rl1 and rl2 are the coordinates of the two poles (The origin of the coordinate system is at the center of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The effect of the patterned substrate is to reorient the SPP’s polarity through a torque, whose forces are indicated by the green vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In this schematic, the size of the SPP is not to scale with the system size and the size of the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' where χ is the area-stretch modulus, A0 is the SPP’s preferred area, and A � {r(l) i } � is the area enclosed by the SPP’s boundary and depends on the coordinates of the beads belonging to the SPP through the shoelace formula, A � {r(l) i } � = 1 2 ���� N � i=1 � x(l) i y(l) i+1 − x(l) i+1y(l) i � ����, (7) with x(l) N+1 = x(l) 1 and y(l) N+1 = y(l) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Finally, the SPPs are confined within a circle of radius R by the interaction potential Uwall (r) = � εwall (r − R + a)n /an if R − a ≤ r < R, 0 if r < R − a, (8) where εwall and a are the strength and range of this interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We choose n = 4 since this value is large enough to prevent the SPPs from crossing the circular confining wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The main difference between this model and prior models for the collective behavior of elongated self-propelled particles is that the present model accounts for the elongation of the self-propelled particles and their flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is in contrast with previous studies wherein particles are either rigid [44–46] or deformable with high aspect ratio and with practically no account for the enclosed volume of the particles [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We consider the case where a region of the substrate is circularly patterned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Experimentally, this would cor- 3 respond, for example, to a substrate that is circularly grooved [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The effect of the substrate’s pattern on an SPP is to align it along the local direction of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is achieved by a simple effective potential energy between the SPP’s poles that produces a torque on the SPP, Usub � r(l) p1 , r(l) p2 � = ks 2 sin2 ϕl, (9) where ks is the strength of the interaction and ϕl is the angle between the polarity Pl = r(l) p2 − r(l) p1 and the local tangent to a circle of radius (r(l) p1 +r(l) p2 )/2 centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This torque tends to align an SPP’s polarity with the local tangent of a circle centered at the origin and passing by the mid-point of the two poles, as schemati- cally shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We focus on the case where the substrate is patterned only within the region (r ≤ Rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Otherwise, the substrate is uniform (non-patterned) for Rp < r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Each SPP is propelled by a motility force of magnitude FD, along its polarity, that is given by fl(t) = FD (Pl(t)/Pl(t)) g (¯vl(t), Pl(t)) , (10) where g(A, B) = +1 or -1 if A·B > 0 or < 0, respectively, and where ¯vl(t) is the SPP’s average velocity over the time interval [t − τm, t], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ¯vl(t) = 1 τm � t t−τm vl(t′)dt′, (11) with vl(t) = (1/N) �N i=1 v(l) i (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (11), we take τm = τ where τ = rb � µ/ε, rb is the preferred bond length, ε is the energy scale and µ is the bead’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Beads are moved according to a molecular dynamics scheme, ˙r(l) i (t) = v(l) i (t), and µ ˙v(l) i (t) = −∇(l) i Unet + fl(t) N − Γv(l) i (t) +Γ √ 2DΞ(l) i (t), (12) where ∇(l) i = (∂x(l) i , ∂y(l) i , ∂z(l) i ) and v(l) i is the instanta- neous velocity of bead i belonging to SPP l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (12), Γ is the friction coefficient, D is the diffusion coefficient of the beads in the ideal limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' in the absence of inter- actions and beads connectivity), and Ξ(l) i (t) is a random vector that has zero-mean and is δ-correlated for the same particle and same component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Ξ(l) i (t) satisfies ⟨Ξ(l) i (t)⟩ = 0, ⟨Ξ(l1) i,α (t) Ξ(l2) j,β (t′)⟩ = δl1l2δijδαβδ (t − t′) , (13) where α, β = x or y, δnm is the Kronecker delta, and δ(t) is the Dirac delta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The equations of motion are integrated using the velocity-Verlet algorithm with a time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='01τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The numerical value of a component of the random force is given by Ξ(l) i,α = � 3 ∆t �1/2 λ(l) i,α, (14) where λ(l) i,α is a random number generated from a uni- form distribution in the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Each SPP is composed of N = 40 beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The values of the param- eters of the model SPPs, which are kept fixed in the present study, are given by k = 100ε/r2 b, κ = 100ε, κ′ = 1000ε, θs = 120◦, ζ = 50ε, rc = rb, χ = 1ε/r2 b, A0 = 100r2 b, τm = τ, D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='0r2 b/τ, and Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='0µ/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (15) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Effects of Patterned Substrate and Motility Force on SPPs’ Collective Behavior We first focus on the combined effect of the patterned substrate and circular confining wall on the SPPs col- lective behavior at an average packing fraction ¯φ = PA0/πR2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398 with R = 200rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This corresponds to P = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Steady-state snapshot (a) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(A) and Movie 1, at FD = 20ε/rb and non-patterned sub- strate (ks = 0), indicate a small amount of clustering and a weak collective motion, in agreement with prior results [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C) shows that at these conditions, the radial distribution of the SPPs packing fraction, φ(r), is almost uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' As FD is increased to 24ε/rb at ks = 0, the motility force drives many SPPs to the boundary leading to their accumulation as shown by snapshot (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(A) and collective unidirectional vortical motion (see Movie 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is also demonstrated by the time de- pendence of the average tangential velocity of the SPPs in an annulus of thickness 10rb near the boundary (red graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) at ks = 0 and FD = 24ε/rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In con- trast, the SPPs motion in an annulus close to the center is fairly turbulent (blue graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) at ks = 0 and FD = 24ε/rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' SPPs accumulation at the boundary is due to the asymmetry between the effect of the motil- ity force, which drives the SPPs toward the boundary, and thermal effects, which drive the SPPs away from the boundary, and has been observed in earlier studies [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In contrast, although the SPPs that are away from the boundary move collectively in clusters, they do not ex- hibit a net vortical motion, as demonstrated by the fluc- tuations around 0 of the average tangential velocity of the SPPs in the annulus close to the center (blue graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) at ks = 0 and FD = 24ε/rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Interaction between the SPPs and the patterned sub- strate leads to a much richer dynamical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Snap- shots (c) and (d) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(A) and their corresponding tangential velocities vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) show that, at ks = 100ε and FD = 18 or 20ε/rb, the patterned 4 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 (A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 0, &\'= 20)/+, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\'= 20)/+, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\'= 24)/+, + [+,] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 0, &\'= 24)/+, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\'= 22)/+, (C) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\'= 18)/+, _.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2[3] 45 +,/3 b !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 0, &\' = 24)/+, c !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' "= 100), &\' = 18 )/+, d !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\' = 20 )/+, (B) e !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100), &\' = 22)/+, a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 0, &\' = 20)/+, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=" + &' = 24)/+, f !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" = 100) (a) (b) (f) (d) (e) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Panel (A): Steady-state snapshots at (a) FD = 20ε/rb and ks = 0, (b) FD = 24ε/rb and ks = 0, (c) FD = 18ε/rb and ks = 100ε, (d) FD = 20ε/rb and ks = 100ε, (e) FD = 22ε/rb and ks = 100ε, and (f) FD = 24ε/rb and ks = 100ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Panel B: Time dependence of the average tangential velocity for different values of ks and FD corresponding to those in Panel (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The blue (red) graphs correspond to SPPs in the blue (red) annulus, shown in snapshot (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Shaded yellow (green) region corresponds to the regime where the vortices in the patterned and non-patterned regions are in same (opposite) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Panel (C): Radial profiles of the packing fraction, ¯φ at values of FD and ks corresponding to those in Panel (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' All data shown in this figure are at ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398, Rp = 100rb and R = 200rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' substrate and the driving force collectively lead to (1) a tangential alignment of the SPPs in the patterned re- gion, (2) their accumulation at the periphery of the pat- terned region, and (3) their exclusion from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' At ks = 100ε and FD = 20ε/rb, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) and Movie 3 show that the SPPs move as a vortex, in the patterned region of the substrate, with very few reversals in its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In contrast, the SPPs outside the patterned region exhibit a weak collective behavior, as demonstrated by the fact that the SPPs’ average tangential velocity in this region fluctuates around 0 (red graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) at ks = 100ε and FD = 20ε/rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' As FD is further increased to FD = 22 or 24ε/rb, at ks = 100ε, the corresponding snapshots (d) or (e), re- spectively, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(A), show that more SPPs are driven to the confining wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is also demonstrated by increased packing fraction next to the boundary at these values of FD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) shows that, at these values of FD, the SPPs exhibit collective vortical motion in both patterned and non-patterned regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These vor- tices can move either in the same direction (shaded yellow regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(B) and Movie 4) or opposite directions (shaded green regions and Movie 5) with frequent rever- sals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Inspection of the vorticity reversals indicates that they are due to collectively moving clusters in the non- patterned region, which collide with the vortices in the patterned region or in the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The SPPs collectivity is quantified through the vortical order parameter defined as Sv = ⟨| P � l=1 σl|⟩/P, (16) where σl = +1 (-1) if the direction of the tangential ve- locity of SPP l is clockwise (counter-clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 3, which depicts Sv vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' FD at ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398, shows that the substrate pattern shifts the onset of vortical collective motion to smaller values of FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Four distinct regimes in the case of ks = 100ε are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In regime I (FD ≲ 16ε/rb), there is no collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In regime II (16ε/rb ≲ FD ≲ 21ε/rb), the collective behavior is dominated by the patterned region, and is characterized by an almost unidirectional vortical motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C) shows that regime II is also characterized by an in- crease in the maximum of the SPPs packing fraction in the patterned region with increasing FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In regime III (21ε/rb ≲ FD ≲ 25ε/rb), both patterned substrate and confining wall independently promote SPPs collec- 10 15 20 25 30 35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" #$ = 0 #$ = 100( (opposite vorticities) #$ = 100( (same vorticities) I II III IV 67 (/9: 0 50 100 150 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" #$ ( 67 = 20(/9: FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' SV vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' FD at ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398, Rp = 100rb and R = 200rb for ks = 0 (red circles) and ks = 100ε (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Full (open) blue circles correspond to Sv at ks = 100ε where the vortices in the patterned and non-patterned regions have same (opposite) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (Inset) Sv vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ks at FD = 20ε/rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The solid lines are simply guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 10 15 20 25 30 35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 1 1 pt 20 32 (A) Circular boundary Periodic boundary conditions !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' " " ["$] (C) 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='08 " ["$] (B) &\' " " ["$] (D) 125"$ 150"$ 175"$ 25"$ 50"$ 75"$ 100"$ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (A) trajectories of a single SPP starting from a po- sition near the center, for differemt values of FD and ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (B) Radial profile of the radial velocity of the SPPs for the case of a circular confining wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (B) Radial profile of the packing fraction for the case of a circular confining wall (solid line) and PBC (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Data shown in (B) and (C) are in the case of FD = 24ε/rb, ks = 100ε, ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398, Rp = 100rb and R = 200rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (D) Radial profiles of the packing fraction for different values of the radius of the patterned region, Rp, indi- cated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These data correspond to FD = 22ε/rb, ks = 100ε, R = 200rb and ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The vertical dashed lines in (B-D) indicate the location of the boundary between the patterned (left) and non-patterned (right) regions of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' tive motion, and lead to vortical motion in both regions with same or opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This results in a bi- furcation of Sv into two branches: one branch with high values of Sv (solid blue circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 3) where the two vortices have same direction, and a second branch with low values of Sv (open blue circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 3) where the two vortices have opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Regime III marks the beginning of the decrease in the value of the maxi- mum of the SPPs’ packing fraction in the patterned re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Finally, in regime IV (FD ≳ 24ε/rb), the major- ity of the SPPs are accumulated near the confining wall, where they move as a unidirectional vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Interestingly, snapshots (c) to (f) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(A) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C) demonstrate that the patterned substrate in- duces an exclusion zone in the center with a diameter that increases with FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is contrasted with the case of a non-patterned substrate, in which the radial profile of the packing fraction is almost uniform, except at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The source of this exclusion zone, is inferred from simulations of a single SPP (dilute regime) at finite values of ks and FD, starting from a location near the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 4(A) (see Movie 6 as well) shows that the SPP’s trajectory is an outward spiral, with a number of turns that increases with increasing ks or decreasing FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Hence, the motility force and the substrate’s pattern co- operatively drive the SPPs away from the patterned re- gion with a rate that increases with FD and decreases with ks, leading to an exclusion zone in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In addition to the exclusion zone in the center, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C) shows that the radial profile of the packing fraction exhibits a broad peak within the patterned re- gion, and close to the boundary between the patterned and non-patterned regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The emergence of this peak is understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The motion of the SPPs within the patterned region is mainly tangential, while in the non-patterned region (but away from the confining wall), the motion is more turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' As a result vp ⊥ < vn ⊥, where vp ⊥ and vn ⊥ are the averages of the magnitudes of the radial components of the SPPs velocities in the pat- terned and non-patterned regions, respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 4(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Steady state requires that the outflux of the SPPs from the patterned must be equal to the influx of the SPPs from the non-patterned regions to the pat- terned region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' φpvp ⊥,out = φnvn ⊥,in, where φp (φn) is the packing fractions of the SPPs in the patterned (non- patterned) region, close the boundary between the pat- terned and non-patterned regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' vp ⊥,out is the average of the radial component of the velocity of the SPPs out- going from the patterned region at the boundary between the patterned and non-patterned regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Likewise, vn ⊥,in is the average of the radial component of the velocity of the SPPs incoming from the non-patterned region at the boundary between the patterned and non-patterned re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Therefore, mass balance between the outflux and influx of the SPPs across this boundary, at steady state, imposes φp > φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Combined with the fact that the inter- play between the motility force and the torque induced by the patterned substrate, which leads to SPPs exclu- sion from the center, the argument above implies that the radial packing fraction profile must exhibit a peak within the patterned region, and close to the boundary between the patterned and non-patterned regions, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' FD enhances the SPPs outflux from the pat- terned region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' it increases vp ⊥, while it decreases the influx from the non-patterned region, due to increased accumulation of the SPPs near the confining wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' As a result, the size of the exclusion zone increases with FD (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 2(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Elimination of SPPs accumulation at the boundary, through imposing periodic boundary conditions (PBC), enhances SPPs influx from the non- patterned region to the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This leads to a decrease in the size of the exclusion zone, as demon- strated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 4(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The results thus far presented correspond to the case of a radius of the patterned region of the substrate, Rp = 100rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' To infer the effect of the size of the patterned region, we performed a series of simulations in the case of ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='398, FD = 22ε/rb, ks = 100ε, and R = 200rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 4(D) shows the radial profile of the packing fraction of these systems with Rp varying between 25rb and 175rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This figure demonstrates that the diameter of the deple- /r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='ks = 100 /rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ks = 160c ε/rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ks = 100c32206 tion zone increases with Rp, which implies that the size depletion of the SPPs from the middle is also affected by the behavior of the SPPs in the non-patterned region of the substrate, in line with the arguments presented in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Effect of SPPs’ Packing Fraction on their Collective Behavior on a Patterned Substrate We now turn to the effect of SPPs packing fraction on their collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We consider the case where FD = 24ε/rb and ks = 100ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The packing fraction is var- ied by changing the number of SPPs from P = 59 to 540, while the radius of the system is kept fixed at R = 138rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Corresponding Sv vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ¯φ, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5, reveals three main regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' For ¯φ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='3, most SPPs accumulate at the boundary where they move as a unidirectional vor- tex (see Movie 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='3 ≲ ¯φ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8, the amount of SPPs is increased in the patterned region, where they move as a vortex with same direction as that in the boundary layer (see Movie 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5 shows that for ¯φ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8, Sv in- creases monotonically with ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Surprisingly, however, Sv decreases with ¯φ for ¯φ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This decrease is interest- ingly correlated with the disappearance of the exclusion zone in the center as demonstrated by the profiles of the packing fraction in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In fact, the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5 shows that an excess of SPPs at the center is induced at ¯φ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Inspection of movies at ¯φ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 reveals an emergence of reversals in the vorticity (demonstrated by SPPs ve- locities snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(A) and by Movie 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These reversals are quantified by the time dependence of vT (t), defined as the average of the tangential velocity of the SPPs in an annulus of thickness 10rb near the system’s boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(B) shows that vT is essentially con- stant in the case of a non-patterned substrate (ks = 0) at FD = 24ε/rb, indicating a unidirectional vortical mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' At ks = 40ε and same FD, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(B) shows that vT exhibits a single reversal during the time interval [20 000τ, 40 000τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In stark contrast, however, vT ex- hibits many reversals at ks = 100ε and same FD dur- ing the same time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Therefore, at high packing fractions, the rate of vorticity reversals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=', number of reversals per unit of time), κ, increases with increasing ks beyond some threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Likewise, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(C) shows that κ increases with ¯φ for ¯φ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The decrease in Sv at ¯φ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5(B), is simply due to coexis- tence of two vortices with opposite directions during the reversal events, as demonstrated by a series of snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' S1 in Supplemental Information [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Correlations between reversal events are inferred from the power spectrum F(ν), defined as the Fourier trans- form of the velocity autocorrelation f(t) = ⟨vT (t0 + t)vT (t0)⟩, where ν is frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(D) shows that, at ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836, F(ν) is peaked at ν ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This indi- cates that reversal events are weakly correlated at pack- ing fractions around this value of ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6(D) shows 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='9 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='" #$ 0 25 50 75 100 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='9 " % % [%\'] " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='887 " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='861 " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836 " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='803 " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='769 " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='736 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Vortical order parameter vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' packing fraction at ks = 100ε, FD = 24ε/rb, Rp = 100rb and R = 138rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Vor- tical motion is dominated by the circular confining wall at low ¯φ (green region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Both circular confining wall and pat- terned substrate contribute to vortical motion at intermediate ¯φ (blue region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' At high ¯φ, vortical motion exhibits reversals (red region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Inset shows radial packing fraction profiles at different values of ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Steady state snapshots at different pack- ing fractions are shown at the top of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The dashed circles in these snapshots indicate the boundary of the pat- terned region of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' that F(ν) exhibits a well-defined peak at ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Therefore, reversal events of the vorticity become inter- estingly quasi-periodic with increasing ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The emergence of quasi-periodic reversals at high densities is also demon- strated by the time dependence of the tangential velocity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Inspection of Movie 9 shows that vorticity reversals always originate from the center of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This concurs with the fact that vorticity reversals are absent at low packing fractions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' when the exclusion zone is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' To demonstrate that the geometry of the confin- ing wall has a weak effect on vorticity reversals, we per- formed a simulation on a system with a square boundary, of linear size Lx = 400rb, and same circular pattern with ks = 100ε, FD = 24ϵ/rb, ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='887 and Rp = 100rb, and found reversals in the vorticity similar to the case with circular boundary and with about same value of κ, as demonstrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' S2 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Likewise, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' S3 [48] shows that systems with periodic boundary conditions, at same values of FD, ks, ¯φ, Rp and Lx, also exhibit vor- ticity reversals, albeit not as correlated as in the case of circular or square boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is due to the fact the periodic boundary conditions induce more turbulent flow of the SPPs in the non-patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' As stated above, reversals in the vorticity are associ- ated with an increase in SPPs packing fraction at the 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='04 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='9 0 1 2 3 4 5 6 20000 25000 30000 35000 40000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 ks=0 ks=40 ks=100 v (t)[rb/ ] t [ ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' = 37000& 38000& 39000& 40000& B B B B (A) (B) (D) (C) + , &-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (×10-2) 4(5) 5 &-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836 + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='887 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (A) Time-sequence of velocity snapshots showing vorticity reversals at FD = 24ε/rb, ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836, Rp = 100rb, R = 138rb and ks = 100ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (B) Tangential velocity vT (t) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' time at FD = 24ε/rb and ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (C) Rate of vorticity reversals vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ¯φ at ks = 100ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (G) The Fourier transform, F(ν), of the velocity autocorrelation function f(t) = ⟨vT (t0 + t)vT (t0)⟩, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' frequency at ks = 100ε at two high values of the packing fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This is found to also be associated with an in- crease in the misalignment between the SPPs polarities and velocities, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' S4 (A) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This re- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 35000 37500 40000 42500 45000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6 ̅"# $ [&\'/)] $[)] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Time dependence of the tangential velocity of an annulus of thickness 10rb near the system’s boundary for the case of FD = 24ε/rb, Rp = 100rb, R = 200rb and ks = 100ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Top and bottom graphs correspond to ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='836 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='887, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' sults in a high degree of fluctuations in the average of the tangential velocity of the SPPs in the center as op- posed to those away from the center, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' S4 (B) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These increased fluctuations at the center leads some SPPs to move in a direction opposite to that of the vortex, and in some cases these SPPs force neighbor- ing SPPs to follow, leading to the observed intermittent vorticity reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Patterned-substrates induced segregation between fast and slow SPPs Our simulations show that at low and intermediate val- ues of the packing fraction, the SPPs spatial distribution depends on their motility force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' One would therefore ex- pect that patterning the substrate may be used as a tool to spatially separate SPPs, based on their motility force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' To verify this hypothesis, we performed a simulation of a binary system, at an average packing fraction ¯φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6, in which half of the SPPs are slow (with Fd = 20ε/rb) and the other half are fast (with FD = 24ε/rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The two types of SPPs are otherwise identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The packing fraction profiles of the two components and a steady-state snap- shot, depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 8(A) and (B), respectively, show that the fast and slow SPPs mostly segregate such that the fast SPPs are highly concentrated in the patterned region and the slow SPPs are more concentrated in the non-patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' In comparison, the two types of SPPs are mixed in the case where the substrate is fully 8 0 25 50 75 100 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 0 25 50 75 100 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='00 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='#] (A) (B) % !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=" &' = 24+/!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content="# &' = 20+/!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='# Overall packing fraction % !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=" &' = 24+/!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content="# &' = 20+/!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='# Overall packing fraction (C) (D) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (A) Radial profile of the packing fraction in the case of a binary system of fast SPPs, with FD = 24ε/rb (blue) and slow SPPs, with FD = 20ε/rb (red), in the case where the average packing fraction is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='6, ks = 100ε, R = 162rb and Rp = 100rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (B) A snapshot of the binary system at steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Blue and red SPPs correspond to fast and slow SPPs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The dashed vertical line and circle in (A) and (B), respectively, indicate the boundary of the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (C) and (D) same as in (A) and (B), respectively, but in the case of a non-patterned substrate (ks = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' uniform, as shown by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 8(C) and (D), except that the fast SPPs are more concentrated at the confining wall than the slow SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The separation between the fast and slow SPPs shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 8 (A) and (B) is counterintuitive in that the coupling between the pattern of the substrate and the motility force tend to expel the SPPs from the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Therefore, one would expect that the fast SPPs are more concentrated in the non-patterned region and that the slow SPPs are more present in the patterned region, as discussed earlier in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='A, which is op- posite to what is observed from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' 8 (A) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The fact that the patterned substrate is able to segre- gate the SPPs based on their motilities is very interesting and potentially very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' However, an explanation of this phenomenon is lacking at the moment and requires further systematic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This segregation could be understood from a balance of the normal stresses ex- erted by the SPPs at the interface between the patterned and non-patterned regions, using for example the Irving- Kirkwood formalism [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This study is planned to be performed by the authors in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Separation between SPPs may also be induced through differences in their interaction strength with the substrate and possibly the degree of their flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS We showed in this article that a complex collective be- havior is exhibited by SPPs that are confined in a circular geometry and that interact with a circularly patterned substrate, which tends to orient the SPPs polarities with the local tangent of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This collective behav- ior is characterized by SPPs vortical motion, accumula- tion in the outer portion of the patterned region and/or the system boundary, and SPPs exclusion from the cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This collective behavior is enhanced with increasing SPPs driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The size of the exclusion zone is de- termined by an interplay between, on one hand, the com- bined effects of the driving force and the patterned sub- strate, which tends to drive the SPPs outward, and, on the other hand, motion of the SPPs in the non-patterned region of the substrate which drives the SPPs into the patterned region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Interestingly, the vortices in the pat- terned and non-patterned regions, at intermediate values of the SPPs packing fraction, may have same or opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Another interesting feature of this system is that at intermediate packing fractions and intermediate values of the motility force, the radial profile of the packing fraction is non-monotonic, with a peak in the patterned region close to its boundary with the non-patterned re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' A simulation of a binary system, composed of slow and fast SPPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=', SPPs with a low and motility forces, respectively) show that they can be segregated such that the fast SPPs are mostly trapped in the patterned region, while the fast SPPs are mainly in the non-patterned re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This implies that SPPs can be segregated based on their motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' With increasing packing fraction, the exclusion zone in the center disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' High misalignment between the SPPs polarities and tangential velocities, in the center of the system, leads to an increased degree of fluctuations in their tangential velocities and reversals in the vorticity that originate from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Interestingly, these rever- sals become quasi-periodic at high packing fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' It is worth noting that while the system exhibits vorticity reversal at both intermediate and high packing fractions, the mechanisms leading to the two types of reversals are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' The results of the present work implies that circular patterning of the substrate can be used as a tool to guide the motion of SPPs into a collective vortical mo- tion, and that at high packing fractions, can be used to create quasi periodic reversals in their vortical motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We also showed that the patterned substrate is able to segregate a binary mixture of slow and fast SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We expect that SPPs can likewise be segregated based on their degrees of adhesion to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This segrega- tion can be enhanced by further increasing the adhesion strength of the fast SPPs to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We note that the present model of SPPs accounts for details often not accounted for in other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' These include elongation of the self-propelled particles, their flexibility, and enclosed area of the SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' It would of 9 course be very desirable to determine the effects of each of these ingredients on the details of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' There is of course a close connection between the SPP dynamics described here with that of swimming bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' How- ever, it is important to note that the estimated value of the Reynolds number based on the parameters used in this study (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' (15)) is about 1, which is much larger than that of swimming bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Using the present ap- proach to investigate the collective motion of cells such as bacteria requires a much smaller Reynolds number which can be achieved by increasing the value of the drag coef- ficient Γ in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' We plan to investigate the effects of these parameters on the observed phenomena in the present study in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' ACKNOWLEDGEMENTS All simulations were performed on computers of the High Performance Computing Facility of the University of Memphis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' This work was funded by the University of Memphis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' [1] S.' metadata={'source': 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+page_content=' Marcq, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' M`ege, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Kabla, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Lim, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} 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8590 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' [47] ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Duman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Isele-Holder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Elgeti, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Gomp- per, Soft Matter 14, 4483 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' [48] See Supplemental Material at [URL will be inserted by publisher] for further results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Irving and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFIT4oBgHgl3EQfWStC/content/2301.11239v1.pdf'} +page_content=' Kirkwood, J.' metadata={'source': 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index 0000000000000000000000000000000000000000..0608835ddfd8fecb9bbe7ea9ec5f676eb96402f0 --- /dev/null +++ b/M9E2T4oBgHgl3EQfBgYX/content/tmp_files/2301.03602v1.pdf.txt @@ -0,0 +1,835 @@ +arXiv:2301.03602v1 [hep-th] 9 Jan 2023 +MIT-CTP/5518 +D-instanton, threshold corrections, and +topological string. +Manki Kima +aCenter for Theoretical Physics, Department of Physics, Massachusetts Institute of +Technology, Cambridge, MA 02139 +In this note, we prove that the one-loop pfaffian of the non-perturbative super- +potential generated by Euclidean D-branes in type II compactifications on orientifolds +of Calabi-Yau threefolds is determined by the moduli integral of the new supersym- +metric index defined by Cecotti, Fendley, Intriligator, and Vafa. As this quantity can +be computed via topological string theory, Chern-Simons theory, matrix models, or by +solving the holomorphic anomaly equation, this result provides a method to directly +compute the one-loop pfaffian of the non-perturbative superpotential. The relation +between the one-loop pfaffian, threshold corrections to the gauge coupling, and the +one-loop partition function of open topological string theory is discussed. +January 11, 2023 + +1 +Introduction +One of the most pressing questions in string theory is to construct or disprove the +existence of (meta)-stable cosmological vacua with features that resemble our own +universe. As an intermediate step towards understanding non-supersymmetric vacua +of string theory, one can study four-dimensional N = 1 supersymmetric vacua of string +theory. As the vacuum structure of such vacua is determined by the K¨ahler potential +and the superpotential of the low-energy supergravity, it is therefore necessary to +understand how to precisely compute the K¨ahler potential and the superpotential. +In this note, we will study the non-perturbative terms in the superpotential gen- +erated by Euclidean D-branes in type II string theory compactified on orientifolds of +Calabi-Yau threefolds [1]. The non-perturbative superpotential due to a Euclidean +D-brane wrapped on a cycle Γ reads +W ⊃ AΓe−TΓ , +(1.1) +where TΓ is the holomorphic worldvolume action of the D-instanton, and AΓ is the +one-loop pfaffian which we will also oftentimes call the one-loop prefactor. The one- +loop pfaffian AD can depend on moduli fields. Non-perturbative corrections to the +superpotential play prominent roles in moduli stabilization scenarios [2, 3], and particle +physics applications of string theory [4–8].1 +Therefore, it is extremely important to carefully examine and make progress on the +D-instanton effects to the superpotential. To simplify the discussion, we will focus on +Euclidean D-branes that only have the universal zero modes describing their positions +in the non-compact directions and their superpartners. Furthermore, we will assume +that the RR tadpoles are saturated by spacetime-filling D-branes to avoid the technical +complications induced by the RR flux. +Despite its importance, it has been extremely challenging to compute the one-loop +prefactor AD. The technical challenge is in part due to the fact that the na¨ıve scattering +amplitudes involving the D-instantons suffer from IR divergences caused by the zero +modes of the D-instantons. This may explain why most of the existing computations of +the variations of the one-loop prefactor relied on indirect approaches, rather than direct +computations. For example, in heterotic string theory, one can determine the functional +form, but not the overall scale, of the one-loop pfaffian by carefully examining the +locus in moduli space at which the number of the zero modes increases [10, 11]. This +result was conjectured to be generalized to F-theory constructions which admit stable +degenerations [12]. Similarly, in type IIB string theory and F-theory counting of the +zero modes from open strings extended between the D-instanton and spacetime filling +D-brane led to the computation of the one-loop prefactor [13–15], but again the overall +1For review on the D-instanton effects in string compactifications, see [9]. +1 + +scale was left undetermined. The result of [13] is confirmed through hard computations +in toroidal orientifold compactifications [16]. If the non-perturbative superpotential in +question is generated by the D(-1)-instantons, then algebro-geometric techniques can +be used to determine the D(-1)-instanton corrections without ambiguities [17]. From +the M-theoretic approach, one can study the intermediate Jacobian of the Euclidean +M5-branes to constrain the one-loop pfaffian [18].2 But this approach also lacks the +capability to determine the overall scale, and at the same time, it is very difficult to +explicitly compute the intermediate Jacobian explicitly except for very special cases +[23, 24]. +Recently, in a series of works on D-instanton amplitudes from the point of view of +string field theory [25–33], it was realized that string field theory can be used to regu- +late the IR divergence of the D-instanton amplitudes. In particular, the IR-regulated +D-instanton amplitudes were found to be in perfect agreement with the predictions +from mirror symmetry, S-duality, and twistorial description of quaternionic geometries +[31, 32]. Prompted by this success, in [34] the D-instanton correction to the superpo- +tential was computed in terms of the open string spectrum. Because of the lack of the +predictive power of dualities and supersymmetry in 4d N = 1 theories, the one-loop +prefactor computed in [34] has not been crosschecked yet. But, as [34] followed the +same prescription as [31, 32], it is strongly suggestive that the overall normalization +obtained by [34] is correct. It is important to perform an independent crosscheck of +the results of [34]. +But, the results of [34] may not be fully satisfactory because, in Calabi-Yau com- +pactifications, the spectrum of strings is not always accessible. In fact, in the case +of Calabi-Yau compactifications, it is not even clear how to approximate the string +spectrum away from the large volume limit. +Therefore, one can reasonably com- +plain that [34] has replaced a practically-impossible-to-compute-quantity with another +practically-impossible-to-compute-quantity. +In this note, we will show that in fact the one-loop prefactor of the non-perturbative +superpotential is more computable than one na¨ıvely would have thought even if the +access to the spectrum of the Calabi-Yau CFT is limited. +A crucial insight comes from the holomorphic anomaly equations and topological +string amplitudes studied by Bershadsky, Cecotti, Ooguri, and Vafa (BCOV) in a series +of papers [35, 36]. +In [36], it was conjectured that the topological string partition +function computes F-terms in the low-energy supergravity theories derived from string +compactifications. +In the closed string case, this conjecture was proved by direct +computations [37] that indeed the topological string partition function at genus g +2For the computation of the intermediate Jacobian using algebro geometric techniques, see for +example [19–22]. +2 + +computes the following F-term +� +d4θFg(W2)g , +(1.2) +where W is the square of the Weyl superfield of N = 2 supergravity multiplet. The +open string version of the conjecture that the open string topological partition function +with h holes computes the F-term of the form +� +d2θF0,h(Tr(W)2)h−1 , +(1.3) +where W is the chiral superfield for the gauge field strength, was confirmed in type +I string theory [38]. In particular, at the one-loop level h = 2, the open topological +partition function is conjectured to compute the threshold correction to the gauge +coupling3 and the one-loop partition function is written as +F0,2 = +� dt +t TrR +� +(−1)FFqL0− 3 +8 +� +, +(1.4) +which is the moduli integral of the new supersymmetric index defined by [41], which +we will call the CFIV index. Although it is conceivable that the open string version +of the BCOV conjecture should hold in all type II compactifications on orientifolds of +Calabi-Yau manifolds, this is not yet confirmed. +The other thread of insight comes from the relationship between the threshold cor- +rection to the gauge coupling on a spacetime filling D-brane and the one-loop prefactor +AΓ of the D-instanton superpotential [4, 5]. Because the D-instanton can be understood +as a gauge instanton on a spacetime filling D-brane [42–45], it is natural to conjecture +that the exponentiated threshold correction is equivalent to the one-loop prefactor of +the D-instanton superpotential. And in fact, in the type II compactifications on orien- +tifolds of Calabi-Yau threefolds, it was proven in [34] that the exponentiated threshold +correction to a probe spacetime filling D-brane wrapped on a cycle Γ is exactly the +same as the one-loop prefactor AΓ.4 +Continuing this train of thought, it is then only natural to conjecture that the one- +loop pfaffian of the non-perturbative superpotential is determined by the CFIV index. +This possibility was already contemplated in [46]. In this note, with the help of the +results of [34] and the character formulas of the extended N = 2 superconformal algebra +studied in [47–49], we will prove that the one-loop prefactor of the nonperturbative +3In the context of heterotic string compactifications with the standard embedding [39], it was +proven in [40] that the closed topological partition function computes the difference in gauge threshold +corrections. +4In [4, 5], this statement was shown for the annuli contributions, but not for the M¨obius strip +contribution. +3 + +superpotential is determined by the new supersymmetric index. This also proves the +BCOV conjecture at the one-loop level for D-brane gauge theories with no matter. +As the moduli integral of the CFIV index is computable via open topological string +theory and holomorphic anomaly equations [46, 50–52], this result paves a way to direct +computation of the one-loop pfaffian AΓ. +This note is organized as follows. In §2, we prove that the one-loop pfaffian is +determined by the CFIV index. In §3 we conclude. In appendices, we collect useful +formulas. In §A, we collect useful formulas involving the Jacobi theta functions and +the Dedekind eta function. In §B, we collect the character formulas of the extended +N = 2 superconformal algebra. +2 +D-instanton superpotential and the CFIV index. +We shall study type II string theory compactified on a Calabi-Yau threefold X. The +worldsheet CFT is decomposed into the b, c, β, γ ghost CFT, the free field CFT with +central charge (c, ¯c) = (6, 6) and (1, 1) supersymmetry describing the four non-compact +directions, and strongly coupled CFT with central charge (c, ¯c) = (9, 9) and (N , ¯ +N) = +(2, 2) supersymmetry describing the Calabi-Yau non-linear sigma model. We will often- +times denote the strongly coupled Calabi-Yau CFT by internal CFT. We will consider +an orientifolding of this worldsheet CFT. The details of the orientifolding can be found +in [34]. +We study a Euclidean D-brane wrapped on a cycle Γ. We will assume that the +Euclidean D-brane only has the universal zero modes and none other to ensure that +the non-perturbative superpotential is generated. In [34], by carefully studying the +D-instanton scattering amplitudes and its relation to the one-loop partition function +of open string field theory, the non-perturbative superpotential generated by the Eu- +clidean D-brane was determined5 +|WΓ| = +κ3 +4 +16π2e−K/2K0Re(TΓ)|e−TΓ| , +(2.1) +where TΓ is the disk level effective action of the D-instanton, K is the tree-level K¨ahler +potential, and we define +K0 := lim +ǫ→0 lim +ǫ′→0 exp +�� 1/ǫ +ǫ′ +dt +2tZA + +� 1/ǫ +ǫ′/4 +dt +2tZM + 3 +� 1/ǫ +0 +dt +2t(e−2πt − 1) +� +. +(2.2) +In (2.2), ZA is the sum of annuli diagrams with at least one end on the D-instanton, +and ZM is the M¨obius strip diagram with the end on the D-instanton. +The extra +factor 1 +2 was introduced due to the orientifold projection. The last term acts as the +5For earlier work, see [6]. +4 + +IR regulator for the IR divergence that arises from the zero modes of the D-instanton. +Therefore, to compute the non-perturbative superpotential precisely, we must compute +the annuli diagrams and the M¨obius strip diagram. This is the focus of this section. +Let us now study the one-loop diagrams with one end on the D-instanton and +the other end on a spacetime filling D-brane.6 In [34], the contribution from an orbit +generated by the integral spectral flow whose highest weight state is a massive state +with (h, Q) of N = 2 superconformal algebra was determined to be +Z(h,Q) +A += (−1)Q +2 +qh− 1+Q +4 ϑ1−Q,0(2τ) . +(2.3) +We can rewrite (2.3) as +(−1)Q+1 +1 +4πη(τ)3qh− 1+Q +4 � +−2πη(τ)3� +ϑ1−Q,0(2τ) . +(2.4) +By using an identity +∂zϑ11(z|τ)|z=0 = −2πη(τ)3 , +(2.5) +we conclude that the following identity holds +Z(h,Q) +A += (−1)Q+1 +4πη(τ)3 qh− 1+Q +4 ϑ1−Q,0(2τ)∂zϑ1,1(z|τ)|z=0 . +(2.6) +Because the character formula in the R sector with the GSO projector insertion is +given by +g(h,Q) +αβ +(τ) = e−iπαβ/2(−1)βQqh− 1+Q +4 +η(τ)3 ϑα,β(τ)ϑα+Q,0(2τ) , +(2.7) +one can easily prove that +1 +2πi∂zg(h,Q) +11 +(z, τ)|z=0 := (−1)Q+1 +2πη(τ)3 qh− 1+Q +4 ϑ1−Q,0(2τ)∂zϑ1,1(z|τ)|z=0 . +(2.8) +Because we assumed that the only zero modes of the D-instanton are the universal +zero modes, massless representations of the annuli diagrams are absent. As the sum of +1 +2πi∂zg(h,Q) +11 +(z, τ)|z=0 is by definition the new supersymmetric index defined by Cecotti, +Fendley, Intriligator, and Vafa [41] +TrR +� +(−1)FFqL0− 3 +8 +� += +� +(h,Q) +1 +2πi∂zg(h,Q) +11 +|z=0 +(2.9) +6The contribution from the annulus diagram with both ends on the D-instanton vanishes due to +higher supersymmetry. This was also proven in [34]. Therefore, we will not explicitly consider this +diagram in this note. +5 + +we arrive at one of our main equations +ZA = +� +(h,Q) +Z(h,Q) +A += TrR +� +(−1)FFqL0− 3 +8 +� +, +(2.10) +where the trace is taken over the states of the internal CFT. As a result, we obtained +that the one-loop partition function is precisely the new supersymmetric index! +We now prove a similar statement for the M¨obius strip diagram. As was proven in +[34], the contribution from a massive representation with (h, Q) to the M¨obius diagram +is given by +Z(h,Q) +M += (−1)1−Qα(h,Q)ˆqh− 1+Q +4 ϑ1−Q,0(2ˆτ) , +(2.11) +where α(h,Q) is a phase induced by the orientifold action, ˆτ = τ + 1/2, and ˆq = +exp(2πiˆτ). We now again see that (2.11) is the same as +− α(h,Q) +2πi ∂zg1,1(z, ˆτ)(h,Q)|z=0 . +(2.12) +Unlike in the case of annuli diagrams, we have included an extra negative sign because +the orientifold action flips the sign of the vacuum. As we assumed that the only zero +modes of the D-instanton are the universal zero modes, we only need to consider the +vacuum representation among the massless representations. Because the character of +the vacuum representation can be written in terms of the character formulas of massive +representations, c.f. (B.16), we again reach the same conclusion. Thus, we conclude +that the following equation holds +ZM = +� +(h,Q) +Z(h,Q) +M += TrR +� +(−1)FFΩqL0− 3 +8 +� +, +(2.13) +where the trace is taken over the states of the internal CFT, and Ω is the orientifold +projection. +Now let us perform a consistency check of the formula (2.13). The most basic +check is to confirm the consistency of the zero modes contributions to (2.13). As we +assumed that the only zero modes of the D-instanton are four bosonic and two fermionic +zero modes, we expect to see 3 in ZM in t → ∞ limit. Let us check this conclusion from +equations (2.11) and (2.12). As was studied in [34], α for the vacuum representation was +determined to be −1, and therefore the contribution from the vacuum representation +is given by +ˆq− 1 +4ϑ1,0(2ˆτ) + ϑ0,0(2ˆτ) . +(2.14) +As in ˆq → 0 limit, ϑ1,0(2ˆτ) = 2ˆq1/4 and ϑ0,0(2ˆτ) = 1, we correctly reproduce 3 in +t → ∞ limit. +Now we shall attempt to reproduce this result by a manual computation in the +6 + +R-sector using (2.13). There are two massless states in the R-sector, and both of which +belong to the orbit connected to the vacuum representation by the half-integral spectral +flow. The massless states are the one with (h, Q) = (3/8, 3/2), and the other with +(h, Q) = (3/8, −3/2). Let XR and ˜XR be the integral spectral flow operators in the R- +sector. We choose a convention such that XR is connected to X with (h, Q) = (3/2, 3), +and ¯XR is connected to ˜X with (h, Q) = (3/2, −3). Then, we have [47, 48] +1 +√ +2XR +0 +���� +3 +8, −3 +2 +� += +���� +3 +8, 3 +2 +� +, +(2.15) +and +1 +√ +2 +˜XR +0 +���� +3 +8, 3 +2 +� += +���� +3 +8, −3 +2 +� +. +(2.16) +As one can choose a convention of the orientifolding, without loss of generality, such +that the orientifolding commutes with X0 and ˜X0 [34], we conclude that the phases +generated by the orientifold action are the same between the states (3/8, 3/2) and +(3/8, −3/2). This leads to a manual computation of the contribution of the massless +state +TrR +� +(−1)FFΩqL0− 3 +8 +� += (−1)−3/2(3/2 − (−3/2)) = 3 , +(2.17) +therefore reproducing 3.7 +3 +Conclusions +In this note, we proved that the one-loop prefactor AΓ of the non-perturbative su- +perpotential is determined by the CFIV index. Because the moduli integral of the +CFIV index can be computed via various techniques including topological string the- +ory, holomorphic anomaly equations, Chern-Simons theory, and matrix models, the +results of this note provide a principled way to evaluate the one-loop prefactor in +generic Calabi-Yau compactifications. +There are a few interesting future directions one can pursue. +• The most imminent next step is to compute the one-loop prefactor of the D- +instanton superpotential in explicit examples. One imminent technical challenge +is to understand the universal behavior of the holomorphic ambiguities around +singular points in the moduli space for annuli and the M¨obius strip diagrams. +• In [18], Witten proved that if a Euclidean M5-brane wrapped on a divisor in a +Calabi-Yau fourfold has a trivial intermediate Jacobian, then the one-loop prefac- +tor does not depend on moduli in the absence of spacetime filling M2-branes. In +7We thank Ashoke for illuminating discussions. +7 + +type IIB compactification on O3/O7-orientifolds, this implies that Euclidean D3- +branes and gaugino condensations which are dual to such Euclidean M5-branes +generate the non-perturbative superpotential terms that do not depend on mod- +uli in the absence of D3-brane contributions. As this statement is inherently +topological, it may be possible to confirm it by using this paper’s result. +• As the one-loop prefactor is computed in heterotic string compactifications [10, +11] and its extension to F-theory compactifications [12], it would be interesting +to crosscheck the results via heterotic/type II dualities. +Acknowledgements +The work of MK was supported by the Pappalardo fellowship. We thank Ashoke Sen +for their valuable comments. We thank Atakan Hilmi Fırat, Liam McAllister, Jakob +Moritz, and Andreas Schachner for discussions. +8 + +A +The Jacobi theta functions +In this section, we collect useful formulas for the Jacobi theta functions and the char- +acter formulas for N = 2 superconformal algebra. In this paper, we will mostly follow +the conventions of [34]. We define +ϑα,β(z|τ) = +� +n∈Z+ α +2 +eiπnβqn2/2yn , +q = e2πiτ , +y = e2πiz . +(A.1) +For ϑα,β(0|τ), we use a shorthand notation ϑα,β(τ) := ϑα,β(0|τ). We collect ϑα,β for +(α, β) = (0, 0), (0, 1), (1, 0), (1, 1) +ϑ0,0(z|τ) = +∞ +� +n=1 +(1 − qn) +� +1 + (y + y−1)qn− 1 +2 + q2n−1� +, +(A.2) +ϑ0,1(z|τ) = +∞ +� +n=1 +(1 − qn) +� +1 − (y + y−1)qn− 1 +2 + q2n−1� +, +(A.3) +ϑ1,0(z|τ) =q +1 +8(y +1 +2 + y− 1 +2) +∞ +� +n=1 +(1 − qn) +� +1 + (y + y−1)qn + q2n� +, +(A.4) +ϑ1,1(z|τ) =iq +1 +8(y +1 +2 − y− 1 +2) +∞ +� +n=1 +(1 − qn) +� +1 − (y + y−1)qn + q2n� +. +(A.5) +The Jacobi theta functions admit quasi-periodicity, +ϑα,β +� +z + 1 +2|τ +� +=ϑα,β+1(z|τ) , +(A.6) +ϑα,β +� +z + τ +2|τ +� +=e−iπβ/2q− 1 +8y− 1 +2ϑα+1,β(z|τ) , +(A.7) +ϑα+2,β(z|τ) =ϑα,β(z|τ) , +(A.8) +ϑα,β+2(z|τ) =eiαπϑα,β(z|τ) . +(A.9) +Theta functions satisfy so-called the Jacobi identity +ϑ0,0(τ)4 − ϑ0,1(τ)4 − ϑ1,0(τ)4 = 0 . +(A.10) +We also record useful relations between the Jacobi theta functions and the Dedekind +eta function. We define the Dedekind eta function as +η(τ) = q +1 +24 +∞ +� +n=1 +(1 − qn) . +(A.11) +9 + +The Dedekind eta function satisfies the following identities +∂zϑ1,1(z|τ)|z=0 = −2πη(τ)3 , +(A.12) +and +ϑ1,0 =2η(2τ)2 +η(τ) +, +(A.13) +ϑ0,1 =η( 1 +2τ)2 +η(τ) , +(A.14) +ϑ0,0 = +η(τ)5 +η( 1 +2τ)2η(2τ)2 . +(A.15) +B +Character formulas for N = 2 superconformal algebra +In this section, we will summarize important properties of N = 2 superconformal +algebra and its associated representation theory [47–49, 53]. In this section will focus +on the left moving sector. The superconformal algebra of the right moving sector can +be obtained by taking the complex conjugate of the algebra of the left moving sector. +As we are interested in Calabi-Yau threefold compactifications, we will restrict to c = 9. +We first collect OPEs for superconformal generators, the energy-momentum tensor +T, super-currents G and ˜G, and U(1) current I, +T(z)T(w) = +c +2(z − w)4 + 2T(w) +(z − w)2 + ∂T(w) +z − w + . . . , +(B.1) +I(z)I(w) = +c +3(z − w)2 + . . . , +(B.2) +I(z)G(w) = +1 +z − wG(w) + . . . , +(B.3) +I(z) ˜G(w) = − +1 +z − w +˜G(w) + . . . , +(B.4) +G(z) ˜G(w) = +2c +3(z − w)3 + +2I(w) +(z − w)2 + +1 +z − w(∂I(w) + 2T(w)) + . . . , +(B.5) +G(z)G(w) =regular , +(B.6) +˜G(z) ˜G(w) =regular . +(B.7) +As was studied in [47], it was shown that N = 2 superconformal algebra extends to +the extended superconformal algebra by including the spectral flow generators X, ˜X +and their superpartners Y, ˜Y .8 +We now summarize the character formulas for the extended superconformal al- +gebra. The character is defined as the partition function of the orbit, generated by +8X and ˜X generate integral shifts of the spectral flow. +10 + +integral spectral flow, of an irreducible representation of the extended superconformal +algebra. For an irreducible representation r, we define the character to be +ch(r) +• +:= Tr•,r +� +qL0− 3 +8yI0� +, +(B.8) +where • can be either Neveu-Schwarz (NS) sector or Ramond (R) sector. We will use +a collection of shorthand notations +g(r) +00 := TrNS,r +� +qL0− 3 +8 +� +, +g(r) +01 := TrNS,r +� +qL0− 3 +8(−1)I0� +, +(B.9) +g(r) +10 := TrR,r +� +qL0− 3 +8 +� +, +g(r) +11 := TrR,r +� +qL0− 3 +8(−1)I0� +. +(B.10) +We will, from now on, identify +F ≡ I0 . +(B.11) +We will first study massive representations. +Because representations in the R +sector can be brought to representations in the NS sector by half-integral spectral +flows, we will focus on irreducible representations in the NS sector. In the NS sector, +the highest weight state of any massive state is constrained to have Q = −1, 0, 1, where +h > |Q|/2. Because the character formula for −Q is the same as the character formula +for Q, we will only consider Q > 0 for simplicity. As a massive state is labeled by +(h, Q), we will denote a massive state by (h, Q). The character formula for all sectors +takes the form +g(h,Q) +αβ += q +3α +8 g +�ατ + β +2 +, τ; h, Q +� +, +(B.12) +where we define +g(z, τ; h, Q) := qh− 1+Q2 +4 +η(τ) +f1,0(z, τ)f2,Q(z, τ) , +(B.13) +fk,Q(z, τ) := +1 +η(τ)q +Q2 +2k yQϑ0,0(kz + Qτ|kτ) . +(B.14) +(B.12) admits a simpler form9 +g(h,Q) +αβ += e−iπαβ/2(−1)βQqh− 1+Q +4 +η(τ)3 ϑα,β(τ)ϑα+Q,0(2τ) . +(B.15) +Let’s now study massless representations. There are three kinds: the vacuum +representation (vac), (+), and (−). The vaccum representation has (h, Q) = (0, 0). +(+) has (h, Q) = (1/2, 1), and (−) has (h, Q) = (1/2, −1). 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B 315 (1989) 193–221. +15 + diff --git a/M9E2T4oBgHgl3EQfBgYX/content/tmp_files/load_file.txt b/M9E2T4oBgHgl3EQfBgYX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..deb49c109e0a5c1abaa375dbc2e6f6e37cb0ef1b --- /dev/null +++ b/M9E2T4oBgHgl3EQfBgYX/content/tmp_files/load_file.txt @@ -0,0 +1,577 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf,len=576 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='03602v1 [hep-th] 9 Jan 2023 MIT-CTP/5518 D-instanton, threshold corrections, and topological string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Manki Kima aCenter for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 In this note, we prove that the one-loop pfaffian of the non-perturbative super- potential generated by Euclidean D-branes in type II compactifications on orientifolds of Calabi-Yau threefolds is determined by the moduli integral of the new supersym- metric index defined by Cecotti, Fendley, Intriligator, and Vafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As this quantity can be computed via topological string theory, Chern-Simons theory, matrix models, or by solving the holomorphic anomaly equation, this result provides a method to directly compute the one-loop pfaffian of the non-perturbative superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The relation between the one-loop pfaffian, threshold corrections to the gauge coupling, and the one-loop partition function of open topological string theory is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' January 11, 2023 1 Introduction One of the most pressing questions in string theory is to construct or disprove the existence of (meta)-stable cosmological vacua with features that resemble our own universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As an intermediate step towards understanding non-supersymmetric vacua of string theory, one can study four-dimensional N = 1 supersymmetric vacua of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As the vacuum structure of such vacua is determined by the K¨ahler potential and the superpotential of the low-energy supergravity, it is therefore necessary to understand how to precisely compute the K¨ahler potential and the superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In this note, we will study the non-perturbative terms in the superpotential gen- erated by Euclidean D-branes in type II string theory compactified on orientifolds of Calabi-Yau threefolds [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The non-perturbative superpotential due to a Euclidean D-brane wrapped on a cycle Γ reads W ⊃ AΓe−TΓ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='1) where TΓ is the holomorphic worldvolume action of the D-instanton, and AΓ is the one-loop pfaffian which we will also oftentimes call the one-loop prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The one- loop pfaffian AD can depend on moduli fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Non-perturbative corrections to the superpotential play prominent roles in moduli stabilization scenarios [2, 3], and particle physics applications of string theory [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='1 Therefore, it is extremely important to carefully examine and make progress on the D-instanton effects to the superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' To simplify the discussion, we will focus on Euclidean D-branes that only have the universal zero modes describing their positions in the non-compact directions and their superpartners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Furthermore, we will assume that the RR tadpoles are saturated by spacetime-filling D-branes to avoid the technical complications induced by the RR flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Despite its importance, it has been extremely challenging to compute the one-loop prefactor AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The technical challenge is in part due to the fact that the na¨ıve scattering amplitudes involving the D-instantons suffer from IR divergences caused by the zero modes of the D-instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This may explain why most of the existing computations of the variations of the one-loop prefactor relied on indirect approaches, rather than direct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' For example, in heterotic string theory, one can determine the functional form, but not the overall scale, of the one-loop pfaffian by carefully examining the locus in moduli space at which the number of the zero modes increases [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This result was conjectured to be generalized to F-theory constructions which admit stable degenerations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Similarly, in type IIB string theory and F-theory counting of the zero modes from open strings extended between the D-instanton and spacetime filling D-brane led to the computation of the one-loop prefactor [13–15], but again the overall 1For review on the D-instanton effects in string compactifications, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 1 scale was left undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The result of [13] is confirmed through hard computations in toroidal orientifold compactifications [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' If the non-perturbative superpotential in question is generated by the D(-1)-instantons, then algebro-geometric techniques can be used to determine the D(-1)-instanton corrections without ambiguities [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' From the M-theoretic approach, one can study the intermediate Jacobian of the Euclidean M5-branes to constrain the one-loop pfaffian [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2 But this approach also lacks the capability to determine the overall scale, and at the same time, it is very difficult to explicitly compute the intermediate Jacobian explicitly except for very special cases [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Recently, in a series of works on D-instanton amplitudes from the point of view of string field theory [25–33], it was realized that string field theory can be used to regu- late the IR divergence of the D-instanton amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In particular, the IR-regulated D-instanton amplitudes were found to be in perfect agreement with the predictions from mirror symmetry, S-duality, and twistorial description of quaternionic geometries [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Prompted by this success, in [34] the D-instanton correction to the superpo- tential was computed in terms of the open string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because of the lack of the predictive power of dualities and supersymmetry in 4d N = 1 theories, the one-loop prefactor computed in [34] has not been crosschecked yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' But, as [34] followed the same prescription as [31, 32], it is strongly suggestive that the overall normalization obtained by [34] is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' It is important to perform an independent crosscheck of the results of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' But, the results of [34] may not be fully satisfactory because, in Calabi-Yau com- pactifications, the spectrum of strings is not always accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In fact, in the case of Calabi-Yau compactifications, it is not even clear how to approximate the string spectrum away from the large volume limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Therefore, one can reasonably com- plain that [34] has replaced a practically-impossible-to-compute-quantity with another practically-impossible-to-compute-quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In this note, we will show that in fact the one-loop prefactor of the non-perturbative superpotential is more computable than one na¨ıvely would have thought even if the access to the spectrum of the Calabi-Yau CFT is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' A crucial insight comes from the holomorphic anomaly equations and topological string amplitudes studied by Bershadsky, Cecotti, Ooguri, and Vafa (BCOV) in a series of papers [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In [36], it was conjectured that the topological string partition function computes F-terms in the low-energy supergravity theories derived from string compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In the closed string case, this conjecture was proved by direct computations [37] that indeed the topological string partition function at genus g 2For the computation of the intermediate Jacobian using algebro geometric techniques, see for example [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 2 computes the following F-term � d4θFg(W2)g , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2) where W is the square of the Weyl superfield of N = 2 supergravity multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The open string version of the conjecture that the open string topological partition function with h holes computes the F-term of the form � d2θF0,h(Tr(W)2)h−1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='3) where W is the chiral superfield for the gauge field strength, was confirmed in type I string theory [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In particular, at the one-loop level h = 2, the open topological partition function is conjectured to compute the threshold correction to the gauge coupling3 and the one-loop partition function is written as F0,2 = � dt t TrR � (−1)FFqL0− 3 8 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='4) which is the moduli integral of the new supersymmetric index defined by [41], which we will call the CFIV index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Although it is conceivable that the open string version of the BCOV conjecture should hold in all type II compactifications on orientifolds of Calabi-Yau manifolds, this is not yet confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The other thread of insight comes from the relationship between the threshold cor- rection to the gauge coupling on a spacetime filling D-brane and the one-loop prefactor AΓ of the D-instanton superpotential [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because the D-instanton can be understood as a gauge instanton on a spacetime filling D-brane [42–45], it is natural to conjecture that the exponentiated threshold correction is equivalent to the one-loop prefactor of the D-instanton superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' And in fact, in the type II compactifications on orien- tifolds of Calabi-Yau threefolds, it was proven in [34] that the exponentiated threshold correction to a probe spacetime filling D-brane wrapped on a cycle Γ is exactly the same as the one-loop prefactor AΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='4 Continuing this train of thought, it is then only natural to conjecture that the one- loop pfaffian of the non-perturbative superpotential is determined by the CFIV index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This possibility was already contemplated in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In this note, with the help of the results of [34] and the character formulas of the extended N = 2 superconformal algebra studied in [47–49], we will prove that the one-loop prefactor of the nonperturbative 3In the context of heterotic string compactifications with the standard embedding [39], it was proven in [40] that the closed topological partition function computes the difference in gauge threshold corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 4In [4, 5], this statement was shown for the annuli contributions, but not for the M¨obius strip contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 3 superpotential is determined by the new supersymmetric index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This also proves the BCOV conjecture at the one-loop level for D-brane gauge theories with no matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As the moduli integral of the CFIV index is computable via open topological string theory and holomorphic anomaly equations [46, 50–52], this result paves a way to direct computation of the one-loop pfaffian AΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This note is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In §2, we prove that the one-loop pfaffian is determined by the CFIV index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In §3 we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In appendices, we collect useful formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In §A, we collect useful formulas involving the Jacobi theta functions and the Dedekind eta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In §B, we collect the character formulas of the extended N = 2 superconformal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 2 D-instanton superpotential and the CFIV index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We shall study type II string theory compactified on a Calabi-Yau threefold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The worldsheet CFT is decomposed into the b, c, β, γ ghost CFT, the free field CFT with central charge (c, ¯c) = (6, 6) and (1, 1) supersymmetry describing the four non-compact directions, and strongly coupled CFT with central charge (c, ¯c) = (9, 9) and (N , ¯ N) = (2, 2) supersymmetry describing the Calabi-Yau non-linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We will often- times denote the strongly coupled Calabi-Yau CFT by internal CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We will consider an orientifolding of this worldsheet CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The details of the orientifolding can be found in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We study a Euclidean D-brane wrapped on a cycle Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We will assume that the Euclidean D-brane only has the universal zero modes and none other to ensure that the non-perturbative superpotential is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In [34], by carefully studying the D-instanton scattering amplitudes and its relation to the one-loop partition function of open string field theory, the non-perturbative superpotential generated by the Eu- clidean D-brane was determined5 |WΓ| = κ3 4 16π2e−K/2K0Re(TΓ)|e−TΓ| , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='1) where TΓ is the disk level effective action of the D-instanton, K is the tree-level K¨ahler potential, and we define K0 := lim ǫ→0 lim ǫ′→0 exp �� 1/ǫ ǫ′ dt 2tZA + � 1/ǫ ǫ′/4 dt 2tZM + 3 � 1/ǫ 0 dt 2t(e−2πt − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2) In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2), ZA is the sum of annuli diagrams with at least one end on the D-instanton, and ZM is the M¨obius strip diagram with the end on the D-instanton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The extra factor 1 2 was introduced due to the orientifold projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The last term acts as the 5For earlier work, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 4 IR regulator for the IR divergence that arises from the zero modes of the D-instanton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Therefore, to compute the non-perturbative superpotential precisely, we must compute the annuli diagrams and the M¨obius strip diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This is the focus of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Let us now study the one-loop diagrams with one end on the D-instanton and the other end on a spacetime filling D-brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='6 In [34], the contribution from an orbit generated by the integral spectral flow whose highest weight state is a massive state with (h, Q) of N = 2 superconformal algebra was determined to be Z(h,Q) A = (−1)Q 2 qh− 1+Q 4 ϑ1−Q,0(2τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='3) We can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='3) as (−1)Q+1 1 4πη(τ)3qh− 1+Q 4 � −2πη(τ)3� ϑ1−Q,0(2τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='4) By using an identity ∂zϑ11(z|τ)|z=0 = −2πη(τ)3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='5) we conclude that the following identity holds Z(h,Q) A = (−1)Q+1 4πη(τ)3 qh− 1+Q 4 ϑ1−Q,0(2τ)∂zϑ1,1(z|τ)|z=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='6) Because the character formula in the R sector with the GSO projector insertion is given by g(h,Q) αβ (τ) = e−iπαβ/2(−1)βQqh− 1+Q 4 η(τ)3 ϑα,β(τ)ϑα+Q,0(2τ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='7) one can easily prove that 1 2πi∂zg(h,Q) 11 (z, τ)|z=0 := (−1)Q+1 2πη(τ)3 qh− 1+Q 4 ϑ1−Q,0(2τ)∂zϑ1,1(z|τ)|z=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='8) Because we assumed that the only zero modes of the D-instanton are the universal zero modes, massless representations of the annuli diagrams are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As the sum of 1 2πi∂zg(h,Q) 11 (z, τ)|z=0 is by definition the new supersymmetric index defined by Cecotti, Fendley, Intriligator, and Vafa [41] TrR � (−1)FFqL0− 3 8 � = � (h,Q) 1 2πi∂zg(h,Q) 11 |z=0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='9) 6The contribution from the annulus diagram with both ends on the D-instanton vanishes due to higher supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This was also proven in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Therefore, we will not explicitly consider this diagram in this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 5 we arrive at one of our main equations ZA = � (h,Q) Z(h,Q) A = TrR � (−1)FFqL0− 3 8 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='10) where the trace is taken over the states of the internal CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As a result, we obtained that the one-loop partition function is precisely the new supersymmetric index!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We now prove a similar statement for the M¨obius strip diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As was proven in [34], the contribution from a massive representation with (h, Q) to the M¨obius diagram is given by Z(h,Q) M = (−1)1−Qα(h,Q)ˆqh− 1+Q 4 ϑ1−Q,0(2ˆτ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='11) where α(h,Q) is a phase induced by the orientifold action, ˆτ = τ + 1/2, and ˆq = exp(2πiˆτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We now again see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='11) is the same as − α(h,Q) 2πi ∂zg1,1(z, ˆτ)(h,Q)|z=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='12) Unlike in the case of annuli diagrams, we have included an extra negative sign because the orientifold action flips the sign of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As we assumed that the only zero modes of the D-instanton are the universal zero modes, we only need to consider the vacuum representation among the massless representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because the character of the vacuum representation can be written in terms of the character formulas of massive representations, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='16), we again reach the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Thus, we conclude that the following equation holds ZM = � (h,Q) Z(h,Q) M = TrR � (−1)FFΩqL0− 3 8 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13) where the trace is taken over the states of the internal CFT, and Ω is the orientifold projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Now let us perform a consistency check of the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The most basic check is to confirm the consistency of the zero modes contributions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As we assumed that the only zero modes of the D-instanton are four bosonic and two fermionic zero modes, we expect to see 3 in ZM in t → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Let us check this conclusion from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As was studied in [34], α for the vacuum representation was determined to be −1, and therefore the contribution from the vacuum representation is given by ˆq− 1 4ϑ1,0(2ˆτ) + ϑ0,0(2ˆτ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='14) As in ˆq → 0 limit, ϑ1,0(2ˆτ) = 2ˆq1/4 and ϑ0,0(2ˆτ) = 1, we correctly reproduce 3 in t → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Now we shall attempt to reproduce this result by a manual computation in the 6 R-sector using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' There are two massless states in the R-sector, and both of which belong to the orbit connected to the vacuum representation by the half-integral spectral flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The massless states are the one with (h, Q) = (3/8, 3/2), and the other with (h, Q) = (3/8, −3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Let XR and ˜XR be the integral spectral flow operators in the R- sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We choose a convention such that XR is connected to X with (h, Q) = (3/2, 3), and ¯XR is connected to ˜X with (h, Q) = (3/2, −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Then, we have [47, 48] 1 √ 2XR 0 ���� 3 8, −3 2 � = ���� 3 8, 3 2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='15) and 1 √ 2 ˜XR 0 ���� 3 8, 3 2 � = ���� 3 8, −3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='16) As one can choose a convention of the orientifolding, without loss of generality, such that the orientifolding commutes with X0 and ˜X0 [34], we conclude that the phases generated by the orientifold action are the same between the states (3/8, 3/2) and (3/8, −3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This leads to a manual computation of the contribution of the massless state TrR � (−1)FFΩqL0− 3 8 � = (−1)−3/2(3/2 − (−3/2)) = 3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='17) therefore reproducing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='7 3 Conclusions In this note, we proved that the one-loop prefactor AΓ of the non-perturbative su- perpotential is determined by the CFIV index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because the moduli integral of the CFIV index can be computed via various techniques including topological string the- ory, holomorphic anomaly equations, Chern-Simons theory, and matrix models, the results of this note provide a principled way to evaluate the one-loop prefactor in generic Calabi-Yau compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' There are a few interesting future directions one can pursue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The most imminent next step is to compute the one-loop prefactor of the D- instanton superpotential in explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' One imminent technical challenge is to understand the universal behavior of the holomorphic ambiguities around singular points in the moduli space for annuli and the M¨obius strip diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In [18], Witten proved that if a Euclidean M5-brane wrapped on a divisor in a Calabi-Yau fourfold has a trivial intermediate Jacobian, then the one-loop prefac- tor does not depend on moduli in the absence of spacetime filling M2-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In 7We thank Ashoke for illuminating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 7 type IIB compactification on O3/O7-orientifolds, this implies that Euclidean D3- branes and gaugino condensations which are dual to such Euclidean M5-branes generate the non-perturbative superpotential terms that do not depend on mod- uli in the absence of D3-brane contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As this statement is inherently topological, it may be possible to confirm it by using this paper’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As the one-loop prefactor is computed in heterotic string compactifications [10, 11] and its extension to F-theory compactifications [12], it would be interesting to crosscheck the results via heterotic/type II dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Acknowledgements The work of MK was supported by the Pappalardo fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We thank Ashoke Sen for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We thank Atakan Hilmi Fırat, Liam McAllister, Jakob Moritz, and Andreas Schachner for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 8 A The Jacobi theta functions In this section, we collect useful formulas for the Jacobi theta functions and the char- acter formulas for N = 2 superconformal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In this paper, we will mostly follow the conventions of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We define ϑα,β(z|τ) = � n∈Z+ α 2 eiπnβqn2/2yn , q = e2πiτ , y = e2πiz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='1) For ϑα,β(0|τ), we use a shorthand notation ϑα,β(τ) := ϑα,β(0|τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We collect ϑα,β for (α, β) = (0, 0), (0, 1), (1, 0), (1, 1) ϑ0,0(z|τ) = ∞ � n=1 (1 − qn) � 1 + (y + y−1)qn− 1 2 + q2n−1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2) ϑ0,1(z|τ) = ∞ � n=1 (1 − qn) � 1 − (y + y−1)qn− 1 2 + q2n−1� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='3) ϑ1,0(z|τ) =q 1 8(y 1 2 + y− 1 2) ∞ � n=1 (1 − qn) � 1 + (y + y−1)qn + q2n� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='4) ϑ1,1(z|τ) =iq 1 8(y 1 2 − y− 1 2) ∞ � n=1 (1 − qn) � 1 − (y + y−1)qn + q2n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='5) The Jacobi theta functions admit quasi-periodicity, ϑα,β � z + 1 2|τ � =ϑα,β+1(z|τ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='6) ϑα,β � z + τ 2|τ � =e−iπβ/2q− 1 8y− 1 2ϑα+1,β(z|τ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='7) ϑα+2,β(z|τ) =ϑα,β(z|τ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='8) ϑα,β+2(z|τ) =eiαπϑα,β(z|τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='9) Theta functions satisfy so-called the Jacobi identity ϑ0,0(τ)4 − ϑ0,1(τ)4 − ϑ1,0(τ)4 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='10) We also record useful relations between the Jacobi theta functions and the Dedekind eta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We define the Dedekind eta function as η(τ) = q 1 24 ∞ � n=1 (1 − qn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='11) 9 The Dedekind eta function satisfies the following identities ∂zϑ1,1(z|τ)|z=0 = −2πη(τ)3 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='12) and ϑ1,0 =2η(2τ)2 η(τ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13) ϑ0,1 =η( 1 2τ)2 η(τ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='14) ϑ0,0 = η(τ)5 η( 1 2τ)2η(2τ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='15) B Character formulas for N = 2 superconformal algebra In this section, we will summarize important properties of N = 2 superconformal algebra and its associated representation theory [47–49, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In this section will focus on the left moving sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The superconformal algebra of the right moving sector can be obtained by taking the complex conjugate of the algebra of the left moving sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As we are interested in Calabi-Yau threefold compactifications, we will restrict to c = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We first collect OPEs for superconformal generators, the energy-momentum tensor T, super-currents G and ˜G, and U(1) current I, T(z)T(w) = c 2(z − w)4 + 2T(w) (z − w)2 + ∂T(w) z − w + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='1) I(z)I(w) = c 3(z − w)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='2) I(z)G(w) = 1 z − wG(w) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='3) I(z) ˜G(w) = − 1 z − w ˜G(w) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='4) G(z) ˜G(w) = 2c 3(z − w)3 + 2I(w) (z − w)2 + 1 z − w(∂I(w) + 2T(w)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='5) G(z)G(w) =regular , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='6) ˜G(z) ˜G(w) =regular .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='7) As was studied in [47], it was shown that N = 2 superconformal algebra extends to the extended superconformal algebra by including the spectral flow generators X, ˜X and their superpartners Y, ˜Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='8 We now summarize the character formulas for the extended superconformal al- gebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The character is defined as the partition function of the orbit, generated by 8X and ˜X generate integral shifts of the spectral flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 10 integral spectral flow, of an irreducible representation of the extended superconformal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' For an irreducible representation r, we define the character to be ch(r) := Tr•,r � qL0− 3 8yI0� , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='8) where • can be either Neveu-Schwarz (NS) sector or Ramond (R) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' We will use a collection of shorthand notations g(r) 00 := TrNS,r � qL0− 3 8 � , g(r) 01 := TrNS,r � qL0− 3 8(−1)I0� , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='9) g(r) 10 := TrR,r � qL0− 3 8 � , g(r) 11 := TrR,r � qL0− 3 8(−1)I0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='10) We will, from now on, identify F ≡ I0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='11) We will first study massive representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because representations in the R sector can be brought to representations in the NS sector by half-integral spectral flows, we will focus on irreducible representations in the NS sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' In the NS sector, the highest weight state of any massive state is constrained to have Q = −1, 0, 1, where h > |Q|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Because the character formula for −Q is the same as the character formula for Q, we will only consider Q > 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' As a massive state is labeled by (h, Q), we will denote a massive state by (h, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The character formula for all sectors takes the form g(h,Q) αβ = q 3α 8 g �ατ + β 2 , τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' h, Q � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='12) where we define g(z, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' h, Q) := qh− 1+Q2 4 η(τ) f1,0(z, τ)f2,Q(z, τ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='13) fk,Q(z, τ) := 1 η(τ)q Q2 2k yQϑ0,0(kz + Qτ|kτ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='14) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='12) admits a simpler form9 g(h,Q) αβ = e−iπαβ/2(−1)βQqh− 1+Q 4 η(τ)3 ϑα,β(τ)ϑα+Q,0(2τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='15) Let’s now study massless representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' There are three kinds: the vacuum representation (vac), (+), and (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The vaccum representation has (h, Q) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (+) has (h, Q) = (1/2, 1), and (−) has (h, Q) = (1/2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' The character formulas for 9Note that in the formula (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='7) in [34] is missing e−iπαβ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' This error does not change the conclu- sions of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 11 the massless representations are obtained by replacing g(z, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' h, Q) with g(vac)(z, τ) =g(z, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 0, 0) − g � z, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 1 2, 1 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='16) g(±)(z, τ) = ± 1 2(f3,1(z, τ) − f3,−1(z, τ)) + 1 2g � z, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' 1 2, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='17) Note that the following identities hold f3,1(z, τ) − f3,−1(z, τ) = 0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='18) for z = 0, 1/2, τ/2, and q 3 8 (f3,1(z, τ) − f3,−1(z, τ)) = 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content='19) for z = (τ + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Witten, Nonperturbative superpotentials in string theory, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' B 474 (1996) 343–360, [hep-th/9604030].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfBgYX/content/2301.03602v1.pdf'} +page_content=' [2] S.' metadata={'source': 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0000000000000000000000000000000000000000..01edabf87801e7380d2fd34f50c516313acef057 --- /dev/null +++ b/MdE1T4oBgHgl3EQftQUl/content/tmp_files/2301.03374v1.pdf.txt @@ -0,0 +1,864 @@ +1 +Internet of Everything (IoE) - From Molecules +to the Universe +Ozgur B. Akan, Fellow, IEEE, Ergin Dinc, Member, IEEE, Murat Kuscu, Member, IEEE, +Oktay Cetinkaya, Member, IEEE, Bilgesu A. Bilgin, Member, IEEE +Abstract—The universe is a vast heterogeneous network of +interconnected entities that continuously generate and exchange +information through various forms of interactions, some of which +are yet to be discovered. Internet of Everything (IoE) framework, +inspired by the ubiquitous and adaptive connectivity and the +seamless interoperability within this universal network, puts +forward a new road map beyond the conventional Internet of +Things (IoT) towards maximizing the resolution of our interface +with the universe to enable unprecedented applications. The first +pillar of this road map is to reveal novel and tangible inter- +connections between seemingly noninteracting branches of IoT, +which we call IoXs with X referring to their application domains, +e.g., Internet of Energy (IoEn), Internet of Vehicles (IoV). The +second pillar is to develop new IoXs that can complement the +existing ones to complete the overall IoE picture and match its +networking traits to that of the universe for a seamless and all- +embracing cyber-physical interface. The objective of this paper is +to evaluate the potential of this holistic IoE approach to expand +the limited application landscape of the current IoT practice +on a scale ranging from molecules to the universe. To this end, +we identify several potential interaction pathways among IoXs +and introduce novel and emerging IoXs that are essential to +the comprehensiveness of IoE. We also discuss the potential +applications that can be enabled by such interconnections within +the IoE framework and identify the associated challenges. +Index Terms—Internet of Everything. +I. INTRODUCTION +Our accumulated scientific knowledge suggests that the uni- +verse is a heterogeneous network of ‘everything’, ranging from +molecules to the planets. Some of the most complex phe- +nomena, e.g., evolution and consciousness, are believed to be +rooted in complex interaction networks that create more infor- +mation than the interacting parts. This ubiquitous connectivity +of the universe and the ‘more than the sum’ characteristics of +the underlying heterogeneous networks are the two main traits +inspiring the emerging Internet of Everything (IoE) approach. +IoE is a big step forward beyond the conventional IoT, +which has long been under the scope of both academia and +O. B. Akan, E. Dinc, and B. A. Bilgin are with the Internet of Everything +(IoE) Group, Department of Engineering, University of Cambridge, Cam- +bridge CB3 0FA, UK (e-mail: {oba21, ed502, bab46}@cam.ac.uk). +M. Kuscu and O. Cetinkaya are with the Department of Electrical and +Electronics Engineering, Koc University, Istanbul 34450, Turkey (e-mail: +{mkuscu, ocetinkaya}@ku.edu.tr). +O. B. Akan is also with the Department of Electrical and Electronics En- +gineering, Koc University, Istanbul 34450, Turkey (e-mail: akan@ku.edu.tr). +This work was supported in part by the ERC project MINERVA (ERC-2013- +CoG #616922), AXA Research Fund (AXA Chair for Internet of Everything +at Koc University), The Scientific and Technological Research Council of +Turkey (TUBITAK) under Grant #120E301, and EU’s Horizon 2020 Research +and Innovation Programme through the Marie Skło-dowska-Curie Individual +Fellowship under Grant Agreement #101028935. +industry, with several applications, e.g., smart meters in energy +grids, industrial and agricultural wireless sensor networks +(WSNs), that found their way into the market. One of the +main challenges of IoT is the lack of interaction between its +branches, i.e., Internet of Xs (IoXs), each targeting only a +specific application domain (X). For example, the Internet of +Vehicles (IoV) aims establishing networks of smart vehicles to +optimize traffic flow at a lower environmental/operational cost. +However, it has no direct liaison with other domains, such as +industrial plants or agricultural fields, which could benefit from +that networking approach of IoV to attain a better efficiency vs +cost index. This apparent disconnection between IoXs leads to +a short-sighted perspective missing out on many opportunities +that lay in the interaction of heterogeneous technologies, which +can generate higher value than individual IoXs. +IoE takes a holistic approach and aims to unify the existing +IoXs based on novel interaction pathways defined between +them. Such interactions include a blockchain-based energy +market enabling consumers to trade energy directly with each +other and with the grid (instead of retailers) via energy tokens, +which is a by-product of the Internet of Money (IoM) and +the Internet of Energy (IoEn) merger. IoV can be combined +with those two to achieve peer-to-peer (P2P) vehicle charging +(using the same tokens), minimizing the waiting times and +pressure on scarce resources through efficient energy coopera- +tion between peers. This ambitious goal, however, requires the +optimal match of donor and recipient vehicles by tracking their +locations and battery levels in real-time. That can be met by +the Internet of Space (IoSP) seamlessly communicating with +the IoM, IoEn, and IoV within the IoE framework. +To match the ubiquitous connectivity and heterogeneous +networking characteristics of the universe, IoE also integrates +new IoXs into its framework. Internet of Nano Things (IoNT), +for example, is poised to increase the resolution of cyber- +physical interfaces and bring connectivity into uncharted ter- +ritories, e.g., inside the human body, with the networks of +smart biological agents. Internet of People and Senses (IoPS), +as another example, refers to the conceptual transfer of infor- +mation and even skills between humans besides the nonverbal +communication of senses, e.g., olfaction and gustation. These +will ultimately enable a seamless cyber-physical interface with +a high spatiotemporal resolution and create unprecedented +opportunities to monitor and control the natural interaction +pathways and develop novel applications. That, of course, +requires overcoming many challenges, such as interoperability, +miniaturization, and energy efficiency. +arXiv:2301.03374v1 [cs.NI] 22 Nov 2022 + +2 +Figure 1. The upcoming IoE landscape with its major components -IoXs. +Our objective in this paper is to bring the upcoming IoE +revolution to attention. Hence, we first discuss the state-of-the- +art in key IoXs (Fig. 1), which are the essential components of +the IoE framework, such as the Industrial Internet of Things +(IIoT), Internet of Agricultural Things (IoAT), IoM, IoV, and +IoEn. We also introduce and discuss emerging IoXs, such as +IoNT, IoPS, and Internet of Space (IoSp), which complement +the existing ones to complete the overall IoE picture. We +identify the opportunities and the challenges in advancing +individual IoXs and creating interconnections between them +within the IoE framework. Lastly, we provide a road map for +the evolution of the IoE framework within the next 30 years. +II. INTERNET OF XS (IOXS) +The introduced IoE vision consists in the seamless interac- +tion of heterogeneous technologies and applications integrating +all the essential elements of the universe, including inanimate +and living entities. This section introduces and reviews these +major technologies as the building blocks of the IoE, i.e., +IoXs, which are categorized based on their application areas +spanning the whole universe, starting from the molecular scale. +A. Internet of Nano Things (IoNT) +Nanotechnology has enabled the manipulation of individual +atoms to develop new nanomaterials with exceptional char- +acteristics and the design of nanoscale machines interfacing +with the physical universe at the atomic level. The idea +of IoNT, as illustrated in Fig. 2, lies in interconnecting +nanomachines of different functionalities to overcome their +resource limitations and increase their operational capabilities, +besides integrating these nanonetworks with the conventional +electromagnetic (EM) wireless networks through nano-macro +and bio-cyber interfaces to enable unprecedented applications, +such as intrabody continuous health monitoring [1]. IoNT will +be the most abundant component of the IoE in terms of the +number of connected nodes and the amount of data generated. +Research in this field has been focusing on physical layer +design, where terahertz (THz)-band EM [2] and molecular +communications (MC) [3] are the most promising approaches +to enable reliable information transfer at the nanoscale. MC, +being already realized by living cells, provides a more biocom- +patible ground for developing artificial nanonetworks; thus, +it has attracted the most attention. IoNT research priori- +tizes developing channel models, nano-transceiver architec- +tures, modulation/detection techniques, and communication +protocols for MC [4]. However, there is still an immense +discrepancy between the complexity of the developed methods +and the resource limitations of nanomachines. Open research +challenges for IoNT are summarized as follows [5]: +• Miniaturization: IoNT requires pushing the size of net- +work nodes down to nanoscale and devising communication +methods compatible with these miniature devices to enable +true cyber-physical interfacing with high spatiotemporal +resolution. However, no IoNT device implementation has +so far achieved all IoNT functionalities, e.g., nanoscale +communication, interface with macroscale networks, and +harvest energy. The emergence of novel nanomaterials, e.g., +graphene, with extraordinary optoelectronic and chemical +properties, is promising for developing novel IoNT interfac- +ing and communication modalities, e.g., the use of plasmons +and molecules for sensing and information exchange. +• Ubiquitous +Connectivity +and +Interoperability: Envi- + +Aero and battery token, P2P vehicle charging +Blockchain-based +Energy +Telepathic +energy market +token +driving +Digital commerce +Car batteries +二 +Internet of Money (loM) +Internet of Vehicles (loV) +Tactile internet & + skill transfer +Reliable +Secure +Brain +update +operation +Industrial loT (lloT) +token +INTERNETOF +Industrialized +Internet of Energy (loEn) +farming +EVERYTHING +Pipeline +monitorihg +High- +performance +grids +Internet of People and Senses (loPS) +Internet of Space (loSp) +Farmland +Y +connectivity +Bio-cyber +Internet of Agricultural Things (loAT) +interfaces +Internet of Nano Things (loNT) +Energy-efficient +and sustainable +farming +Health monitoring of plants and cattle3 +Figure 2. (a) Conceptual drawing of a continuous health monitoring application of IoNT. (b) MC among engineered bacteria within IoNT. (c) Graphene-based +nanoscale MC transmitter & receiver architectures. (d) EH nanomachine architecture for IoNT. (e) Graphene plasmonic nanoscale THz transceiver architecture. +sioned IoNT applications span various harsh environments, +e.g., intrabody, imposing several connectivity challenges, +which cannot be overcome by conventional communica- +tion methods, thus necessitates novel bio-inspired tech- +niques like MC. Developing bio-cyber interfaces to connect +nanonetworks, including those composed of bacteria- and +nanomaterial-based networks, with each other and to the +Internet is another challenge towards interoperability, requir- +ing fundamental research on exploiting novel nanomaterials +to create seamless interfaces and transceivers accommodat- +ing both molecular and EM communication modalities. +• Self-sufficiency: Conventional means of energy supply and +storage are not feasible at nanoscale since miniaturization +introduces strict limitations, especially for storage. This +challenge can be tackled by developing more bio-inspired +techniques (e.g., energy-efficient MC), novel energy harvest- +ing (EH) methods (e.g., intrabody chemical EH from glu- +cose), low-complex communication protocols, and compe- +tent molecular or wireless power transfer (WPT) techniques. +• Big Data: Envisioned IoNT applications require developing +novel data analytics tools to exploit the unprecedentedly +big data generated by billions of densely deployed nanoma- +chines, which consist of mostly unstructured data because +of the limited computation capabilities of the nanomachines. +The interconnection of IoNT with other IoXs can introduce +an infinite variety of new directions to the IoE realm. For +example, pipeline monitoring in industrial plants (w/IIoT) and +oil&gas distribution systems (w/IoEn) with nanoscale sensors +detecting corrosion, leaks, blockages, or impurities can sat- +isfy regulatory requirements. Similarly, networked nanorobots +injected into the circulatory system of humans can timely +diagnose any disease or implication, e.g., blood clots, tumors, +and treat/remove them with dedicated drug delivery or actuat- +ing capabilities. IoNT can also cooperate with IoPS towards +achieving bio-cyber interfaces, translating biochemical signals +delivered by intra-body nanonetworks into electrical terms, and +vice versa, for seamless biotic-abiotic interactions. +B. Internet of People and Senses (IoPS) +Sharing human cognitive functionalities and senses through +the Internet, i.e., IoPS, can lead to the most groundbreaking +applications of the IoE framework. The interconnection of +people’s brains, i.e., Brainets, for an advanced network-scale +consciousness leading to higher-level intelligence and for new +forms of direct conceptual communication and collaboration +among people, is the ultimate goal of the IoPS vision [6]. +However, IoPS adopts many other technologies gaining matu- +rity. For example, the Tactile Internet, i.e., real-time sharing of +touch and actuation, enabling the transfer of skills and labour, +has already found applications in remote healthcare, education, +and gaming. Digital communication of smell and taste is also +gaining momentum, promising new forms of social networking +based on non-verbal communication modalities [7]. Yet, the +IoPS faces fundamental research challenges explained below: +• Miniaturization: Similar to the IoNT, IoPS calls for sub- +stantial efforts to develop miniaturized and biocompatible +bio-cyber/neural interfaces that are capable of transducing +sense patterns and cognitive process outcomes into digital +signals and recreating them at the receiving end. In this +direction, the emergence of nanomaterials and the IoNT +technology are promising for developing interfaces with +living cells at very high spatiotemporal resolutions. +• Big Data: Exploiting the big data generated by nanoscale +bio-cyber/neural interfaces requires a comprehensive under- +standing of the complex human brain and isolating useful +patterns therein. This challenge has been targeted by the +Human Brain Project1 and the Brain Initiative2, the two +major research projects respectively supported by the EU +and US, ominating further efforts in the area. +Under the IoPS vision, we can connect ourselves to the +Internet, not only for health monitoring, etc., but also for +gaining new capabilities, e.g., mind control over electronic +devices, Internet access by thought, and sensing in a wider +EM and acoustic spectrum. Moreover, our bodily functions, +such as body heat/fluids and brain activities, can be used to +validate blockchain transactions and thus mine cryptocurren- +cies or brain tokens (w/IoM). Last but not least, we can bring +telepathic driving (w/IoV) into existence via brain-to-vehicle +technology, revolutionizing the autonomous car industry with +mind-controlled steering, powertrain, and brake systems. +1The Human Brain Project. Source: https://www.humanbrainproject.eu/en/ +2The Brain Initiative. Source: https://braininitiative.nih.gov/ + +Graphene plasmonic + Nanoactuator +THz nano-transceiver +(a) +(b) +(d) Graphene +Alzheimer & Epilepsy Monitoring +nanosensor +and Brain Stimulation + Nanonetworks +Interconnected +Nano-memory +Body-Area Nanosensor +Networks +a +'Piezoelectric zinc oxide +人 +energy harvesting unit +Heart +Information +Incoming + Monitoring +(c) +(e) +INTERNET + molecules +THz-EM Wave +Networks +Insulated +Graphene +gold contacts +Cancer +Plasmonic Waves +Monitoring/ +Drug Delivery +Networks +Bio-cyber +Health-care +Dielectric +Nanoporous +Reservoir + provider +interface + graphene Functionalized +Metal +walls +Electrical stimuli +Graphene +graphene +responsive hydrogel +Nanoribbon4 +Figure 3. History of the industrial revolution, revealing how the adaptation of the Internet marked the new epoch (Industry 4.0) almost a half-century earlier. +C. Internet of Industrial Things (IIoT) +The fourth industrial revolution, namely Industry 4.0 [8], +strives for the combination of Internet and future-oriented +technologies with already electrified and automated industrial +machinery (Fig. 3). It aims to improve industry services via +digitizing the manufacturing process involved by means of +increasing the effectiveness of collaboration between machines +and by improving product and service quality. To achieve this, +the whole connected system of physical entities, e.g., factory +equipment and products together with cyber-entities, collec- +tion of software performing optimal control of the physical +processes, referred to as Cyber-Physical Systems (CPSs), are +deployed into factories/industrial plants within the IoE context. +Furthermore, the issues like safety, security, and surveillance +at those places are significantly improved via WSNs. The +forthcoming Industry 5.0 Era is expected to better fit into +this purpose as well as the IoE framework by interconnecting +several IoXs. Although the industry is among the first adopting +the IoE, many challenges hinder widespread IIot utilization: +• Interoperability: The integration of heterogeneous devices +comprising the CPSs to be used, which vary with the type +of industry and the processes involved in manufacturing, is +a significant challenge. Since the efficacy and efficiency of +the deployed CPSs heavily rely on the seamless cooperation +of involved devices, problems in the interoperability of these +devices directly translate into financial consequences. +• Big Data: Product line and quality optimization of manufac- +turing require big data analytics to be performed on CPSs, +which are very case-sensitive; hence, require customized +solutions for each industry and even for each workplace. +• Security and Privacy: Automation of product lines come +with increased safety risks as the CPSs are prone to mal- +functions and cyber attack-driven failures. Furthermore, the +surveillance required for process control has the potential +pitfall of restricting the privacy of workers at the workplace. +IIoT is closely associated with many IoXs. For example, the +digitalization of farming within the Industry 4.0 era enabled +precision agriculture with advanced industrialized farming +tools. Today, the interaction between IIoT and IoAT is moving +towards an integrated system of systems solution through +the seamless cooperation of weather data, farm equipment, +and irrigation systems. IIoT in energy sectors (IoEn), as +another combination, can minimize downtimes, balance sup- +ply&demand, and achieve predictive maintenance. Similarly, +the merger of IIoT with digital commerce (IoM) can avoid +disruptions in the supply chain through data-driven insights, +besides keeping better inventory and maintaining quality, re- +ferring to ever-efficient asset optimization and tracking. +D. Internet of Vehicles (IoV) +The number of devices connected to the Internet is projected +to reach 41 billion by 20273, of which vehicles represent a sub- +stantial portion. The integration of vehicles in the IoT domain +and their interaction with other vehicles, pedestrians, (network) +infrastructure, and road-side units, also termed vehicle-to- +everything (V2X) communications, has converted the old +Vehicular Ad Hoc Networks (VANETs) into the IoV concept +[9]. The ultimate goal of IoV is to establish a network of +“smart” vehicles that exchange information for coordinating +their automated behaviour to minimize risks and maximize +traffic flow at lower emission, cost, and energy consumption. +IoV gathers specific hardware, software, network technolo- +gies, and third-party services to offer novel safety, mobility, +and infotainment applications, which require reliable com- +munication between vehicles and their surroundings. Since +cities become more interconnected and intelligent as days +pass, they offer the ideal conditions for IoV proliferation +while helping connected vehicles gradually transform into +autonomous entities. However, several issues need addressing +before IoV reaches its full potential: +• Security: IoV integrates various technologies, standards, +and services for the flawless operation of its components, +which makes it vulnerable to malicious acts. As the IoV +allows remote access to in-vehicle sensors, GPS, brakes, +etc., successful attacks may result in serious casualties. +• Reliability: Reliable connection and perpetual connectivity +are of utmost importance for the IoV, and network failures, +malevolent attacks, and other bottlenecks can severely im- +pact the whole infrastructure. Thus, the highly mobile and +dynamic nature of the IoV necessitates spatial and temporal +endurance to the changes in external factors, such as speed, +location, and attackers, to ensure that the IoV components +communicate without interruptions. +• Big Data: Currently, connected vehicles process about 1GB +of data per second, but that number is expected to grow as +more infrastructure goes online and becomes interconnected. +Insufficient storage capacity or network latency can hamper +cloud computing and damage the systems. +3The Internet of Things 2020 Report. Source: https://www.businessinsider. +com/internet-of-things-report?r=US&IR=T. + +>Industry 1.0 +> Industry 2.0 +>Industry3.0 +>Industry4.0 +Industry 5.0 +-1784- +-1870- +-1969- +-2011- +-Near Future- +Mechanization +Electrification +Automation +Digitalization +Personalization +·Mechanical production +·Mass production +·Automated production + Cyber physical systems +Cyber-physical cognitive systems +Water and steam power +· Diversion of labour +·Computers +·(Industrial) loT +·Mass customization +. 1st mechanical loom +. 1st assembly line +·Electronics + Robotics and Al +. Human-robot co-working +. IT systems +•Big Data +· Exoskeletons +.1st PLC +· Cloud Computing +·6G and beyond +. Industrial blockchain +. Mixed reality5 +One of many novel applications emerging from IoV and +IoX cooperation is P2P charge trading between electric ve- +hicles (w/IoM & IoEn). Safe and accurate vehicle-assistance +systems, as another example, are achieved by networked +miniature sensors (w/IoNT) spreading into every part of a +vehicle to track proximity and environmental conditions in +real-time. The farming industry also benefits from the IoV, +with drones creating phenotypes by determining growth/health +status through dynamic aerial monitoring of plants (w/IoAT). +IoV’s intersection with IoPS has introduced Social IoV, which +gathers common interest groups, i.e., drivers, passengers, and +transport authorities, to exchange traffic information for a +better driving experience. +E. Internet of Money (IoM) +Cryptocurrencies have emerged to tackle the inefficiencies +of the conventional banking system, e.g., long account opening +and transaction processing times. They utilize blockchain +technology [10], which is based on a distributed ledger system +with no central ledger, i.e., a bank approving transactions. +Here, the cryptocurrency miners serve as the distributed ledger +to generate cryptocurrencies, which offers improved reliability, +flexibility, and security. Cryptocurrency wallets can be ob- +tained immediately anywhere by anyone, and transactions can +be made online in minutes without any border or cost. Hence, +cryptocurrencies are regarded as the money of the Internet, i.e., +IoM, and their value is stored by the connectivity of distributed +ledger. Although cryptocurrency market capitalization once +reached almost $3T, IoM faces significant challenges: +• Scarcity of Resources: The number of cryptocurrency +transactions is significantly increasing, and that causes an +increase in the approval times. This issue can be tackled by +increasing network resources, e.g., the number of cryptocur- +rency miners, subject to ongoing GPU shortage. +• Energy Consumption: Increasing network resources leads +to higher energy consumption. According to the University +of Cambridge researchers, bitcoin mining alone consumes +97.3TWh of energy annually -nearly as much as Pakistan- as +of Q3 20224. Developing energy-efficient integrated circuits +and algorithms can overcome this. +• Standardization: Since governments perceive cryptocurren- +cies as an enabler for illegal money traffic, there is no +standardization for cryptocurrencies yet. +The distributed ledger system, underpinned by IoE devices, +can offer many brand new applications. For example, a +blockchain-based energy market can enable consumers to trade +energy directly from the grid (instead of retailers) in a more +secure way via energy tokens, a by-product of IoM and IoEn +merger. IoV can be combined with those two to achieve P2P +vehicle charging via battery tokens. Speaking of tokens, brain +tokens can be used as a denomination for skill transfer between +humans (w/IoPS), as mentioned earlier. Lastly, blockchain +technology can assist better tracking of manufactured products +and avoid disruptions in the supply chain (w/IIoT), thereby +maximizing efficiencies. +4Cambridge Bitcoin Energy Consumption Index. Source: https://cbeci.org/ +Figure 4. Smart and intelligent power grid in the IoEn framework. +F. Internet of Energy (IoEn) +IoEn defines the modernization of energy production, trans- +mission, distribution, and consumption by upgrading the ex- +isting energy infrastructure via the IoE. The current setting +adopts a centralized approach, causing vast amounts of waste +during transmission or production due to supply&demand +mismatch. Furthermore, it is not flexible to accommodate +renewable energy plants at any scale. Thus, the Smart Grid +concept is proposed to transform the existing grid into a +decentralized and intelligent one using smart meters, actuators, +and WSNs [11]. This concept automates grids through real- +time monitoring of supply&demand, as depicted in Fig. 4, +achieving higher efficiencies. +IoEn also covers powering WSNs, essential for IoE real- +ization, where the limited battery life is a key issue. Non- +deterministic battery depletion threatens sensor reliability, +replacement of which means high maintenance costs and +frequent disruptions. EH, together with WPT, can mitigate this +by making IoE devices self-sufficient, i.e., energy-autonomous. +However, the availability of harvestable sources, e.g., solar, +often depends on external factors, imposing another challenge. +One way to tackle that is to adopt multiple EH mechanisms, +enabling higher reliability and self-sufficiency of sensors [12]. +Yet, energy-efficient sensing, computation and communication +techniques should also be considered for uninterrupted opera- +tions in the IoE. The other challenges of IoEn are as follows: +• Interoperability: Integrating small- and large-scale sup- +ply&demand from various sources into decentralized and +deregulated energy production increases the complexity of +power grids. Thus, interoperability of energy systems and +their monitoring via the WSNs is an important issue. +• Privacy: The privacy of customers can be exploited by +analyzing their real-time electricity usage behavior; thus, +information-centric networking solutions assuring data pro- +tection have to be put into practice to protect user identities. +• Security: In a fully autonomous power grid, any cyber- +attack or failure in information technologies can result in +billion-dollar losses; hence, security is another crucial issue. +As Fig. 1 depicts, IoEn is essential for all IoXs. Electric +vehicles, as one example, impose a growing electricity de- +mand, requiring efficient planning and management of energy +from generation to utilization. The battery state and location of +vehicles together with nearby charging stations’ capacity/status +can be tracked in real-time through the IoV to optimally match +vehicles with stations, minimizing the waiting times and the + +Generation +Transmission +Distribution & Consumption +loEn +Distributed +Generation6 +pressure on scarce resources. Increasing the share of renewable +sources in energy consumption can help achieve this goal +while lowering greenhouse gas emissions and thus positively +contribute to global Net Zero commitments. +G. Internet of Space (IoSp) +Communication satellites (CSs) placed in Earth’s orbits play +a significant role in supporting the Internet and are expected to +be the backbone of future IoE [13]. However, the existing CS +infrastructure cannot support the exploding demand for data +traffic with plausible rates, and to mend this, new-age high- +throughput satellites (HTSs) equipped with spot-beam tech- +nology are being launched. Besides the conventional large +(>5tonnes) HTSs deployed in geostationary orbit (GEO), new- +generation small (<500kg) CSs are increasingly being de- +ployed into low earth orbit (LEO), and constellations of small +CSs which network together into a dynamic web of global +coverage, such as those embarked upon SpaceX and OneWeb. +Recent advancements in space expedition capabilities intro- +duced by SpaceX have made populating space with human +artefacts a cheaper and faster process to the extent that a +human colonization effort of Mars is no longer an unfeasible +prospect. Such an effort would include CSs joining Mars +Reconnaissance and Odyssey orbiters relaying information +received from the Curiosity Rover with 1Gbit/day capacity, +which is insufficient to support this effort. With CSs orbiting +Mars deployed, the Internet on Mars can be established to join +the Internet on Earth as part of a greater Internet, namely IoSp. +The key challenges in IoSP realization are as below: +• Ubiquitous Connectivity: Primary communication chal- +lenge in the operation of an interplanetary link involves +establishing ubiquitous connectivity in a network with very +high inter-node distance, which requires developing delay- +and disruption-tolerant communication protocols, besides +the strategic deployment of relay stations to minimize the +inter-node communication disruption. +• Energy Resources: Communication devices deployed into +space need to harvest energy, typically from the Sun, to +perform their tasks uninterrupted. Within the IoSp frame- +work, many communication nodes will need to be located +in isolated corners of space away from the Sun, where +sunlight intensity is insufficient to support communications +by contemporary EH techniques. Thus, advancements in +IoSp call for more efficient EH methods and low-cost +communication protocols. +IoSp interacts with various IoXs to improve the performance +of terrestrial applications, among which the most notable ones +are secure software updates for autonomous & connected cars +(w/IoV) and reliable operation of the crypto market (w/IoM), +avoiding cyber-attacks and government censorship. +H. Internet of Agricultural Things (IoAT) +Over the last decade, we have seen the rise of smart +agriculture technologies, e.g., farmland WSNs, due to the +increasing food demand, food security, and health concerns. +As part of the IoE framework, the IoAT covers the existing +and developing information and communication technology +(ICT) applications in agriculture, ranging from smart farming +to smart food logistics, processing, and awareness, reinforced +and diversified with the introduction of novel IoE concepts, +e.g., Internet of Plants and Animals (IoPA), IoNT, IoV, that +aim at the integration of every component of agriculture. +Existing IoAT applications include precision agriculture +with high-accuracy weather forecasts and livestock health +monitoring enabled by wearable sensors, water-efficient irri- +gation systems, real-time tracking of individual products for +food awareness and supply chain planning [14]. Developing +predictive analytics tools for analyzing the voluminous data +generated by the heterogeneous components of the IoAT is a +major topic. One of the large-scale research projects in this +direction was the Internet of Food and Farm 2020, aiming to +maintain safe and healthy food through IoT technologies5. +Despite the advancements in the area, there are still major +challenges hampering the realization of a full-fledged IoAT: +• Ubiquitous Connectivity and Interoperability: IoAT re- +quires connectivity for network components in hard-to-reach +environments, e.g., remote farmlands, and seamless inter- +operability among heterogeneous IoAT components, e.g., +intrabody IoNT nodes monitoring livestock health status, the +network of plants and animals, through bio-cyber interfaces. +• Big Data: The big data generated by the IoAT components +will be highly heterogeneous as it includes those generated +by living entities, e.g., plants, animals, and humans, neces- +sitating the development of novel predictive analytics tools. +Applications can be diversified with the close collaboration +of IoAT with other IoE components. For example, IoNT can +enable real-time and continuous health/quality monitoring of +crops and livestock with molecular precision; energy-neutral +drones [15] (IoV) can enable automated multimedia farm- +monitoring and cattle-tracking; IoSp can offer more secure, +reliable, and fast farmland connectivity; IoPA, using vari- +ous communication modalities, e.g., acoustic, chemical, can +provide better understanding and control of the biosphere +increasing the efficiency of the agricultural processes. +III. CONCLUSIONS +This paper brings the upcoming IoE revolution to attention +and defines its most promising applications. There is a greater +opportunity brought by the IoE vision that is still untouched +because all existing IoX applications have been targeting a +single domain, not requiring broad connectivity or interaction +with other IoXs. To realize this opportunity, individual IoXs +will be seamlessly interacting under the IoE umbrella and +continuously feeding each other. In this way, we can close the +technological gap between our communication infrastructure +and the universe, thus fulfilling the holistic vision of the IoE. +The long-standing quest to develop ICT to better interact +with the entire universe has enabled the large-scale deploy- +ment of WSNs over several domains starting from the early +2000s, as illustrated in Fig. 5. Since we started to exploit the +data collected through WSNs more effectively with advanced +5Internet of Food and Farm 2020. Source: https://www.iof2020.eu/ + +7 +Figure 5. IoE roadmap: past, present, and future of IoE technologies. +data analytics tools and cloud computing, we have seen the +emergence of more intelligent applications, such as semi- +autonomous driving and smart automation, starting from the +early 2010s. Yet, this progress is not disruptive enough for +developing fully automated processes covering heterogeneous +technologies, e.g., fully autonomous grid, driving, farming, +and manufacturing. This ambitious goal requires overcoming +substantial challenges, such as universal-scale connectivity, +interoperability, standardization of heterogeneous IoXs, self- +sustainability, and processing of big and heterogeneous data. +We have already seen some progress towards addressing the +key challenges of the IoE, such as the development of brain- +machine interfaces, and neural implants for interconnecting +different physical domains, e.g., living cells and artificial +devices; bio-compatible EH methods for self-sufficient sen- +sors/actuators even at nanoscale; data analytics and machine +learning algorithms to make use of big data for improving +services. At this point, more practical steps can be taken by +developing new communication techniques orthogonal to the +conventional EM to extend the connectivity, devising universal +transceivers that can support multiple communication modali- +ties to provide interoperability and hybrid EH methods that can +adaptively operate in different environments. Given the pace of +technological advances, we are in a good position to envision +a fully connected universe, including all living and artificial +things, realized by 2050 via the emerging IoE approach. +REFERENCES +[1] I. F. Akyildiz and J. M. Jornet, “The internet of nano-things,” IEEE +Wireless Communications, vol. 17, no. 6, pp. 58-63, 2010. +[2] F. Lemic et al., “Survey on terahertz nanocommunication and networking: +A top-down perspective,” IEEE Journal on Selected Areas in Communi- +cations, vol. 39, no. 6, pp. 1506-1543, 2021. +[3] O. B. Akan et al., “Fundamentals of molecular information and commu- +nication science,” Proc. IEEE, vol. 105, no. 2, pp. 306-318, 2016. +[4] M. Kuscu et al., “Transmitter and receiver architectures for molecular +communications: A survey on physical design with modulation, coding, +and detection techniques,”Proc. IEEE, vol. 107, no. 7, pp. 1302-1341, 2019. +[5] M. Civas et al.,“Universal Transceivers: Opportunities and Future Directions +for the Internet of Everything (IoE),” Front. Comms. Net., vol. 2, 2021. +[6] L. Jiang et al., “BrainNet: a multi-person brain-to-brain interface for direct +collaboration between brains,” Sci. Rep., vol. 9, no. 1, pp. 1-11, 2019. +[7] E. Calvi et al., “The scent of emotions: A systematic review of human +intra-and interspecific chemical communication of emotions,” Brain and +behavior, vol. 10, no. 5, pp. e01585, 2020. +[8] A. G. Frank et al., “Industry 4.0 technologies: Implementation patterns in +manufacturing companies,” Int. J. Prod. Econ., vol. 210, pp. 15-26, 2019. +[9] B. Ji et al., “Survey on the internet of vehicles: Network architectures and +applications,” IEEE Commun. Stand. Mag., vol. 4, no. 1, pp. 34-41, 2020. +[10] H.-N. Dai et al., “Blockchain for Internet of Things: A survey,” IEEE +Internet of Things Journal, vol. 6, no. 5, pp. 8076-8094, 2019. +[11] K. Wang et al., “A survey on energy internet: Architecture, approach, and +emerging technologies,” IEEE Syst. J, vol. 12, no. 3, pp. 2403-2416, 2017. +[12] O. B. Akan et al., “Internet of Hybrid Energy Harvesting Things,” IEEE +Internet of Things Journal, vol. 5, no. 2, pp. 736-746, 2017. +[13] I. Akyildiz and A. Kak, “The Internet of Space Things/CubeSats: A +ubiquitous cyber-physical system for the connected world,” Computer +Networks, vol. 150, pp. 134-149, 2019. +[14] N. Holden et al., “Review of the sustainability of food systems and transi- +tion using the Internet of Food,” NPJ Sci. Food, vol. 2, no. 1, pp. 1-7, 2018. +[15] T. Long et al., “Energy neutral internet of drones,” IEEE Communica- +tions Magazine, vol. 56, no. 1, pp. 22-28, 2018. +Ozgur B. Akan received his PhD degree from the Georgia Institute of +Technology, Atlanta, GA, USA, in 2004. He is currently the Head of +the Internet of Everything Group, Department of Engineering, University +of Cambridge, UK, and the Director of the Next-generation and Wireless +Communications Laboratory, Department of Electrical and Electronics Engi- +neering, Koc University, Turkey. His research interests include wireless, nano, +molecular, and neural communications, and the Internet of Everything. +Ergin Dinc received his PhD degree in Electrical and Electronics Engineering +from Koc University, Turkey, in 2016. After his PhD, he held postdoctoral +positions at KTH Royal Institute of Technology, Sweden, and University of +Cambridge, UK. His research interests are molecular communications, neural +communication, and cyber–physical systems. +Murat Kuscu received his PhD degrees in Engineering from University of +Cambridge, UK, in 2020, and in Electrical and Electronics Engineering from +Koc University, Turkey, in 2017. He is currently an Assistant Professor at +the Department of Electrical and Electronics Engineering, Koc University. +His research interests include Internet of Bio-Nano Things, nanomaterials, +biosensors, and microfluidics. +Oktay Cetinkaya received his PhD degree in Electrical and Electronics +Engineering from Koc University, Turkey, in 2018. After his PhD, he worked +as a Research Fellow at the University of Southampton, UK, a Research +Associate at the University of Sheffield, UK, and a Senior Research Associate +at the University of Oxford, UK. His research focuses on energy-neutral +communications in the Internet of Things. +Bilgesu A. Bilgin received his PhD degree in Mathematics from Koc +University, Turkey, in 2015. After his PhD, he worked as a postdoctoral +researcher at Koc University and the University of Cambridge, UK. His +research interests include molecular communication, intrabody nanonetworks, +and dynamical systems. + +2000 +2010 +2020 +2030 +2040 +2050 +Fully-autonomous +Smart +loEn +01234 +Metering +Grid +Grid +Vehicular +Fully-autonomous +Semi-autonomous +Flying Cars +lov +O +Driving +Networks +Driving +First +Mars Colony + LEO Communication +loSp +Interplenatary +Satellites +Interplanetary Internet +5 +Internet +Harvesting +Agricultural +Agricultural +Q +Fully-autonomous +IoAT +WSNS +UAVs +Robots +Farming +Smart +Industrial +IloT +Industry4.0 +Exoscelatons +Industry5.0 +WSNs +Automation +B +Cryptocurrency +Blockchain& +loM +Ethereum +Bitcoin +Standardization +Molecular +Synthetic Bacteria +Bio-cyber +IoNT +Nanonetworks +Networks +Interface +Communication +Brain-machine +Neural +Skill +Sense +loPS +Telepathy +Interface +Implants +Transfer +Transfer \ No newline at end of file diff --git a/MdE1T4oBgHgl3EQftQUl/content/tmp_files/load_file.txt b/MdE1T4oBgHgl3EQftQUl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b406e3e6aace49a94a8aa93707c22c16acbaacd7 --- /dev/null +++ b/MdE1T4oBgHgl3EQftQUl/content/tmp_files/load_file.txt @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf,len=552 +page_content='1 Internet of Everything (IoE) - From Molecules to the Universe Ozgur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Akan, Fellow, IEEE, Ergin Dinc, Member, IEEE, Murat Kuscu, Member, IEEE, Oktay Cetinkaya, Member, IEEE, Bilgesu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Bilgin, Member, IEEE Abstract—The universe is a vast heterogeneous network of interconnected entities that continuously generate and exchange information through various forms of interactions, some of which are yet to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Everything (IoE) framework, inspired by the ubiquitous and adaptive connectivity and the seamless interoperability within this universal network, puts forward a new road map beyond the conventional Internet of Things (IoT) towards maximizing the resolution of our interface with the universe to enable unprecedented applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The first pillar of this road map is to reveal novel and tangible inter- connections between seemingly noninteracting branches of IoT, which we call IoXs with X referring to their application domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', Internet of Energy (IoEn), Internet of Vehicles (IoV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The second pillar is to develop new IoXs that can complement the existing ones to complete the overall IoE picture and match its networking traits to that of the universe for a seamless and all- embracing cyber-physical interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The objective of this paper is to evaluate the potential of this holistic IoE approach to expand the limited application landscape of the current IoT practice on a scale ranging from molecules to the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' To this end, we identify several potential interaction pathways among IoXs and introduce novel and emerging IoXs that are essential to the comprehensiveness of IoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' We also discuss the potential applications that can be enabled by such interconnections within the IoE framework and identify the associated challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Index Terms—Internet of Everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' INTRODUCTION Our accumulated scientific knowledge suggests that the uni- verse is a heterogeneous network of ‘everything’, ranging from molecules to the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Some of the most complex phe- nomena, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', evolution and consciousness, are believed to be rooted in complex interaction networks that create more infor- mation than the interacting parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This ubiquitous connectivity of the universe and the ‘more than the sum’ characteristics of the underlying heterogeneous networks are the two main traits inspiring the emerging Internet of Everything (IoE) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoE is a big step forward beyond the conventional IoT, which has long been under the scope of both academia and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Akan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Dinc, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Bilgin are with the Internet of Everything (IoE) Group, Department of Engineering, University of Cambridge, Cam- bridge CB3 0FA, UK (e-mail: {oba21, ed502, bab46}@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Kuscu and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Cetinkaya are with the Department of Electrical and Electronics Engineering, Koc University, Istanbul 34450, Turkey (e-mail: {mkuscu, ocetinkaya}@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Akan is also with the Department of Electrical and Electronics En- gineering, Koc University, Istanbul 34450, Turkey (e-mail: akan@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This work was supported in part by the ERC project MINERVA (ERC-2013- CoG #616922), AXA Research Fund (AXA Chair for Internet of Everything at Koc University), The Scientific and Technological Research Council of Turkey (TUBITAK) under Grant #120E301, and EU’s Horizon 2020 Research and Innovation Programme through the Marie Skło-dowska-Curie Individual Fellowship under Grant Agreement #101028935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' industry, with several applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', smart meters in energy grids, industrial and agricultural wireless sensor networks (WSNs), that found their way into the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' One of the main challenges of IoT is the lack of interaction between its branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', Internet of Xs (IoXs), each targeting only a specific application domain (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, the Internet of Vehicles (IoV) aims establishing networks of smart vehicles to optimize traffic flow at a lower environmental/operational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, it has no direct liaison with other domains, such as industrial plants or agricultural fields, which could benefit from that networking approach of IoV to attain a better efficiency vs cost index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This apparent disconnection between IoXs leads to a short-sighted perspective missing out on many opportunities that lay in the interaction of heterogeneous technologies, which can generate higher value than individual IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoE takes a holistic approach and aims to unify the existing IoXs based on novel interaction pathways defined between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Such interactions include a blockchain-based energy market enabling consumers to trade energy directly with each other and with the grid (instead of retailers) via energy tokens, which is a by-product of the Internet of Money (IoM) and the Internet of Energy (IoEn) merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoV can be combined with those two to achieve peer-to-peer (P2P) vehicle charging (using the same tokens), minimizing the waiting times and pressure on scarce resources through efficient energy coopera- tion between peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This ambitious goal, however, requires the optimal match of donor and recipient vehicles by tracking their locations and battery levels in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' That can be met by the Internet of Space (IoSP) seamlessly communicating with the IoM, IoEn, and IoV within the IoE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' To match the ubiquitous connectivity and heterogeneous networking characteristics of the universe, IoE also integrates new IoXs into its framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Nano Things (IoNT), for example, is poised to increase the resolution of cyber- physical interfaces and bring connectivity into uncharted ter- ritories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', inside the human body, with the networks of smart biological agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of People and Senses (IoPS), as another example, refers to the conceptual transfer of infor- mation and even skills between humans besides the nonverbal communication of senses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', olfaction and gustation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' These will ultimately enable a seamless cyber-physical interface with a high spatiotemporal resolution and create unprecedented opportunities to monitor and control the natural interaction pathways and develop novel applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' That, of course, requires overcoming many challenges, such as interoperability, miniaturization, and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='03374v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='NI] 22 Nov 2022 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The upcoming IoE landscape with its major components -IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Our objective in this paper is to bring the upcoming IoE revolution to attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Hence, we first discuss the state-of-the- art in key IoXs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1), which are the essential components of the IoE framework, such as the Industrial Internet of Things (IIoT), Internet of Agricultural Things (IoAT), IoM, IoV, and IoEn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' We also introduce and discuss emerging IoXs, such as IoNT, IoPS, and Internet of Space (IoSp), which complement the existing ones to complete the overall IoE picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' We identify the opportunities and the challenges in advancing individual IoXs and creating interconnections between them within the IoE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Lastly, we provide a road map for the evolution of the IoE framework within the next 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' INTERNET OF XS (IOXS) The introduced IoE vision consists in the seamless interac- tion of heterogeneous technologies and applications integrating all the essential elements of the universe, including inanimate and living entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This section introduces and reviews these major technologies as the building blocks of the IoE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', IoXs, which are categorized based on their application areas spanning the whole universe, starting from the molecular scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Nano Things (IoNT) Nanotechnology has enabled the manipulation of individual atoms to develop new nanomaterials with exceptional char- acteristics and the design of nanoscale machines interfacing with the physical universe at the atomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The idea of IoNT, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 2, lies in interconnecting nanomachines of different functionalities to overcome their resource limitations and increase their operational capabilities, besides integrating these nanonetworks with the conventional electromagnetic (EM) wireless networks through nano-macro and bio-cyber interfaces to enable unprecedented applications, such as intrabody continuous health monitoring [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoNT will be the most abundant component of the IoE in terms of the number of connected nodes and the amount of data generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Research in this field has been focusing on physical layer design, where terahertz (THz)-band EM [2] and molecular communications (MC) [3] are the most promising approaches to enable reliable information transfer at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' MC, being already realized by living cells, provides a more biocom- patible ground for developing artificial nanonetworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' thus, it has attracted the most attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoNT research priori- tizes developing channel models, nano-transceiver architec- tures, modulation/detection techniques, and communication protocols for MC [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, there is still an immense discrepancy between the complexity of the developed methods and the resource limitations of nanomachines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Open research challenges for IoNT are summarized as follows [5]: Miniaturization: IoNT requires pushing the size of net- work nodes down to nanoscale and devising communication methods compatible with these miniature devices to enable true cyber-physical interfacing with high spatiotemporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, no IoNT device implementation has so far achieved all IoNT functionalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', nanoscale communication, interface with macroscale networks, and harvest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The emergence of novel nanomaterials, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', graphene, with extraordinary optoelectronic and chemical properties, is promising for developing novel IoNT interfac- ing and communication modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', the use of plasmons and molecules for sensing and information exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Ubiquitous Connectivity and Interoperability: Envi- Aero and battery token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' P2P vehicle charging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Blockchain-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Telepathic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='energy market ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='driving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Digital commerce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Car batteries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='二 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Money (loM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Vehicles (loV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Tactile internet & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='skill transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Reliable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Secure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='operation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Industrial loT (lloT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='INTERNETOF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Industrialized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Energy (loEn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='farming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='EVERYTHING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Pipeline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='monitorihg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='High- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='grids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of People and Senses (loPS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Space (loSp) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Farmland ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Bio-cyber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Agricultural Things (loAT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='interfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Internet of Nano Things (loNT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Energy-efficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='and sustainable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='farming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Health monitoring of plants and cattle3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' (a) Conceptual drawing of a continuous health monitoring application of IoNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' (b) MC among engineered bacteria within IoNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' (c) Graphene-based nanoscale MC transmitter & receiver architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' (d) EH nanomachine architecture for IoNT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' (e) Graphene plasmonic nanoscale THz transceiver architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' sioned IoNT applications span various harsh environments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', intrabody, imposing several connectivity challenges, which cannot be overcome by conventional communica- tion methods, thus necessitates novel bio-inspired tech- niques like MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Developing bio-cyber interfaces to connect nanonetworks, including those composed of bacteria- and nanomaterial-based networks, with each other and to the Internet is another challenge towards interoperability, requir- ing fundamental research on exploiting novel nanomaterials to create seamless interfaces and transceivers accommodat- ing both molecular and EM communication modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Self-sufficiency: Conventional means of energy supply and storage are not feasible at nanoscale since miniaturization introduces strict limitations, especially for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This challenge can be tackled by developing more bio-inspired techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', energy-efficient MC), novel energy harvest- ing (EH) methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', intrabody chemical EH from glu- cose), low-complex communication protocols, and compe- tent molecular or wireless power transfer (WPT) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Big Data: Envisioned IoNT applications require developing novel data analytics tools to exploit the unprecedentedly big data generated by billions of densely deployed nanoma- chines, which consist of mostly unstructured data because of the limited computation capabilities of the nanomachines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The interconnection of IoNT with other IoXs can introduce an infinite variety of new directions to the IoE realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, pipeline monitoring in industrial plants (w/IIoT) and oil&gas distribution systems (w/IoEn) with nanoscale sensors detecting corrosion, leaks, blockages, or impurities can sat- isfy regulatory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Similarly, networked nanorobots injected into the circulatory system of humans can timely diagnose any disease or implication, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', blood clots, tumors, and treat/remove them with dedicated drug delivery or actuat- ing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoNT can also cooperate with IoPS towards achieving bio-cyber interfaces, translating biochemical signals delivered by intra-body nanonetworks into electrical terms, and vice versa, for seamless biotic-abiotic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of People and Senses (IoPS) Sharing human cognitive functionalities and senses through the Internet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', IoPS, can lead to the most groundbreaking applications of the IoE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The interconnection of people’s brains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', Brainets, for an advanced network-scale consciousness leading to higher-level intelligence and for new forms of direct conceptual communication and collaboration among people, is the ultimate goal of the IoPS vision [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, IoPS adopts many other technologies gaining matu- rity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, the Tactile Internet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', real-time sharing of touch and actuation, enabling the transfer of skills and labour, has already found applications in remote healthcare, education, and gaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Digital communication of smell and taste is also gaining momentum, promising new forms of social networking based on non-verbal communication modalities [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Yet, the IoPS faces fundamental research challenges explained below: Miniaturization: Similar to the IoNT, IoPS calls for sub- stantial efforts to develop miniaturized and biocompatible bio-cyber/neural interfaces that are capable of transducing sense patterns and cognitive process outcomes into digital signals and recreating them at the receiving end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' In this direction, the emergence of nanomaterials and the IoNT technology are promising for developing interfaces with living cells at very high spatiotemporal resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Big Data: Exploiting the big data generated by nanoscale bio-cyber/neural interfaces requires a comprehensive under- standing of the complex human brain and isolating useful patterns therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This challenge has been targeted by the Human Brain Project1 and the Brain Initiative2, the two major research projects respectively supported by the EU and US, ominating further efforts in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Under the IoPS vision, we can connect ourselves to the Internet, not only for health monitoring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', but also for gaining new capabilities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', mind control over electronic devices, Internet access by thought, and sensing in a wider EM and acoustic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Moreover, our bodily functions, such as body heat/fluids and brain activities, can be used to validate blockchain transactions and thus mine cryptocurren- cies or brain tokens (w/IoM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Last but not least, we can bring telepathic driving (w/IoV) into existence via brain-to-vehicle technology, revolutionizing the autonomous car industry with mind-controlled steering, powertrain, and brake systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1The Human Brain Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Source: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='humanbrainproject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='eu/en/ 2The Brain Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Source: https://braininitiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='gov/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Graphene plasmonic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Nanoactuator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='THz nano-transceiver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='(d) Graphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Alzheimer & Epilepsy Monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='nanosensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='and Brain Stimulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Nanonetworks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Interconnected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Nano-memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Body-Area Nanosensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content="'Piezoelectric zinc oxide " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='人 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='energy harvesting unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Heart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Incoming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='INTERNET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='molecules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='THz-EM Wave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Insulated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Graphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='gold contacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Cancer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Plasmonic Waves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Monitoring/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Drug Delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Bio-cyber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Health-care ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Dielectric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Nanoporous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Reservoir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='provider ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='graphene Functionalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Metal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='walls ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Electrical stimuli ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Graphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='graphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='responsive hydrogel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Nanoribbon4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' History of the industrial revolution, revealing how the adaptation of the Internet marked the new epoch (Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0) almost a half-century earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Industrial Things (IIoT) The fourth industrial revolution, namely Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 [8], strives for the combination of Internet and future-oriented technologies with already electrified and automated industrial machinery (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' It aims to improve industry services via digitizing the manufacturing process involved by means of increasing the effectiveness of collaboration between machines and by improving product and service quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' To achieve this, the whole connected system of physical entities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', factory equipment and products together with cyber-entities, collec- tion of software performing optimal control of the physical processes, referred to as Cyber-Physical Systems (CPSs), are deployed into factories/industrial plants within the IoE context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Furthermore, the issues like safety, security, and surveillance at those places are significantly improved via WSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The forthcoming Industry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 Era is expected to better fit into this purpose as well as the IoE framework by interconnecting several IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Although the industry is among the first adopting the IoE, many challenges hinder widespread IIot utilization: Interoperability: The integration of heterogeneous devices comprising the CPSs to be used, which vary with the type of industry and the processes involved in manufacturing, is a significant challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Since the efficacy and efficiency of the deployed CPSs heavily rely on the seamless cooperation of involved devices, problems in the interoperability of these devices directly translate into financial consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Big Data: Product line and quality optimization of manufac- turing require big data analytics to be performed on CPSs, which are very case-sensitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' hence, require customized solutions for each industry and even for each workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Security and Privacy: Automation of product lines come with increased safety risks as the CPSs are prone to mal- functions and cyber attack-driven failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Furthermore, the surveillance required for process control has the potential pitfall of restricting the privacy of workers at the workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IIoT is closely associated with many IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, the digitalization of farming within the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 era enabled precision agriculture with advanced industrialized farming tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Today, the interaction between IIoT and IoAT is moving towards an integrated system of systems solution through the seamless cooperation of weather data, farm equipment, and irrigation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IIoT in energy sectors (IoEn), as another combination, can minimize downtimes, balance sup- ply&demand, and achieve predictive maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Similarly, the merger of IIoT with digital commerce (IoM) can avoid disruptions in the supply chain through data-driven insights, besides keeping better inventory and maintaining quality, re- ferring to ever-efficient asset optimization and tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Vehicles (IoV) The number of devices connected to the Internet is projected to reach 41 billion by 20273, of which vehicles represent a sub- stantial portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The integration of vehicles in the IoT domain and their interaction with other vehicles, pedestrians, (network) infrastructure, and road-side units, also termed vehicle-to- everything (V2X) communications, has converted the old Vehicular Ad Hoc Networks (VANETs) into the IoV concept [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The ultimate goal of IoV is to establish a network of “smart” vehicles that exchange information for coordinating their automated behaviour to minimize risks and maximize traffic flow at lower emission, cost, and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoV gathers specific hardware, software, network technolo- gies, and third-party services to offer novel safety, mobility, and infotainment applications, which require reliable com- munication between vehicles and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Since cities become more interconnected and intelligent as days pass, they offer the ideal conditions for IoV proliferation while helping connected vehicles gradually transform into autonomous entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, several issues need addressing before IoV reaches its full potential: Security: IoV integrates various technologies, standards, and services for the flawless operation of its components, which makes it vulnerable to malicious acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' As the IoV allows remote access to in-vehicle sensors, GPS, brakes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', successful attacks may result in serious casualties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Reliability: Reliable connection and perpetual connectivity are of utmost importance for the IoV, and network failures, malevolent attacks, and other bottlenecks can severely im- pact the whole infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Thus, the highly mobile and dynamic nature of the IoV necessitates spatial and temporal endurance to the changes in external factors, such as speed, location, and attackers, to ensure that the IoV components communicate without interruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Big Data: Currently, connected vehicles process about 1GB of data per second, but that number is expected to grow as more infrastructure goes online and becomes interconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Insufficient storage capacity or network latency can hamper cloud computing and damage the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 3The Internet of Things 2020 Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Source: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='businessinsider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' com/internet-of-things-report?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='r=US&IR=T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' >Industry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 > Industry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 >Industry3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 >Industry4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 Industry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 1784- 1870- 1969- 2011- Near Future- Mechanization Electrification Automation Digitalization Personalization Mechanical production Mass production Automated production Cyber physical systems Cyber-physical cognitive systems Water and steam power Diversion of labour Computers (Industrial) loT Mass customization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1st mechanical loom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1st assembly line Electronics Robotics and Al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Human-robot co-working .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IT systems Big Data Exoskeletons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='1st PLC Cloud Computing 6G and beyond .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Industrial blockchain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Mixed reality5 One of many novel applications emerging from IoV and IoX cooperation is P2P charge trading between electric ve- hicles (w/IoM & IoEn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Safe and accurate vehicle-assistance systems, as another example, are achieved by networked miniature sensors (w/IoNT) spreading into every part of a vehicle to track proximity and environmental conditions in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The farming industry also benefits from the IoV, with drones creating phenotypes by determining growth/health status through dynamic aerial monitoring of plants (w/IoAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoV’s intersection with IoPS has introduced Social IoV, which gathers common interest groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', drivers, passengers, and transport authorities, to exchange traffic information for a better driving experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Money (IoM) Cryptocurrencies have emerged to tackle the inefficiencies of the conventional banking system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', long account opening and transaction processing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' They utilize blockchain technology [10], which is based on a distributed ledger system with no central ledger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', a bank approving transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Here, the cryptocurrency miners serve as the distributed ledger to generate cryptocurrencies, which offers improved reliability, flexibility, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Cryptocurrency wallets can be ob- tained immediately anywhere by anyone, and transactions can be made online in minutes without any border or cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Hence, cryptocurrencies are regarded as the money of the Internet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', IoM, and their value is stored by the connectivity of distributed ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Although cryptocurrency market capitalization once reached almost $3T, IoM faces significant challenges: Scarcity of Resources: The number of cryptocurrency transactions is significantly increasing, and that causes an increase in the approval times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This issue can be tackled by increasing network resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', the number of cryptocur- rency miners, subject to ongoing GPU shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Energy Consumption: Increasing network resources leads to higher energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' According to the University of Cambridge researchers, bitcoin mining alone consumes 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='3TWh of energy annually -nearly as much as Pakistan- as of Q3 20224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Developing energy-efficient integrated circuits and algorithms can overcome this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Standardization: Since governments perceive cryptocurren- cies as an enabler for illegal money traffic, there is no standardization for cryptocurrencies yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The distributed ledger system, underpinned by IoE devices, can offer many brand new applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, a blockchain-based energy market can enable consumers to trade energy directly from the grid (instead of retailers) in a more secure way via energy tokens, a by-product of IoM and IoEn merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoV can be combined with those two to achieve P2P vehicle charging via battery tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Speaking of tokens, brain tokens can be used as a denomination for skill transfer between humans (w/IoPS), as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Lastly, blockchain technology can assist better tracking of manufactured products and avoid disruptions in the supply chain (w/IIoT), thereby maximizing efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 4Cambridge Bitcoin Energy Consumption Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Source: https://cbeci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='org/ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Smart and intelligent power grid in the IoEn framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Energy (IoEn) IoEn defines the modernization of energy production, trans- mission, distribution, and consumption by upgrading the ex- isting energy infrastructure via the IoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The current setting adopts a centralized approach, causing vast amounts of waste during transmission or production due to supply&demand mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Furthermore, it is not flexible to accommodate renewable energy plants at any scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Thus, the Smart Grid concept is proposed to transform the existing grid into a decentralized and intelligent one using smart meters, actuators, and WSNs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This concept automates grids through real- time monitoring of supply&demand, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 4, achieving higher efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoEn also covers powering WSNs, essential for IoE real- ization, where the limited battery life is a key issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Non- deterministic battery depletion threatens sensor reliability, replacement of which means high maintenance costs and frequent disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' EH, together with WPT, can mitigate this by making IoE devices self-sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', energy-autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, the availability of harvestable sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', solar, often depends on external factors, imposing another challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' One way to tackle that is to adopt multiple EH mechanisms, enabling higher reliability and self-sufficiency of sensors [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Yet, energy-efficient sensing, computation and communication techniques should also be considered for uninterrupted opera- tions in the IoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The other challenges of IoEn are as follows: Interoperability: Integrating small- and large-scale sup- ply&demand from various sources into decentralized and deregulated energy production increases the complexity of power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Thus, interoperability of energy systems and their monitoring via the WSNs is an important issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Privacy: The privacy of customers can be exploited by analyzing their real-time electricity usage behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' thus, information-centric networking solutions assuring data pro- tection have to be put into practice to protect user identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Security: In a fully autonomous power grid, any cyber- attack or failure in information technologies can result in billion-dollar losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' hence, security is another crucial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1 depicts, IoEn is essential for all IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Electric vehicles, as one example, impose a growing electricity de- mand, requiring efficient planning and management of energy from generation to utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The battery state and location of vehicles together with nearby charging stations’ capacity/status can be tracked in real-time through the IoV to optimally match vehicles with stations, minimizing the waiting times and the Generation Transmission Distribution & Consumption loEn Distributed Generation6 pressure on scarce resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Increasing the share of renewable sources in energy consumption can help achieve this goal while lowering greenhouse gas emissions and thus positively contribute to global Net Zero commitments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Space (IoSp) Communication satellites (CSs) placed in Earth’s orbits play a significant role in supporting the Internet and are expected to be the backbone of future IoE [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' However, the existing CS infrastructure cannot support the exploding demand for data traffic with plausible rates, and to mend this, new-age high- throughput satellites (HTSs) equipped with spot-beam tech- nology are being launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Besides the conventional large (>5tonnes) HTSs deployed in geostationary orbit (GEO), new- generation small (<500kg) CSs are increasingly being de- ployed into low earth orbit (LEO), and constellations of small CSs which network together into a dynamic web of global coverage, such as those embarked upon SpaceX and OneWeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Recent advancements in space expedition capabilities intro- duced by SpaceX have made populating space with human artefacts a cheaper and faster process to the extent that a human colonization effort of Mars is no longer an unfeasible prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Such an effort would include CSs joining Mars Reconnaissance and Odyssey orbiters relaying information received from the Curiosity Rover with 1Gbit/day capacity, which is insufficient to support this effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' With CSs orbiting Mars deployed, the Internet on Mars can be established to join the Internet on Earth as part of a greater Internet, namely IoSp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The key challenges in IoSP realization are as below: Ubiquitous Connectivity: Primary communication chal- lenge in the operation of an interplanetary link involves establishing ubiquitous connectivity in a network with very high inter-node distance, which requires developing delay- and disruption-tolerant communication protocols, besides the strategic deployment of relay stations to minimize the inter-node communication disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Energy Resources: Communication devices deployed into space need to harvest energy, typically from the Sun, to perform their tasks uninterrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Within the IoSp frame- work, many communication nodes will need to be located in isolated corners of space away from the Sun, where sunlight intensity is insufficient to support communications by contemporary EH techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Thus, advancements in IoSp call for more efficient EH methods and low-cost communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoSp interacts with various IoXs to improve the performance of terrestrial applications, among which the most notable ones are secure software updates for autonomous & connected cars (w/IoV) and reliable operation of the crypto market (w/IoM), avoiding cyber-attacks and government censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Internet of Agricultural Things (IoAT) Over the last decade, we have seen the rise of smart agriculture technologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', farmland WSNs, due to the increasing food demand, food security, and health concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' As part of the IoE framework, the IoAT covers the existing and developing information and communication technology (ICT) applications in agriculture, ranging from smart farming to smart food logistics, processing, and awareness, reinforced and diversified with the introduction of novel IoE concepts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', Internet of Plants and Animals (IoPA), IoNT, IoV, that aim at the integration of every component of agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Existing IoAT applications include precision agriculture with high-accuracy weather forecasts and livestock health monitoring enabled by wearable sensors, water-efficient irri- gation systems, real-time tracking of individual products for food awareness and supply chain planning [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Developing predictive analytics tools for analyzing the voluminous data generated by the heterogeneous components of the IoAT is a major topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' One of the large-scale research projects in this direction was the Internet of Food and Farm 2020, aiming to maintain safe and healthy food through IoT technologies5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Despite the advancements in the area, there are still major challenges hampering the realization of a full-fledged IoAT: Ubiquitous Connectivity and Interoperability: IoAT re- quires connectivity for network components in hard-to-reach environments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', remote farmlands, and seamless inter- operability among heterogeneous IoAT components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', intrabody IoNT nodes monitoring livestock health status, the network of plants and animals, through bio-cyber interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Big Data: The big data generated by the IoAT components will be highly heterogeneous as it includes those generated by living entities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', plants, animals, and humans, neces- sitating the development of novel predictive analytics tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Applications can be diversified with the close collaboration of IoAT with other IoE components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' For example, IoNT can enable real-time and continuous health/quality monitoring of crops and livestock with molecular precision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' energy-neutral drones [15] (IoV) can enable automated multimedia farm- monitoring and cattle-tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoSp can offer more secure, reliable, and fast farmland connectivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoPA, using vari- ous communication modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', acoustic, chemical, can provide better understanding and control of the biosphere increasing the efficiency of the agricultural processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' CONCLUSIONS This paper brings the upcoming IoE revolution to attention and defines its most promising applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' There is a greater opportunity brought by the IoE vision that is still untouched because all existing IoX applications have been targeting a single domain, not requiring broad connectivity or interaction with other IoXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' To realize this opportunity, individual IoXs will be seamlessly interacting under the IoE umbrella and continuously feeding each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' In this way, we can close the technological gap between our communication infrastructure and the universe, thus fulfilling the holistic vision of the IoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' The long-standing quest to develop ICT to better interact with the entire universe has enabled the large-scale deploy- ment of WSNs over several domains starting from the early 2000s, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Since we started to exploit the data collected through WSNs more effectively with advanced 5Internet of Food and Farm 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Source: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='iof2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='eu/ 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' IoE roadmap: past, present, and future of IoE technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' data analytics tools and cloud computing, we have seen the emergence of more intelligent applications, such as semi- autonomous driving and smart automation, starting from the early 2010s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Yet, this progress is not disruptive enough for developing fully automated processes covering heterogeneous technologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', fully autonomous grid, driving, farming, and manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' This ambitious goal requires overcoming substantial challenges, such as universal-scale connectivity, interoperability, standardization of heterogeneous IoXs, self- sustainability, and processing of big and heterogeneous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' We have already seen some progress towards addressing the key challenges of the IoE, such as the development of brain- machine interfaces, and neural implants for interconnecting different physical domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=', living cells and artificial devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' bio-compatible EH methods for self-sufficient sen- sors/actuators even at nanoscale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' data analytics and machine learning algorithms to make use of big data for improving services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' At this point, more practical steps can be taken by developing new communication techniques orthogonal to the conventional EM to extend the connectivity, devising universal transceivers that can support multiple communication 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 22-28, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Ozgur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Akan received his PhD degree from the Georgia Institute of Technology, Atlanta, GA, USA, in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' He is currently the Head of the Internet of Everything Group, Department of Engineering, University of Cambridge, UK, and the Director of the Next-generation and Wireless Communications Laboratory, Department of Electrical and Electronics Engi- neering, Koc University, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' His research interests include wireless, nano, molecular, and neural communications, and the Internet of Everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Ergin Dinc received his PhD degree in Electrical and Electronics Engineering from Koc University, Turkey, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' After his PhD, he held postdoctoral positions at KTH Royal Institute of Technology, Sweden, and University of Cambridge, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' His research interests are molecular communications, neural communication, and cyber–physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Murat Kuscu received his PhD degrees in Engineering from University of Cambridge, UK, in 2020, and in Electrical and Electronics Engineering from Koc University, Turkey, in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' He is currently an Assistant Professor at the Department of Electrical and Electronics Engineering, Koc University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' His research interests include Internet of Bio-Nano Things, nanomaterials, biosensors, and microfluidics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Oktay Cetinkaya received his PhD degree in Electrical and Electronics Engineering from Koc University, Turkey, in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' After his PhD, he worked as a Research Fellow at the University of Southampton, UK, a Research Associate at the University of Sheffield, UK, and a Senior Research Associate at the University of Oxford, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' His research focuses on energy-neutral communications in the Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Bilgesu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' Bilgin received his PhD degree in Mathematics from Koc University, Turkey, in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' After his PhD, he worked as a postdoctoral researcher at Koc University and the University of Cambridge, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' His research interests include molecular communication, intrabody nanonetworks, and dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content=' 2000 2010 2020 2030 2040 2050 Fully-autonomous Smart loEn 01234 Metering Grid Grid Vehicular Fully-autonomous Semi-autonomous Flying Cars lov O Driving Networks Driving First Mars Colony LEO Communication loSp Interplenatary Satellites Interplanetary Internet 5 Internet Harvesting Agricultural Agricultural Q Fully-autonomous IoAT WSNS UAVs Robots Farming Smart Industrial IloT Industry4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 Exoscelatons Industry5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} +page_content='0 WSNs Automation B Cryptocurrency Blockchain& loM Ethereum Bitcoin Standardization Molecular Synthetic Bacteria Bio-cyber IoNT Nanonetworks Networks Interface Communication Brain-machine Neural Skill Sense loPS Telepathy Interface Implants Transfer Transfer' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQftQUl/content/2301.03374v1.pdf'} diff --git a/PtAyT4oBgHgl3EQftfn_/vector_store/index.faiss b/PtAyT4oBgHgl3EQftfn_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9cd4b066ffe5bb6e1beb10ea2a7f1c82411f955a --- /dev/null +++ b/PtAyT4oBgHgl3EQftfn_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4449ffa5b5d701f46303619a0a438576cbf95037ea97750cb71acc5ec2e7c8f8 +size 5767213 diff --git a/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/2301.01777v1.pdf.txt b/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/2301.01777v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8fc43c2024b242a0c8c0dab72d49203704ef43f --- /dev/null +++ b/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/2301.01777v1.pdf.txt @@ -0,0 +1,2680 @@ +Intrinsically-multilayer moir´e heterostructures +Aaron Dunbrack1 and Jennifer Cano1, 2 +1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11974, USA +2Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA +(Dated: January 6, 2023) +We introduce trilayer and multilayer moir´e heterostructures that cannot be viewed from the +“moir´e-of-moir´e” perspective of helically-twisted trilayer graphene. +These “intrinsically trilayer” +moir´e systems feature periodic modulation of a local quasicrystalline structure. +They open the +door to realizing moir´e heterostructures with vastly more material constituents because they do not +constrain the lattice constants of the layers. In this manuscript, we define intrinsically multilayer +patterns, provide a recipe for their construction, derive their local configuration space, and connect +the visual patterns to physical observables in material systems. +I. +INTRODUCTION +The observation of superconductivity and correlated +insulators in twisted bilayer graphene [1, 2] launched the +study of “moir´e materials,” where two-dimensional ma- +terials with the same [1–35] or similar [36–45] lattice con- +stants are stacked at a small relative twist angle. This +paradigm is naturally extended to trilayer stacking and +beyond, both with some layers aligned [46–52] and with +multiple twist angles [53–58]. Recently it has also been +extended to stacking at angles nearby a large commensu- +rate twist angle [59, 60]. In all cases, the moir´e pattern +is obtained from layers with either the same or similar +lattice constant (or a commensurate supercell). In this +paper, we lift that restriction. +We introduce moir´e patterns made from stacking more +than two layers in which no two layers separately dis- +play a moir´e pattern. We call these patterns “intrinsi- +Types of +moir´e +patterns +Small twist +Large twist +Two layers +Twisted bilayer +graphene +Near-commensurate +TBLG +Three or +more layers +Twisted trilayer +graphene +Intrinsically trilayer +moir´e +TABLE I: Summary of moir´e heterostructures: the +“intrinsically trilayer” moir´e patterns we introduce +occur at large twist angle and with three or more layers. +cally trilayer moir´e” (or more generally, “intrinsically N- +layer moire”) because, unlike twisted trilayer graphene, +the moir´e pattern disappears if any one layer is removed. +As we will explain, intrinsically trilayer moir´e patterns +cannot be viewed from the “moir´e of moir´e” perspective +often used to describe twisted trilayer graphene [53]. +Intrinsically N-layer moir´e patterns have an important +advantage over bilayer moir´e patterns because they do +not impose a constraint on lattice constants. This vastly +increases the space of possible material combinations. +Specifically, moir´e patterns in bilayer systems require the +constituent materials to have nearly the same lattice con- +stant or to be nearly commensurate. In contrast, intrin- +sically N-layer moir´e patterns can be constructed from +virtually arbitrary combinations of materials. +In the present work, we focus on the crystal structure of +intrinsically N-layer moir´e heterostructures, postponing +a study of electronic structure to future work. +We begin by reviewing the origin of moir´e patterns. In +Sec. II, we provide an intuitive picture of how moir´e pat- +terns arise in real space. We explain the construction for +bilayers and then offer a na¨ıve generalization to multilay- +ers. In Sec. III, we argue that reciprocal space provides +a more natural and concise characterization, from which +we derive both bilayer and N-layer moir´e patterns. +We then focus on multilayer heterostructures. +In +Sec. IV, we return to real space to resolve an appar- +ent contradiction: the momentum-space perspective im- +plies that periodic moir´e patterns of more than two lay- +ers exist, but the na¨ıve generalization of bilayer config- +uration space [61, 62] fails to indicate these patterns, in +part because the local structure is generally quasicrys- +talline rather than crystalline. Consequently, we develop +a more nuanced notion of configuration space, in which +some apparent degrees of freedom disappear on moir´e +wavelengths. We discuss physical properties that are a +function of this configuration space; lattice relaxation is +one example. +Finally, in Sec. V, we discuss experimental probes +and propose physical realizations of intrinisically N-layer +moir´e patterns. +Throughout, we assume a three-, four-, or six-fold ro- +tation symmetry shared between all layers of the moir´e +arXiv:2301.01777v1 [cond-mat.mes-hall] 4 Jan 2023 + +2 +−10 +0 +10 +−10 +0 +10 +FIG. 1: A moir´e lattice of two square layers twisted at +6.7329◦. Commensurate lattice in red, moir´e lattice in +blue. +heterostructure. +In the absence of this symmetry, the +generic moir´e pattern will be stripes rather than a 2D +pattern. +II. +CONFIGURATION SPACE FOR BILAYERS: +MOIR´E PATTERNS IN REAL SPACE +Moir´e patterns are intuitively understood in real space +as a slow modulation of the local lattice structure. The +set of all possible local environments is known as config- +uration space [61, 62]. The configuration space approach +extends beyond linear transformations of perfectly rigid +crystals to include lattice relaxation effects. However, the +approach becomes subtle for heterostructures of multiple +layers or different lattice constants. +In this section, we review configuration space in the +simplest case of bilayers with near-identical lattices. We +then extend the formalism to bilayer systems perturbed +from a commensurate stacking. Finally, we offer a “na¨ıve +configuration space” for trilayer systems, and briefly dis- +cuss how it leads to the complex patterns observed in +twisted trilayer graphene. (Later, in Sec. IV, we will pro- +vide a more complete accounting of configuration space +in systems with more than two layers and explain the +breakdown of the na¨ıve configuration space.) +A. +Two square lattices +Consider two stacked periodic layers. There are two +cases to consider: when the two layers share a common +(larger) period, and when they do not. If they do share +a common period, we call the structures commensurate. +If they do not, we call them incommensurate. +In Fig. 1, we illustrate a small commensurate pattern +formed by two square lattices at a relative twist angle of +approximately 6.7◦ about a square corner. This aligns +the square corners of the unit cell (8,9) of one layer with +(9,8) of the other, forming the commensurate superlattice +outlined in red. +However, in the center of each red supercell is a lo- +cation that looks very similar to the corners, where the +unit cells are also aligned at the center of the square cells +rather than at a vertex. This smaller grid of locations +where the square-centers are aligned defines the moir´e +lattice, outlined in blue. +Thus, the visual moir´e cell, +which enjoys an approximate translation symmetry, is +smaller than the commensurate unit cell, which exhibits +an exact translation symmetry. +In general, the visual +pattern will either be the same size or smaller than the +commensurate cell (although for two identical square lat- +tices, the moir´e cell is always smaller by at least a factor +of +√ +2, regardless of twist angle. +The commensurate cell size is highly sensitive to angle +and exists only on a dense subset of angles. Computing +the size of a commensurate cell as a function of twist +angle is analogous to determining the size of the minimal +denominator of a fraction as a function of the value of +that fraction, as explained in Supplement 1. +The moir´e cell, however, varies smoothly with twist +angle for small twist angles. At sufficiently large twist +angles, the moir´e cell becomes smaller than a unit cell, +which indicates that the moir´e pattern ceases to exist and +no visual pattern arises. +This example shows how a moir´e pattern arises from +the two layers being stacked at different “local relative +translations” at different positions, i.e., in the brighter +regions, the lattices are stacked atom-on-atom, while in +the darker regions, the lattices are stacked atom-on-void. +The moir´e lattice is defined by the collection of points +where the two layers align in either configuration. +B. +Local configuration space: two identical layers +The space of relative translations of the aligned lay- +ers defines the local configuration space. For instance, +TBLG exhibits regions of AA and AB stacking, as well +as intermediate regions, as illustrated in Fig. 2. +For two identical layers, the local configuration space +is defined with respect to relative translations of the two +untwisted layers, as we will now describe. Although the +idea is intuitive in this case, developing the mathemat- +ical infrastructure carefully here will elucidate the more +complicated situations we consider later. +1. +Configuration space as differences of relative coordinates +In the simplest setup where the two untwisted layers +have identical lattice vectors, we define the local configu- +ration C(x) in terms of the relative coordinates xi of each +layer. The relative coordinate xi(x) is a two-component +vector that specifies where the position x resides in the + +3 +−30 +−15 +0 +15 +30 +−30 +−15 +0 +15 +30 +FIG. 2: A moir´e lattice of two hexagonal layers with +unit-length interatomic distance stacked with a relative +twist angle of 5◦. Red and blue circles indicate an +“AA-stacked” region where hexagons align and an +“AB-stacked” region where they are offset, respectively. +unit cell of layer i. Thus, xi is determined by the ma- +trix Ai, whose columns are the (twisted) lattice vectors +of layer i, as +xi(x) = A−1 +i x +mod I +(1) +where “mod I” means “modulo the columns of I” (i.e., +mod {(1, 0), (0, 1)}). The local configuration is then de- +fined as the difference between the two relative coordi- +nates +C(x) = x2(x) − x1(x) +mod I +(2) += (A−1 +2 +− A−1 +1 )x +mod I +(3) +While the functions xi vary on the scale of the original +lattice, for a small twist or lattice mismatch, C(x) varies +much more slowly, and the period of C(x) defines the +moir´e lattice. +Therefore, the moir´e lattice vectors are +given by the columns of the matrix +AM = (A−1 +2 +− A−1 +1 )−1 +(4) +in the case where the inverse exists. If the inverse does +not exist, then there is not a 2D moir´e pattern. +In the case where the two layers are identical and +twisted by a relative angle θ, one can simplify further +by writing A1,2 = R(±θ/2)A, where R(θ) is the rotation +matrix. The moir´e lattice vectors then simplify to +AM = [R(θ/2) − R(−θ/2)]−1 A = +1 +2 sin(θ/2)R +�π +2 +� +A. +(5) +In other words, the moir´e lattice vectors are rotated by +π/2 compared to the original lattice vectors A and scaled +up by a factor of 1/(2 sin(θ/2)). +The same formalism applies to aligned layers with a +small difference in their lattice constants. For example, +if A2 = (1+δ)A1, then Eq. (4) can be simplified without +any matrix algebra to AM = 1+δ +δ A1 (neglecting the over- +all sign). Generalizing to the case of two layers with a +small lattice mismatch arranged with a slight twist angle +yields Eq. (1) in Ref. 63. +Eq. (4) in this paper also allows for anisotropic lattice +mismatch, as might be induced by a strain. +2. +Configuration space as a quotient of translation groups +More abstractly, configuration space is equivalently de- +fined as the space of nontrivial translations of the lattices +before twisting, as we now explain. +A combination of +translations is “trivial” if it differs from zero translation +of each layer by the simultaneous translation of all layers +by the same amount. +In other words: consider the two identical lattices be- +fore twisting. Denote the group of translations of each +layer modulo lattice translations by Ti. (Note Ti will be +isomorphic to the torus T 2 = R2/Z2.) Similarly denote +the group of translations of the two lattices simultane- +ously (modulo translations that preserve the shared pre- +twist lattice) as T12. The space of configurations is the +space of translations of each layer, modulo simultaneous +translations of the two layers: +Tconfig = T1 × T2/T12. +(6) +This space of configurations is itself a torus. +We now relate this space to the moir´e pattern. Suppose +we transform each layer by a linear transformation Mi, +e.g., for twist, Mi = R(θi). In terms of the matrices of +lattice vectors before and after twisting, +Mi = AiA−1. +(7) +We now interpret this transformation as a position- +dependent translation, which will give the Ti-coordinate +in Eq. (6). +To find the translation of one layer associated with a +point x0 in real space, consider the map which first trans- +forms physical space, then transforms back but centered +at x0. (E.g., for a twist by θ, first twist about the origin +by θ, then twist back around x0 by −θ.) Conceptually, +the first transformation sets up the twisted system, and +the latter re-aligns the layers without further translating +x0. +Algebraically, understanding that “transform around +x0” can be written as “translate x0 to the origin, trans- +form, then translate back,” the translation is given by +x → M −1 +i +(Mix − x0) + x0 = x − (M −1 +i +− I)x0, +(8) +which is a translation because it takes the form x → +x − a. This translation is then taken modulo the pre- +twist lattice vectors to get the element of T1. + +4 +−20 +0 +20 +−20 +0 +20 +FIG. 3: Moir´e pattern from two square lattices with +side lengths 1 and +√ +2 arranged with a relative twist +angle of 42◦. +Doing this for each layer yields the translation opera- +tors that determine a point in configuration space defined +by Eq. (6). Modding out by simultaneous translations in +Eq. (6) yields the relative translation difference between +the two layers, +˜C(x) = (M −1 +2 +− M −1 +1 )x +mod A +(9) +where A is the shared lattice before twisting. +This is +in one-to-one correspondence with the characterization +of configuration space in Eq. (3). The moir´e unit cell is +given by +AM = (M −1 +2 +− M −1 +1 )−1A, +(10) +which is exactly Eq. (4). Written in this way, the moir´e +lattice is “factored” into one term, M −1 +2 +−M −1 +1 , that de- +pends on the transformations but not the original lattice, +and another term, A, that depends on the lattice but not +the transformations. The second term can be interpreted +as the size of configuration space and the first as the rate +at which the moir´e pattern explores that space. +C. +Generalization to near-commensurate twisting +Now instead of two identical layers, consider two layers +that form a small (i.e., not moir´e) commensurate super- +cell. Applying a small twist or lattice mismatch produces +a moir´e pattern. For instance, two square lattices whose +side lengths differ by a factor of +√ +2 form a commensu- +rate supercell when arranged at a 45◦ relative orientation; +when twisted by an angle near 45◦, they form a moir´e +pattern as illustrated in Fig. 3. A second example is two +identical honeycomb lattices twisted near a commensu- +rate angle that is not a multiple of 60◦, as discussed in +Ref. 60; near-21.8◦ TBLG is shown in Fig. 4. +The abstract description of configuration space de- +scribed in Eq. (6) extends to this case with only one minor +modification: instead of considering the translations as +acting on the lattices at zero twist, consider them at the +relevant commensurate stacking. Hence, the Ti are now +defined modulo the individual lattices at the commensu- +rate stacking, whereas T12 is defined modulo the lattice +vectors of the commensurate structure. +An argument for the size of the moir´e pattern comes +from Eq. (9) and the subsequent discussion. +A lin- +ear transformation (e.g. +twist) performed on a near- +commensurate structure explores the configuration space +at the same rate as the structure formed by performing +the same transformation on a zero-degree stacked struc- +ture. However, the configuration space of the former is +(perhaps counterintuitively) smaller, for reasons we now +explain heuristically. +The size of configuration space in the case of two lay- +ers stacked to form a supercell can be sensibly guessed +from Eq. (6). Let Ai, AC, Acs and AM denote the areas +of the unit cell of layer i, the commensurate supercell, +configuration space, and the moir´e unit cell, respectively. +Replacing each translation group in Eq. (6) by the area +of the corresponding torus yields +Acs = A1A2 +AC +. +(11) +Exploiting the fact that Ai = | det(Ai)| and guided by +the intuition that A in Eq. (10) should be generalized to +some “configuration space lattice,” the area of the moir´e +cell is +AM = +1 +| det +� +M −1 +2 +− M −1 +1 +� +| +A1A2 +AC +. +(12) +The intuition that we should use the configuration space +lattice follows from factoring Eq. (10) as described in the +text following that equation. +(We give a rigorous de- +scription of how to find the “configuration space lattice +vectors” Acs in Appendix B and prove that they are in- +deed the analogue of A in Eq. (10).) +As a concrete example, consider two identical lattices +twisted at an angle θ away from a commensurate stack- +ing where the commensurate cell is a factor of N larger +in area than the original unit cell (for instance, in near- +21.8◦ TBLG, the commensurate cell is 7 times larger in +area than the original graphene cell). The size of Ti does +not depend on how the layers are stacked, but T12 will +be a factor of N larger in area when they are twisted +θ away from the commensurate stacking compared to +when the layers are stacked at an overall twist angle of +θ. +Therefore, according to Eq. (11), the configuration +space, which is defined modulo T12, would be a factor of +N smaller. Since the matrices M1,2 in the denominator +of Eq. (12) depend only on θ and not on the supercell +or original lattice, it follows that, contrary to the most +obvious intuition, for two specified 2D layers, the larger +the commensurate cell, the smaller the moir´e pattern. + +5 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−6 +−4 +−2 +0 +2 +4 +6 +−6 +−4 +−2 +0 +2 +4 +6 +−6 +−4 +−2 +0 +2 +4 +6 +14 +16 +18 +20 +22 +24 +26 +FIG. 4: A moir´e pattern formed by two unit triangular lattices arranged with a relative twist of 22.4◦ (21.8◦ + 0.6◦). +The resulting triangular moir´e lattice has a unit cell of side length 36.1, shown in green. The moir´e pattern is subtle, +alternating between regions with individual sixfold-symmetric “centers” (red) and regions with triplets of “centers” +connected in a triangle (blue). A larger picture of the moir´e pattern is shown in Fig. S2-1. +In Appendix B, in addition to formally deriving +Eq. (11), the relative coordinates of heterostructures +nearby a supercell configuration are derived, generaliz- +ing Eq. (9). +D. +A na¨ıve approach to configuration space with +more than two layers +We now try to apply the idea of configuration space as +the translation of each layer modulo overall translations +to heterostructures with more than two layers. We call +this notion “na¨ıve configuration space” (in contrast to a +more nuanced notion to be given in Sec. IV). For instance, +in the case of three identical layers near zero stacking, as +in twisted trilayer graphene, the local configuration space +is a four-dimensional torus: +Tconfig = T1 × T2 × T3/T123 +(13) +In general, the local configuration space of N arbitrarily- +twisted layers (with respect to a reference configuration) +is a (2N − 2)-dimensional torus: +Tconfig = +�� +i +Ti +� +/Tall +(14) +Because this configuration space has dimension greater +than two, we do not generally expect that it is fully ex- +plored. The consequence is a complex structure of over- +lapping moir´e patterns (illustrated for twisted trilayer +graphene in Fig. 1b of Ref. 64), and the four-dimensional +space will generally be the correct parameter space for +many layers twisted near a single commensurate struc- +ture of all layers (as can be seen in, e.g., Ref. 54). +As the next section will show, however, there are moir´e +patterns that arise when multilayer structures are twisted +near special incommensurate configurations. +In these +cases, more care is required to define which configura- +tions are distinct in a way that will manifest on moir´e +lengthscales: Tconfig as written in Eq. (14) is not correct +because Tall is not the correct space by which to mod +out. + +6 +III. +MOIR´E IN FREQUENCY SPACE +An alternative to defining a moir´e pattern in real space +is to define it by the appearance of low-frequency modes +in momentum space. This approach is discussed at length +in Ref. 65; here we summarize by focusing on the modes +of a black-and-white image. +However, the content is +much more general; see Appendix A for details. +Consider a layered material as a set of transparencies +placed over a light source. The atomic structure defines +a local transmission coefficient Ti(x) that specifies how +much light layer i lets through at point x. For a black- +and-white image, Ti(x) = 1 wherever the layer’s image +is white and Ti(x) = 0 where it is black; this paradigm +extends to grayscale images using opacities between zero +and one. +By the definition of the transmission function, given +Ti(x) in each layer i, the resulting transmission function +of the layered structure is given by: +T(x) = +� +i +Ti(x), +(15) +which defines how the resulting multilayer pattern is +formed from the patterns of the individual layers. +The moir´e-scale physics emerges by extracting the low- +frequency modes. In each periodic layer i, the Fourier +transform is defined by: +Ti(x) = +� +n +ci,n exp (iki,n · x) +(16) +where the sum is over the reciprocal lattice vectors ki,n. +Fourier transforming Eq. (15) yields: +ˆT(k) = [ ˆT1 ∗ ˆT2 ∗ . . . ∗ ˆTN](k), +(17) +where ∗ denotes the discretized convolution: +[f ∗ g](k) = +� +n,m +cndmδ(k − kn − k′ +m), +(18) +so that +[T1 ∗ . . . ∗ TN](k) = +� +n1,...,nN +��� +i +ci,ni +� +δ(k − +� +i +ki,ni) +� +(19) +Therefore, a low-frequency (small-k) mode requires +there exist a collection of modes ni so that � +i ki,ni ≈ 0. +This sum is the moir´e wavevector, +kM = +� +i +ki,ni, +(20) +which in turn yields the moir´e wavelength and orienta- +tion. +Such a collection of modes arise naturally by consider- +ing a small deformation (twist, stretch, etc.) away from +a reference configuration where � +i ki,ni = 0 exactly. For +a bilayer system, k1,n + k2,m = 0 is precisely a commen- +surability condition. +The case n = m corresponds to +the familiar near-zero-degree moir´e pattern for nearly- +identical lattices. On the other hand, the case n ̸= m +corresponds to a near-commensurate moir´e, which can +result when the two lattices differ in size (illustrated in +Fig. 3) or are arranged near a commensurate angle (il- +lustrated in Figs. 4 and 5). +A. +Near-commensurate example +As a concrete example, consider two square lattices +arranged with a twist angle near the 36.9◦ commensurate +angle, as illustrated in Fig. 5. The lowest Fourier modes +before twisting are illustrated in Fig. 6; note the (1,2) +mode of one layer coincides with the (2,1) mode of the +other. The magnitude of the wave vector of these modes +is |k36.9| = +√ +5k0, where k0 is the magnitude of the wave +vector of the lowest mode of a single layer. +In general, if two modes with a wave vector of magni- +tude |k| are initially aligned before twisting, then after a +relative twist by an angle θ, the difference between the +two wave vectors has magnitude +|kM| = 2 sin(θ/2)|k|, +(21) +as is seen geometrically in Fig. 7 and can be derived +mathematically by taking k1 = −R(θ)k2 in Eq. (20). +Accordingly, the moir´e pattern at 36.9◦ + θ is a factor +of +√ +5 smaller in real space than the moir´e pattern at +0◦ + θ because +|k36.9+θ +M +| = 2 sin(θ/2)|k36.9| += 2 sin(θ/2) +√ +5|k0| += +√ +5|k0+θ +M |. +(22) +The same result was obtained in Sec. II C through more +complicated arguments in real space. +The moir´e patterns obtained from twisting near a com- +mensurate angle, as illustrated in Figs. 4 and 5, are +fainter than those for the corresponding structures near +zero degrees in Figs. 2 and 1, respectively. +The faint +pattern occurs because the higher-frequency modes have +smaller amplitudes than the lowest mode, and therefore +the coefficients cndm in Eq. (18) are smaller. (The range +of visibility of different near-commensurate moir´e pat- +terns is also illustrated in Fig. 3.2 of Ref. 65.) +B. +Intrinsically multilayer moir´e +The moir´e formalism in reciprocal space, i.e. Eq. (20), +also provides a requirement for a moir´e pattern to exist +in a multilayer heterostructure: there must exist a linear +combination of reciprocal lattice vectors in the different +layers that adds up to a vector much smaller than the +reciprocal lattice vectors of the original layers. +In the + +7 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−10 −8 +−6 +−4 +−2 +0 +2 +4 +6 +8 +10 +−10 +−8 +−6 +−4 +−2 +0 +2 +4 +6 +8 +10 +−10 −8 +−6 +−4 +−2 +0 +2 +4 +6 +8 +10 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +FIG. 5: A moir´e lattice formed by two unit square lattices arranged at a relative twist of 37.5◦ (36.9◦ + 0.6◦), with a +42.7 side length moir´e cell (green square). There is a resulting pattern of “holey regions” (red square) and “knitted +regions” (blue square). A larger unannotated picture of the moir´e pattern is presented in Fig. S2-2. +following, we provide a recipe for meeting this condition +that is analogous to twisting near commensurate struc- +tures. +First, find a stacking arrangement of the layers such +that a reciprocal lattice vector can be chosen in each layer +so that the sum over the chosen reciprocal lattice vectors +in all layers is zero, i.e., � +i ki,ni = 0, where ki,ni is the +chosen reciprocal lattice vector in layer i. We call such +a configuration singular (following the terminology from +Ref. 65), which is a generalization of a commensurate +configuration. +Note this notation differs from Ref. 61, +where incommensurate is defined as non-singular in our +terminology. +Once a singular configuration is identified, a small +twist or stretch of each layer away from the singular con- +figuration results in the same sum of reciprocal lattice +vectors being nonzero but small. This small sum of the +lattice vectors is precisely a reciprocal lattice vector of +the moir´e lattice, as defined in Eq. (20). +We call a moire pattern “intrinsically n-layer” if it orig- +inates from a singular configuration where no two lay- +ers are singular. In other words, an intrinsically n-layer +moir´e material is one whose singular configuration is a +sum of reciprocal lattice vectors from all layers that add +to zero, but no two vectors from that sum add to zero by +themselves. Notice this is distinct from, e.g., helically- +twisted trilayer graphene [53–56]; there the singular pat- +tern is at zero twist angle, where any two layers have +reciprocal lattice vectors which add to zero.(Patterns +where some layers are aligned, such as alternating-twisted +trilayer[46, 47] and twisted double bilayer graphene[48– +51], often have patterns that arise from only two mis- +aligned sets of layers, rather than more than two; more- +over, such patterns are always singular in themselves.) +An example of an intrinsically trilayer moir´e pattern +is three square lattices twisted near 120◦, illustrated in +Fig. 8. The sum of the n = (1, 0) lattice vectors from each +layer vanishes, so at 120◦ there is a singular structure. +Notice that this singular structure is not commensurate; +in fact, it is a twelvefold-symmetric quasicrystal. In gen- +eral, the singular structures will be quasicrystalline, but +not necessarily with higher rotational symmetries. + +8 +FIG. 6: Reciprocal space of two square lattices stacked +at a commensurate 36.9◦ twist angle. Red(blue) open +circles indicate the reciprocal lattice vectors of the +top(bottom) layer; black filled circles indicate shared +reciprocal lattice vectors. Thick lines shows that the +(1,2) mode of the blue layer coincides with the (2,1) +mode of the red layer. Light gray indicates the +reciprocal commensurate lattice. +k1 +k2 +kM +FIG. 7: Lowest frequency modes of two square lattices +at a small relative twist. Red and blue circles indicate +reciprocal lattice vectors of each layer. The small +difference between the lowest modes k1 − k2 gives the +moir´e wavevector kM, from which Eq. (21) follows. +1. +What is a singular structure? +Since the notion of a “singular structure” is not a stan- +dard notion of the physics literature (although it has ap- +peared in the mathematical literature on moir´e patterns; +see Ref. 65), it is worth spending a moment highlighting +both how it is different from a commensurate structure +and how it is different from a general twist angle. +First, a multilayer system is commensurate if the com- +bined system has exact translation symmetries. In other +words, there must exist lattice vectors a1,2 for the multi- +layer system such that, for each layer i with lattice vec- +tors a(i) +1,2, the vectors a1,2 are integer linear combinations +of a(i) +1,2. As shown in Appendix D, this definition of com- +mensurate is equivalent to every layer being individually +commensurate with the first layer. Therefore, in an N +layer system with threefold or fourfold rotational symme- +try, commensurability imposes 2N − 2 scalar constraints +(from N − 1 vector constraints) on the size and orienta- +tion of the lattice vectors. +By +contrast, +consider +the +singularity +condition +� +i ki,ni = 0, where ki,ni are each reciprocal lattice vec- +tors of layer i. This imposes only two scalar constraints +(one vector constraint) on the orientations of layers, re- +gardless of the number of layers. For a bilayer system, +the singularity condition is equivalent to commensurabil- +ity, but with more than two layers, commensurability is +a strictly stronger condition. +Now contrast that situation with generic twist an- +gles. Singular structures have a property unusual among +twisted systems: the average, long-distance properties of +the system are sensitive to relative translations of the +layers, as we now explain. +Given a system with a local property f(x), the av- +erage value of that property over an area A is given +by +1 +|A| +� +A f(x)d2x. If that area becomes very large, un- +der appropriate convergence conditions on f, the average +value converges to the Fourier transform of f at the ori- +gin, ˆf(0). +Suppose now that f(x) can be written as a product +of functions of each layer; e.g., for a trilayer system, +f(x) = f1(x)f2(x)f3(x), where fi(x) is periodic with the +periodicity of layer i. Notice the transmission function +defined in Eq. (15) has this property. +The zeroth Fourier mode of f is determined by Fourier +modes ˆfi(ki) of each layer such that � +i ki = 0, as shown +in Eq. (19). If the layers are not stacked in a singular +structure, the only solution to � +i ki = 0 is when ki = 0 +in each layer. Therefore, the average value of f in the +multilayer is a product of the average values of f in each +individual layer; relative translations of the layers have +no impact on this zeroth Fourier mode. +By contrast, for a singular structure, there exists a +nontrivial combination of Fourier modes in each layer +that contribute to the average value of f. For instance, +consider a trilayer system with reciprocal lattice vectors +ki in each layer such that � +i ki = 0. Further suppose + +9 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−10 +−5 +0 +5 +10 +−10 +−5 +0 +5 +10 +15 +20 +25 +30 +35 +15 +20 +25 +30 +35 +G2 +G3 +G1 +G2 +G3 +G1 +GM +FIG. 8: A moir´e lattice of three unit square lattices at a relative twist of 119.3◦, resulting in a moir´e unit cell of side +length 47 (drawn in green). Local structures are shown at right. Top right illustrates the reciprocal lattice vectors at +exactly 120◦ (left) and after the 0.7◦ deviation from the singular structure (right, deviation exaggerated for +illustration purposes), resulting in the moir´e reciprocal lattice vector GM shown in green. A larger unannotated +picture of the moir´e pattern is presented in Fig. S2-3. +fi = c0,i + 2c1,i cos(ki · x), for some coefficients c0,i, c1,i. +From Eq. (19), the zeroth Fourier mode of f is +ˆf(0) = c0,1c0,2c0,3 + 2c1,1c1,2c1,3 +(23) +where the factor of 2 derives from the positive and neg- +ative contributions of the cosine. (If fi had a rotation +symmetry instead of being a 1D cosine, the factor of 2 +would turn into a 4 or 6.) Now translating each layer i +by ai transforms the zeroth Fourier mode into +ˆf(0) = c0,1c0,2c0,3 + 2c1,1c1,2c1,3 cos +�� +ki · ai +� +, +(24) +which is different for generic choices of ai. +Thus, the physical consequence of a singular structure +is that local properties of the multilayer are sensitive to +relative translations. This is also true for commensurate +structures, but is not true for a general non-singular or +non-commensurate stacking. However, notice that for a +fixed set of ki, Eq. (24) is invariant under the special set +of translations ai which satisfy � kiai = 0. These special +translations will be important in developing our notion +of configuration space for multilayer systems in Sec. IV. +As discussed in Sec. III, the condition that the physical +quantity of interest is a product of properties in each +layer, i.e., f = f1f2f3 for a trilayer system, simplifies +the discussion, but can also be relaxed significantly. The +more general description is given in Appendix A. +2. +Labelling singular structures +We now provide a convenient labelling schema for sin- +gular structures. Since a singular structure is specified +by a combination of reciprocal lattice vectors that adds +up to zero, it can be conveniently labelled by the integer +indices of the reciprocal lattice vectors. +Let bi,1 and bi,2 be the basis of reciprocal lattice vectors + +10 +in layer i. Then a singular structure will be specified by +a set of ni,j that satisfy the singularity condition +� +i,j +ni,jbi,j = 0. +(25) +For a trilayer system, the singular structure given by +ni,j is labelled as (n1,1, n1,2; n2,1, n2,2; n3,1, n3,2). +This +description can be generalized to any number of layers, +including bilayers. Note that the labelling depends on +the choice of reciprocal lattice vectors; thus, a set of ni,j +combined with knowledge of the reciprocal lattice vectors +in each layer determines the singular structure. +The ni,j for an N-layer system naturally live in Z2N. +The singularity condition in Eq. (25) defines a 1D sub- +lattice in this space. Assuming rotational symmetry, one +choice of ni,j yields another linearly-independent ni,j af- +ter rotation. Thus, combined there is a 2D sublattice in +Z2N satisfying the singularity condition. It is also possi- +ble for the sublattice to have a higher even dimension, as +we will show for trilayer graphene in Sec. III C. Regard- +less of dimension, we call the ni,j that satisfy the singu- +larity condition the zero mode lattice, because they cor- +respond to combinations of Fourier modes in each layer +that contribute to the k = 0 Fourier mode of the sin- +gular structure. Under the assumption that the sublat- +tice is 2D and that the degree of rotational symmetry +is known, each singular structure can be labelled by a +single set of ni,j that defines one of the basis vectors of +the zero mode lattice; the other basis vector follows from +rotational symmetry. +As a few concrete examples: the standard near-zero +moir´e pattern of two layers is the (1, 0; −1, 0) moir´e pat- +tern because b1,1 − b2,1 = 0. +The near-21.8◦ struc- +ture shown in Fig. 4 and the near-36.9◦ structure in +Fig. 5 are both (1, 2; −2, −1) moir´e patterns because +b1,1 + 2b1,2 − 2b2,1 − b2,2 = 0 in both cases, despite +their different rotational symmetry. Finally, the intrin- +sically trilayer pattern illustrated in Fig. 8 would be the +(1, 0; 1, 0; 1, 0) moir´e, assuming the first basis vector of +the three layers are chosen 120 degrees apart. +3. +Degeneracy of singular structures +We now consider how singular structures arise in the +manifold of possible twists and lattice mismatches be- +tween the layers, which we call deformation space. (More +generally, we could also include strains that break rota- +tional symmetries in our deformations; we call this gener- +alization anisotropic deformation space. However, since +such deformations can result in 1D instead of 2D moir´e +patterns, we neglect such transformations here and sim- +plify our discussion by referring to our space of isotropic +deformations by the shorter term.) +Commensurate structures of bilayer systems are spe- +cial among singular structures because they are zero- +dimensional manifolds in deformation space: no small +G1 +G2 +G3 +G1 +G2 +G3 +G1 +G2 +G3 +FIG. 9: Starting from a particular singular structure, a +small twist away combined with a corresponding strain +results in another singular structure. These +transformations yield a manifold of singular structures +rather than an isolated point, as occurs for bilayers. +deformation of a bilayer singular structure yields the +same singular structure. For instance, in the simple case +of aligned layers (corresponding to the (1, 0; −1, 0) com- +mensurate structure), no combination of small relative +mismatch or twist of the two layers will yield another +(1, 0; −1, 0) commensurate structure. +This is not, however, the case for singular struc- +tures with more than two layers. With N layers there +are 2N − 2 possible isotropic deformations (twists and +isotropic strains) of the layers relative to each other: each +layer beyond the first adds two additional parameters +(namely, strain and mismatch with respect to the first +layer). The singular structure then adds two constraints +(Eq. (25) and its rotated counterpart) on this deforma- +tion space, meaning that it forms a (2N −4)-dimensional +manifold in this space of deformations. +Intuitively, this is because there is a continuum of ways +to change the sides of the triangle that keep it a trian- +gle. For example, given a triangle formed by reciprocal +lattice vectors, one can deform two of the lattices by a +combination of twists and (isotropic) strains while leav- +ing the third fixed and still have a triangle, as illustrated +in Fig. 9. In contrast, the only way to deform the layers +and preserve a singular digon formed by the reciprocal +lattice vectors of a bilayer is to perform an overall twist +or isotropic stretch of both layers simultaneously. +These singularity-preserving deformations are at the +crux of understanding what the na¨ıve configuration space +description in Sec. II D fails to see about intrinsically tri- +layer moir´e patterns, namely, why the effective param- +eter space seems to be periodically spanned by the two +dimensional moir´e pattern even though the na¨ıve param- +eter space is four-dimensional. The connection between +these pictures will be explained in Sec. IV B. +C. +The doubly-singular structure of twisted +trilayer graphene +We now examine twisted trilayer graphene from the +perspective of singular structures. +Twisted trilayer +graphene arises at the intersection of two singular struc- +tures: +the (1, 0; −1, 0; 0, 0) singular structure and the +(0, 0; 1, 0; −1, 0) singular structure. +In this sense, it is +“doubly-singular”; therefore, with four singularity con- +straints instead of the two considered in the previous + +11 +θ12 +θ23 +(1,0,-1,0,0,0) +(0,0,1,0,-1,0) +(1,0,0,0,-1,0) +(a) δ12 = δ32 = 0 +δ12 = δ32 +θ12 = θ23 +(1, 0, −2, 0, 1, 0) +(b) θ12 = θ23 and δ12 = δ32 +FIG. 10: Several singular structures of TTLG along two +specific slices of the four-dimensional parameter space +(θ12, θ23, δ12, δ32) indicated by solid colored lines. The +dashed green line represents the constraint of +helically-twisted trilayer graphene. The bilayer singular +structures shown in the left figure deviate from +helically-twisted trilayer graphene to order θ, but the +trilayer singular structure shown in the right figure only +deviates to order θ2. Hence, the green singular structure +produces a moir´e pattern at 1/θ2-scale, whereas the +bilayer singular structures plotted in blue/red/purple +produce (competing) moir´e pattern at 1/θ scale. +section, the combination of singular structures is zero- +dimensional, not 2D like the intrinsically trilayer pattern +(the dimension is 2N −6 instead of 2N −4, where N = 3 +for three layers). +Twisting relative to the singular structure in this case +be understood as generating multiple moir´e patterns si- +multaneously. Without a fine-tuned combination of twist +and mismatch, the overlapping structure of the multiple +moir´e patterns yields complex and unclear-scale patterns, +as illustrated in Ref. [64]. +In the special case where the twist angles of the first +and third layers are equal and opposite, however, some- +thing special happens: at +1 +θ2 length scales, a single regu- +lar moir´e pattern is observed. This pattern is referred to +as a “moir´e of moir´e,” since it arises from a moir´e pattern +induced by the two competing 1 +θ-scale moir´e patterns. +This +1 +θ2 -order pattern can be understood as the pat- +tern arising from the (1, 0; −2, 0; 1, 0) singular structure. +Specifically, defining k0 to be a smallest reciprocal lattice +vector of graphene, the trilayer structure where the first +and third layers are twisted a small amount in opposite +directions away from the middle layer can be described +by k1 = R(θ)k0, k2 = −2k0, and k3 = R(−θ)k0. Per +Eq. (20), the moir´e wave vector is given by +kM = [R(θ) + R(−θ) − 2I]k0 = 2(cos(θ) − 1)Ik0, +(26) +which is of order θ2 for small θ. Hence, the moir´e wave- +length is of order +1 +θ2 . +Moreover, since the order-θ2 deviation is only from this +particular singular structure, and not from the “doubly- +singular” structure, it exhibits a single 2D moir´e pattern +rather than complex overlapping structures. +The rele- +vant singular structures are illustrated in Fig. 10. +IV. +CONFIGURATION SPACE OF +INTRINSICALLY TRILAYER MOIR´E PATTERNS +There is an apparent contradiction between the na¨ıve +configuration space described in Sec. II D, which indi- +cates that trilayers have complex moir´e patterns that +cannot possibly fit on a lattice, and the intrinsically tri- +layer moir´e patterns presented in Sec. III, which very +clearly do so. We seek to resolve this contradiction by a +more nuanced description of the configuration space. +The missing ingredient from the na¨ıve configuration +space given in Eq. (14) is a collection of “nontrivial triv- +ial transformations,” which are nontrivial in that they +do not correspond to overall translations, but trivial in +that they do not change the local moir´e structure. The +correct configuration space of the moir´e pattern is the set +of translations of each layer modulo overall translations +(i.e., simultaneous translations of all layers by the same +amount) and these new transformations. +We now describe how to find these additional transfor- +mations. We do so in a way that naturally derives not +only the dimensionality of the true configuration space, +but also explains why it is toroidal. +The intuition of the argument derives from the charac- +terization of singular structures provided in Sec. III B 1: +singular structures are those structures for which cer- +tain relative translations of the layers change the aver- +age value of local quantities by providing phases between +different contributions to the zeroth Fourier mode of the +quantity of interest, as in Eq. (24). A moir´e heterostruc- +ture can be viewed as resulting from these different possi- +ble phases: different regions in the moir´e heterostructure +correspond to different relative translations of the singu- +lar structure. +The nontrivial trivial transformations we seek to find +derive from the converse of that identification: any rela- +tive translation which does not result in a phase will make +no impact on average properties. Such relative transla- +tions that do not result in phases, therefore, are precisely +the nontrivial trivial transformations. +We find the nontrivial trivial transformations formally +using in the frequency picture described in Sec. III. +For simplicity, +we take as a concrete example the +(1, 0; 1, 0; 1, 0)-moir´e on the square lattice (illustrated in +Fig. 8). The Fourier modes are indexed by Z6, but the +moir´e modes arise from the zero mode lattice described in +Sec. III B 2. In this specific case, the zero mode lattice is +spanned by the vectors (1, 0, 1, 0, 1, 0) and (0, 1, 0, 1, 0, 1), +which we call n(1) and n(2) (each of which also have in- +dices, n(1,2) +i,j +). +A translation of layer i by ai (not necessarily a lattice +vector) will multiply the Fourier mode with indices ni,j +by a phase exp +�� +i,j ni,jbi,j · ai +� +, which follows from the +discrete Fourier transform in Eq. (19). For the relative +translations which preserve the moir´e lattice, this phase +vanishes when evaluated on the zero mode lattice. +Clearly, translating each layer by the same amount, + +12 +ai = a, results in this phase vanishing on the zero-mode +lattice, where � n(k) +i,j bi,j = 0 for both k. This imposes +two constraints on the six-dimensional space. +The additional constraints are found by setting a1 = 0, +at which point the constraint is b2,i · a2 = −b3,i · a3; the +simplest two basis solutions are {a2 = b3,1, a3 = −b2,1} +and {a2 = b3,2, a3 = −b2,2}. These extra translations +are most of the “nontrivial trivial transformations” we +were searching for, and suffice to reduce the dimension- +ality of the configuration space from four to two. Note +that this two-dimensional space is periodic, i.e., a torus +rather than a plane, because the sum � +i,j ni,jbi,j · ai +need not vanish identically for the phase to vanish; in- +stead, it can be a multiple of 2π. This periodicity ensures +that the final phase space is indeed a torus. +Therefore, our final and most general characterization +of the phase space is as the collection of relative transla- +tions of the layers modulo those which act trivially on the +zero mode lattice (i.e., on the combinations of modes that +contribute to the zero mode in the singular structure). +Note that in multiply-singular structures, such as +TTLG, the moir´e-generating lattice is greater than +two-dimensional. +Therefore, there is at least a four- +dimensional manifold defining the configuration space. +Consequently, one cannot regard this configuration space +as being periodically fully explored in real space. This +explains the difference between the complex patterns in +TTLG and the periodic moir´e in intrinsically trilayer sys- +tems. +A. +Configuration space and lattice relaxation +To illustrate the usefulness of configuration space, we +consider lattice relaxation in intrinsically trilayer moir´e +systems. Lattice relaxation is usually computed by tak- +ing an average energy density of any particular stacking +configuration, then enlarging regions of low-energy stack- +ing while shrinking regions of high-energy stackings [66]. +We claim that the average energy density of a singular +structure on long wavelengths does not change under a +nontrivial trivial transformation. That is to say, struc- +tures in the na¨ıve configuration space (Sec. II D) that dif- +fer by a nontrivial trivial transformation have the same +energy density. +We now justify this claim. Consider the energy den- +sity of a singular structure, ρ(x), and consider the +energy density over some large region of radius R, +1 +πR2 +� +|x| 0, and set the lattice constant of the smaller lattice to one. (The two-chain model this yields has been +studied; see, e.g., [S1-1].) +We now derive the moir´e scale in real space. +From the characterization of moir´e patterns given in +Sec. II B 1, the moir´e lattice is the set of points x where the relative coordinates of the two unit cells are +the same. The relative coordinate for the unit-length lattice is x mod 1, and the relative coordinate for the +(1 + δ)-length lattice is +x +1+δ mod 1. The moir´e lattice is therefore given by the set of x where +x = +x +1 + δ +mod 1. +(S1-1) +This simplifies to +(1 + δ)x = x +mod 1 + δ +(S1-2) +δx = 0 +mod 1 + δ, +(S1-3) +which is satisfied when δx = k(1 + δ) for any integer k. Taking k = 1 gives the moir´e lengthscale +x = 1 + δ +δ += 1 + 1 +δ . +(S1-4) +In the case that δ = m +n , it follows that the moir´e scale is n+m +m , i.e., the moir´e scale is +1 +m of the commen- +surate scale. In the specific case where m = 1, the commensurate unit cell coincides with the moir´e unit cell. +This is illustrated in Fig. S1-1. +Figure S1-1: 1D moir´e patterns illustrating commensurate and moir´e unit cells. The relative lattice lengths +are δ = −1/18, −2/19, and −3/20, which all share a commensurate supercell (indicated by longer black +lines). In the first row the commensurate and moir´e unit cells coincide but in the second and third rows, the +moir´e pattern is one-half and one-third the commensurate unit cell, respectively. +1 + +-2 +-1 +0 +1 +2 +Figure S1-2: 1D moir´e patterns near a 1:3 ratio, with δ = 0.03. Top: moir´e pattern from such a near- +matching. Bottom: frequency modes of each layer (in red/blue) with arrows illustrating the resultant moir´e +frequencies. +Alternatively, the moir´e unit cell can be deduced by the frequency picture described in Sec. III. The unit +lattice has unit frequency, while the larger lattice has frequency +1 +1+δ. Therefore, the moir´e frequency is +1 − +1 +1 + δ = +δ +1 + δ . +(S1-5) +This is precisely the moir´e lattice size given in Eq. (S1-4). +S1-2 +Near 1:p matching +We now consider the case where one lattice has a unit length unit cell and the other has a cell of size p(1+δ). +To be concrete, we consider the case p = 3, with the moir´e pattern illustrated in the top half of Fig. S1-2. +Using the frequency picture described in Sec. III, consider the pth mode of the p-length lattice and the +first mode of the unit lattice, as illustrated in the bottom half of Fig. S1-2. These frequencies match at +commensurability and therefore are slightly displaced after a mismatch by δ. Their difference determines +the moir´e lattice size to be 1 + 1 +δ , the same size as in the 1:1 case. +We now characterize the same pattern in real space, following Appendix B 1. Instead of the moir´e unit +cell being determined by the set of points where the relative coordinates are equal, it is defined by the set of +points where p times the relative coordinate of the larger lattice equals the relative coordinate of the smaller +lattice (mod 1). To derive this intuition, we reframe the 1:1 case in a way that will generalize. +In the 1:1 case, two points x, y in the moir´e pattern are equivalent if the difference between the relative +coordinates of the two layers at x is the same as at y, because then the two points have the same local +stacking. Equivalently, consider the local stacking arrangement at point x and then ask whether the relative +coordinates from each layer at y appear in the stacking at x. If so, x and y are equivalent points in the moir´e +pattern; otherwise, they are different. +Comparing the relative coordinates within the commensurate unit cell at two points naturally extends to +the 1:p case; however, this approach gives the wrong result for moir´e pattern size. There are p points in the +commensurate unit cell that correspond to any particular point in the smaller unit cell (as illustrated for the +1:3 case in Fig. S1-3a). Given a particular commensurate stacking, points in the moir´e pattern where the +relative coordinates of the two layers appear in that stacking are p times more common than points where +the relative coordinates in the commensurate cell are the same. These extra points are, in fact, equivalent +points on the moir´e pattern; accounting for these extra equivalent points yields the correct condition on +relative coordinates. +Using that condition, it is possible to compute moir´e pattern size in the real space picture. At a particular +point x, the relative coordinate of the unit lattice is x mod 1, and the relative coordinate of the p(1+δ)-length +2 + +0 +1 +2 +3 +(a) 1:3 matching +0 +1 +2 +3 +4 +5 +6 +1 +0 +(b) 2:3 matching +Figure S1-3: Commensurate unit cell for specified matchings. Shading indicates the relative coordinate in +each corresponding layer. When one layer is stretched by a small amount δ, the moir´e lattice is determined +by the set of points where the relative coordinates of each layer match one of the combinations shown here +for the commensurate cell. For 1:3 matching, this condition is equivalent to Eq. (S1-6). +’ +-2 +-1 +0 +1 +2 +Figure S1-4: 1D moir´e patterns near a 2:3 ratio, with δ = 0.01. Above bar: moir´e pattern from such a near- +matching. Below bar: frequency modes of each layer (in red/blue), with arrows illustrating the resultant +moir´e frequencies. +lattice is +x +p(1+δ) mod 1. Thus, the moir´e lattice vectors occur at points x satisfying +x = p +� +x +p(1 + δ) +� +mod 1. +(S1-6) +We find the same moir´e pattern size as in the 1:1 case, 1 + 1 +δ . +S1-3 +Near q:p matching +Now, we consider the commensurate structure with q : p matching, where the first lattice has length q and +the second lattice has length p(1 + δ). Without loss of generality, we consider p and q relatively prime, so +that δ = 0 yields a commensurate structure at pq. +In frequency space, the difference between the pth mode of the p-length lattice and the qth mode of the +q-length lattice yields a moir´e lattice size of 1 + 1 +δ again. The case of q = 2 and p = 3 is shown in the lower +half of Fig. S1-4. The moir´e lattice is determined by gcd(p, q) = 1. [I.e., the moir´e lattice is ∼ 1 +δ , rather than +p +δ or q +δ, to leading order in δ.] In the framework of Appendix B, this is because the Minkowski sum of two +1D lattices with lattice constants a, b has lattice constant gcd(a, b). +In real space, the matching condition is equivalent to the 1:p case: a moir´e lattice vector occurs at any +position where the relative coordinate of the top and bottom layer match any combination that would occur +in the commensurate cell. The 2:3 case is illustrated in Fig. S1-3b. This definition yields the same moir´e +length of 1 + 1 +δ . +3 + +Figure S1-5: Intrinsically trilayer 1D moir´e patterns. A 1:ϕ:(ϕ − 1) ratio of frequencies, where ϕ = 1+ +√ +5 +2 +is the golden ratio, produces a singular structure. We deviate from that singular structure by varying the +first lattice constant by δ = 0.02. The top image overlaps all three layers, showing a periodic structure with +approximately eight periods in the displayed range. (A smaller bilayer moir´e pattern is also visible.) The +bottom shows the individual pairs of bilayers (from top to bottom: 1:(ϕ − 1), 1:ϕ, ϕ:(ϕ − 1)), which do not +exhibit any moir´e patterns on the longer lengthscale. +S1-4 +Trilayer moir´e +Intrinsically trilayer moir´e is also possible in 1D. In Fig. S1-5, we illustrate a moir´e pattern between three +layers with periods 1 + δ, ϕ, and ϕ − 1, where ϕ = 1+ +√ +5 +2 +is the golden ratio. The singular structure at δ = 0 +results because 1 +ϕ = ϕ − 1. In addition to the intrinsically trilayer pattern, there are bilayer moir´e patterns +that are also visible in the figure. +References +[S1-1] Qiang Gao and Eslam Khalaf. +Symmetry origin of lattice vibration modes in twisted multilayer +graphene: Phasons versus moir´e phonons. Phys. Rev. B, 106:075420, Aug 2022. +4 + +Supplement 2: Raw Moir´e Images +In this supplement, we re-present several figures from the main text without the guides to the eye so that +the visual arrangements can be viewed more objectively. These are presented as Figs. S2-1, S2-2, and S2-3. +1 + +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +Figure S2-1: A larger and unannotated illustration of Fig. 4, showing two unit square lattices at a relative +twist of 0.6◦ away from the 36.9◦ commensurate angle. +2 + +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +Figure S2-2: A larger and unannotated illustration of Fig. 5, showing two unit square lattices at a relative +twist of 0.6◦ away from the 36.9◦ commensurate angle. +3 + +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +Figure S2-3: A larger and unannotated illustration of Fig. 8, showing three unit square lattices near 60.7◦. +4 + diff --git a/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/load_file.txt b/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c27b730314f1ef4f41bde621c038e52af03b479 --- /dev/null +++ b/QdAzT4oBgHgl3EQfz_5Y/content/tmp_files/load_file.txt @@ -0,0 +1,1840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf,len=1839 +page_content='Intrinsically-multilayer moir´e heterostructures Aaron Dunbrack1 and Jennifer Cano1, 2 1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11974, USA 2Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA (Dated: January 6, 2023) We introduce trilayer and multilayer moir´e heterostructures that cannot be viewed from the “moir´e-of-moir´e” perspective of helically-twisted trilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' These “intrinsically trilayer” moir´e systems feature periodic modulation of a local quasicrystalline structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' They open the door to realizing moir´e heterostructures with vastly more material constituents because they do not constrain the lattice constants of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In this manuscript, we define intrinsically multilayer patterns, provide a recipe for their construction, derive their local configuration space, and connect the visual patterns to physical observables in material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' INTRODUCTION The observation of superconductivity and correlated insulators in twisted bilayer graphene [1, 2] launched the study of “moir´e materials,” where two-dimensional ma- terials with the same [1–35] or similar [36–45] lattice con- stants are stacked at a small relative twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This paradigm is naturally extended to trilayer stacking and beyond, both with some layers aligned [46–52] and with multiple twist angles [53–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Recently it has also been extended to stacking at angles nearby a large commensu- rate twist angle [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In all cases, the moir´e pattern is obtained from layers with either the same or similar lattice constant (or a commensurate supercell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In this paper, we lift that restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We introduce moir´e patterns made from stacking more than two layers in which no two layers separately dis- play a moir´e pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We call these patterns “intrinsi- Types of moir´e patterns Small twist Large twist Two layers Twisted bilayer graphene Near-commensurate TBLG Three or more layers Twisted trilayer graphene Intrinsically trilayer moir´e TABLE I: Summary of moir´e heterostructures: the “intrinsically trilayer” moir´e patterns we introduce occur at large twist angle and with three or more layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' cally trilayer moir´e” (or more generally, “intrinsically N- layer moire”) because, unlike twisted trilayer graphene, the moir´e pattern disappears if any one layer is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' As we will explain, intrinsically trilayer moir´e patterns cannot be viewed from the “moir´e of moir´e” perspective often used to describe twisted trilayer graphene [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Intrinsically N-layer moir´e patterns have an important advantage over bilayer moir´e patterns because they do not impose a constraint on lattice constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This vastly increases the space of possible material combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Specifically, moir´e patterns in bilayer systems require the constituent materials to have nearly the same lattice con- stant or to be nearly commensurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In contrast, intrin- sically N-layer moir´e patterns can be constructed from virtually arbitrary combinations of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In the present work, we focus on the crystal structure of intrinsically N-layer moir´e heterostructures, postponing a study of electronic structure to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We begin by reviewing the origin of moir´e patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II, we provide an intuitive picture of how moir´e pat- terns arise in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We explain the construction for bilayers and then offer a na¨ıve generalization to multilay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III, we argue that reciprocal space provides a more natural and concise characterization, from which we derive both bilayer and N-layer moir´e patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We then focus on multilayer heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV, we return to real space to resolve an appar- ent contradiction: the momentum-space perspective im- plies that periodic moir´e patterns of more than two lay- ers exist, but the na¨ıve generalization of bilayer config- uration space [61, 62] fails to indicate these patterns, in part because the local structure is generally quasicrys- talline rather than crystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Consequently, we develop a more nuanced notion of configuration space, in which some apparent degrees of freedom disappear on moir´e wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We discuss physical properties that are a function of this configuration space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' lattice relaxation is one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' V, we discuss experimental probes and propose physical realizations of intrinisically N-layer moir´e patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Throughout, we assume a three-, four-, or six-fold ro- tation symmetry shared between all layers of the moir´e arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='01777v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='mes-hall] 4 Jan 2023 2 −10 0 10 −10 0 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1: A moir´e lattice of two square layers twisted at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='7329◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Commensurate lattice in red, moir´e lattice in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In the absence of this symmetry, the generic moir´e pattern will be stripes rather than a 2D pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' CONFIGURATION SPACE FOR BILAYERS: MOIR´E PATTERNS IN REAL SPACE Moir´e patterns are intuitively understood in real space as a slow modulation of the local lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The set of all possible local environments is known as config- uration space [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The configuration space approach extends beyond linear transformations of perfectly rigid crystals to include lattice relaxation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, the approach becomes subtle for heterostructures of multiple layers or different lattice constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In this section, we review configuration space in the simplest case of bilayers with near-identical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We then extend the formalism to bilayer systems perturbed from a commensurate stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Finally, we offer a “na¨ıve configuration space” for trilayer systems, and briefly dis- cuss how it leads to the complex patterns observed in twisted trilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (Later, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV, we will pro- vide a more complete accounting of configuration space in systems with more than two layers and explain the breakdown of the na¨ıve configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Two square lattices Consider two stacked periodic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' There are two cases to consider: when the two layers share a common (larger) period, and when they do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' If they do share a common period, we call the structures commensurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' If they do not, we call them incommensurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, we illustrate a small commensurate pattern formed by two square lattices at a relative twist angle of approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='7◦ about a square corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This aligns the square corners of the unit cell (8,9) of one layer with (9,8) of the other, forming the commensurate superlattice outlined in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, in the center of each red supercell is a lo- cation that looks very similar to the corners, where the unit cells are also aligned at the center of the square cells rather than at a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This smaller grid of locations where the square-centers are aligned defines the moir´e lattice, outlined in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Thus, the visual moir´e cell, which enjoys an approximate translation symmetry, is smaller than the commensurate unit cell, which exhibits an exact translation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In general, the visual pattern will either be the same size or smaller than the commensurate cell (although for two identical square lat- tices, the moir´e cell is always smaller by at least a factor of √ 2, regardless of twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The commensurate cell size is highly sensitive to angle and exists only on a dense subset of angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Computing the size of a commensurate cell as a function of twist angle is analogous to determining the size of the minimal denominator of a fraction as a function of the value of that fraction, as explained in Supplement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e cell, however, varies smoothly with twist angle for small twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' At sufficiently large twist angles, the moir´e cell becomes smaller than a unit cell, which indicates that the moir´e pattern ceases to exist and no visual pattern arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This example shows how a moir´e pattern arises from the two layers being stacked at different “local relative translations” at different positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', in the brighter regions, the lattices are stacked atom-on-atom, while in the darker regions, the lattices are stacked atom-on-void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e lattice is defined by the collection of points where the two layers align in either configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Local configuration space: two identical layers The space of relative translations of the aligned lay- ers defines the local configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For instance, TBLG exhibits regions of AA and AB stacking, as well as intermediate regions, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For two identical layers, the local configuration space is defined with respect to relative translations of the two untwisted layers, as we will now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Although the idea is intuitive in this case, developing the mathemat- ical infrastructure carefully here will elucidate the more complicated situations we consider later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Configuration space as differences of relative coordinates In the simplest setup where the two untwisted layers have identical lattice vectors, we define the local configu- ration C(x) in terms of the relative coordinates xi of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The relative coordinate xi(x) is a two-component vector that specifies where the position x resides in the 3 −30 −15 0 15 30 −30 −15 0 15 30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 2: A moir´e lattice of two hexagonal layers with unit-length interatomic distance stacked with a relative twist angle of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Red and blue circles indicate an “AA-stacked” region where hexagons align and an “AB-stacked” region where they are offset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' unit cell of layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Thus, xi is determined by the ma- trix Ai, whose columns are the (twisted) lattice vectors of layer i, as xi(x) = A−1 i x mod I (1) where “mod I” means “modulo the columns of I” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', mod {(1, 0), (0, 1)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The local configuration is then de- fined as the difference between the two relative coordi- nates C(x) = x2(x) − x1(x) mod I (2) = (A−1 2 − A−1 1 )x mod I (3) While the functions xi vary on the scale of the original lattice, for a small twist or lattice mismatch, C(x) varies much more slowly, and the period of C(x) defines the moir´e lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, the moir´e lattice vectors are given by the columns of the matrix AM = (A−1 2 − A−1 1 )−1 (4) in the case where the inverse exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' If the inverse does not exist, then there is not a 2D moir´e pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In the case where the two layers are identical and twisted by a relative angle θ, one can simplify further by writing A1,2 = R(±θ/2)A, where R(θ) is the rotation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e lattice vectors then simplify to AM = [R(θ/2) − R(−θ/2)]−1 A = 1 2 sin(θ/2)R �π 2 � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (5) In other words, the moir´e lattice vectors are rotated by π/2 compared to the original lattice vectors A and scaled up by a factor of 1/(2 sin(θ/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The same formalism applies to aligned layers with a small difference in their lattice constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For example, if A2 = (1+δ)A1, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (4) can be simplified without any matrix algebra to AM = 1+δ δ A1 (neglecting the over- all sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Generalizing to the case of two layers with a small lattice mismatch arranged with a slight twist angle yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (1) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (4) in this paper also allows for anisotropic lattice mismatch, as might be induced by a strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Configuration space as a quotient of translation groups More abstractly, configuration space is equivalently de- fined as the space of nontrivial translations of the lattices before twisting, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A combination of translations is “trivial” if it differs from zero translation of each layer by the simultaneous translation of all layers by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In other words: consider the two identical lattices be- fore twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Denote the group of translations of each layer modulo lattice translations by Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (Note Ti will be isomorphic to the torus T 2 = R2/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') Similarly denote the group of translations of the two lattices simultane- ously (modulo translations that preserve the shared pre- twist lattice) as T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The space of configurations is the space of translations of each layer, modulo simultaneous translations of the two layers: Tconfig = T1 × T2/T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6) This space of configurations is itself a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We now relate this space to the moir´e pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Suppose we transform each layer by a linear transformation Mi, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', for twist, Mi = R(θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In terms of the matrices of lattice vectors before and after twisting, Mi = AiA−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (7) We now interpret this transformation as a position- dependent translation, which will give the Ti-coordinate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' To find the translation of one layer associated with a point x0 in real space, consider the map which first trans- forms physical space, then transforms back but centered at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', for a twist by θ, first twist about the origin by θ, then twist back around x0 by −θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') Conceptually, the first transformation sets up the twisted system, and the latter re-aligns the layers without further translating x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Algebraically, understanding that “transform around x0” can be written as “translate x0 to the origin, trans- form, then translate back,” the translation is given by x → M −1 i (Mix − x0) + x0 = x − (M −1 i − I)x0, (8) which is a translation because it takes the form x → x − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This translation is then taken modulo the pre- twist lattice vectors to get the element of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4 −20 0 20 −20 0 20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 3: Moir´e pattern from two square lattices with side lengths 1 and √ 2 arranged with a relative twist angle of 42◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Doing this for each layer yields the translation opera- tors that determine a point in configuration space defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Modding out by simultaneous translations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6) yields the relative translation difference between the two layers, ˜C(x) = (M −1 2 − M −1 1 )x mod A (9) where A is the shared lattice before twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This is in one-to-one correspondence with the characterization of configuration space in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e unit cell is given by AM = (M −1 2 − M −1 1 )−1A, (10) which is exactly Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Written in this way, the moir´e lattice is “factored” into one term, M −1 2 −M −1 1 , that de- pends on the transformations but not the original lattice, and another term, A, that depends on the lattice but not the transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The second term can be interpreted as the size of configuration space and the first as the rate at which the moir´e pattern explores that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Generalization to near-commensurate twisting Now instead of two identical layers, consider two layers that form a small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', not moir´e) commensurate super- cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Applying a small twist or lattice mismatch produces a moir´e pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For instance, two square lattices whose side lengths differ by a factor of √ 2 form a commensu- rate supercell when arranged at a 45◦ relative orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' when twisted by an angle near 45◦, they form a moir´e pattern as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A second example is two identical honeycomb lattices twisted near a commensu- rate angle that is not a multiple of 60◦, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' near-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='8◦ TBLG is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The abstract description of configuration space de- scribed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6) extends to this case with only one minor modification: instead of considering the translations as acting on the lattices at zero twist, consider them at the relevant commensurate stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Hence, the Ti are now defined modulo the individual lattices at the commensu- rate stacking, whereas T12 is defined modulo the lattice vectors of the commensurate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' An argument for the size of the moir´e pattern comes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (9) and the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A lin- ear transformation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' twist) performed on a near- commensurate structure explores the configuration space at the same rate as the structure formed by performing the same transformation on a zero-degree stacked struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, the configuration space of the former is (perhaps counterintuitively) smaller, for reasons we now explain heuristically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The size of configuration space in the case of two lay- ers stacked to form a supercell can be sensibly guessed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Let Ai, AC, Acs and AM denote the areas of the unit cell of layer i, the commensurate supercell, configuration space, and the moir´e unit cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Replacing each translation group in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (6) by the area of the corresponding torus yields Acs = A1A2 AC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (11) Exploiting the fact that Ai = | det(Ai)| and guided by the intuition that A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (10) should be generalized to some “configuration space lattice,” the area of the moir´e cell is AM = 1 | det � M −1 2 − M −1 1 � | A1A2 AC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (12) The intuition that we should use the configuration space lattice follows from factoring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (10) as described in the text following that equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (We give a rigorous de- scription of how to find the “configuration space lattice vectors” Acs in Appendix B and prove that they are in- deed the analogue of A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') As a concrete example, consider two identical lattices twisted at an angle θ away from a commensurate stack- ing where the commensurate cell is a factor of N larger in area than the original unit cell (for instance, in near- 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='8◦ TBLG, the commensurate cell is 7 times larger in area than the original graphene cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The size of Ti does not depend on how the layers are stacked, but T12 will be a factor of N larger in area when they are twisted θ away from the commensurate stacking compared to when the layers are stacked at an overall twist angle of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (11), the configuration space, which is defined modulo T12, would be a factor of N smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Since the matrices M1,2 in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (12) depend only on θ and not on the supercell or original lattice, it follows that, contrary to the most obvious intuition, for two specified 2D layers, the larger the commensurate cell, the smaller the moir´e pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 5 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 14 16 18 20 22 24 26 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4: A moir´e pattern formed by two unit triangular lattices arranged with a relative twist of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='4◦ (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='8◦ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='6◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The resulting triangular moir´e lattice has a unit cell of side length 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='1, shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e pattern is subtle, alternating between regions with individual sixfold-symmetric “centers” (red) and regions with triplets of “centers” connected in a triangle (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A larger picture of the moir´e pattern is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' S2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In Appendix B, in addition to formally deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (11), the relative coordinates of heterostructures nearby a supercell configuration are derived, generaliz- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A na¨ıve approach to configuration space with more than two layers We now try to apply the idea of configuration space as the translation of each layer modulo overall translations to heterostructures with more than two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We call this notion “na¨ıve configuration space” (in contrast to a more nuanced notion to be given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' in the case of three identical layers near zero stacking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' as in twisted trilayer graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' the local configuration space is a four-dimensional torus: Tconfig = T1 × T2 × T3/T123 (13) In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' the local configuration space of N arbitrarily- twisted layers (with respect to a reference configuration) is a (2N − 2)-dimensional torus: Tconfig = �� i Ti � /Tall (14) Because this configuration space has dimension greater than two,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' we do not generally expect that it is fully ex- plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The consequence is a complex structure of over- lapping moir´e patterns (illustrated for twisted trilayer graphene in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1b of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 64), and the four-dimensional space will generally be the correct parameter space for many layers twisted near a single commensurate struc- ture of all layers (as can be seen in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' As the next section will show, however, there are moir´e patterns that arise when multilayer structures are twisted near special incommensurate configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In these cases, more care is required to define which configura- tions are distinct in a way that will manifest on moir´e lengthscales: Tconfig as written in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (14) is not correct because Tall is not the correct space by which to mod out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' MOIR´E IN FREQUENCY SPACE An alternative to defining a moir´e pattern in real space is to define it by the appearance of low-frequency modes in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This approach is discussed at length in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 65;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' here we summarize by focusing on the modes of a black-and-white image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, the content is much more general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Consider a layered material as a set of transparencies placed over a light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The atomic structure defines a local transmission coefficient Ti(x) that specifies how much light layer i lets through at point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For a black- and-white image, Ti(x) = 1 wherever the layer’s image is white and Ti(x) = 0 where it is black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' this paradigm extends to grayscale images using opacities between zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' By the definition of the transmission function, given Ti(x) in each layer i, the resulting transmission function of the layered structure is given by: T(x) = � i Ti(x), (15) which defines how the resulting multilayer pattern is formed from the patterns of the individual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e-scale physics emerges by extracting the low- frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In each periodic layer i, the Fourier transform is defined by: Ti(x) = � n ci,n exp (iki,n · x) (16) where the sum is over the reciprocal lattice vectors ki,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Fourier transforming Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (15) yields: ˆT(k) = [ ˆT1 ∗ ˆT2 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' ∗ ˆTN](k), (17) where ∗ denotes the discretized convolution: [f ∗ g](k) = � n,m cndmδ(k − kn − k′ m), (18) so that [T1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' ∗ TN](k) = � n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=',nN ��� i ci,ni � δ(k − � i ki,ni) � (19) Therefore, a low-frequency (small-k) mode requires there exist a collection of modes ni so that � i ki,ni ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This sum is the moir´e wavevector, kM = � i ki,ni, (20) which in turn yields the moir´e wavelength and orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Such a collection of modes arise naturally by consider- ing a small deformation (twist, stretch, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') away from a reference configuration where � i ki,ni = 0 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For a bilayer system, k1,n + k2,m = 0 is precisely a commen- surability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The case n = m corresponds to the familiar near-zero-degree moir´e pattern for nearly- identical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' On the other hand, the case n ̸= m corresponds to a near-commensurate moir´e, which can result when the two lattices differ in size (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 3) or are arranged near a commensurate angle (il- lustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Near-commensurate example As a concrete example, consider two square lattices arranged with a twist angle near the 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9◦ commensurate angle, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The lowest Fourier modes before twisting are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' note the (1,2) mode of one layer coincides with the (2,1) mode of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The magnitude of the wave vector of these modes is |k36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9| = √ 5k0, where k0 is the magnitude of the wave vector of the lowest mode of a single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In general, if two modes with a wave vector of magni- tude |k| are initially aligned before twisting, then after a relative twist by an angle θ, the difference between the two wave vectors has magnitude |kM| = 2 sin(θ/2)|k|, (21) as is seen geometrically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 7 and can be derived mathematically by taking k1 = −R(θ)k2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Accordingly, the moir´e pattern at 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9◦ + θ is a factor of √ 5 smaller in real space than the moir´e pattern at 0◦ + θ because |k36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9+θ M | = 2 sin(θ/2)|k36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9| = 2 sin(θ/2) √ 5|k0| = √ 5|k0+θ M |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (22) The same result was obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II C through more complicated arguments in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The moir´e patterns obtained from twisting near a com- mensurate angle, as illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4 and 5, are fainter than those for the corresponding structures near zero degrees in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 2 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The faint pattern occurs because the higher-frequency modes have smaller amplitudes than the lowest mode, and therefore the coefficients cndm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (18) are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (The range of visibility of different near-commensurate moir´e pat- terns is also illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Intrinsically multilayer moir´e The moir´e formalism in reciprocal space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (20), also provides a requirement for a moir´e pattern to exist in a multilayer heterostructure: there must exist a linear combination of reciprocal lattice vectors in the different layers that adds up to a vector much smaller than the reciprocal lattice vectors of the original layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In the 7 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 20 22 24 26 28 30 32 34 36 38 40 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 5: A moir´e lattice formed by two unit square lattices arranged at a relative twist of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='5◦ (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9◦ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='6◦), with a 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='7 side length moir´e cell (green square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' There is a resulting pattern of “holey regions” (red square) and “knitted regions” (blue square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A larger unannotated picture of the moir´e pattern is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' S2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' following, we provide a recipe for meeting this condition that is analogous to twisting near commensurate struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' First, find a stacking arrangement of the layers such that a reciprocal lattice vector can be chosen in each layer so that the sum over the chosen reciprocal lattice vectors in all layers is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', � i ki,ni = 0, where ki,ni is the chosen reciprocal lattice vector in layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We call such a configuration singular (following the terminology from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 65), which is a generalization of a commensurate configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Note this notation differs from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 61, where incommensurate is defined as non-singular in our terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Once a singular configuration is identified, a small twist or stretch of each layer away from the singular con- figuration results in the same sum of reciprocal lattice vectors being nonzero but small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This small sum of the lattice vectors is precisely a reciprocal lattice vector of the moir´e lattice, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We call a moire pattern “intrinsically n-layer” if it orig- inates from a singular configuration where no two lay- ers are singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In other words, an intrinsically n-layer moir´e material is one whose singular configuration is a sum of reciprocal lattice vectors from all layers that add to zero, but no two vectors from that sum add to zero by themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Notice this is distinct from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', helically- twisted trilayer graphene [53–56];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' there the singular pat- tern is at zero twist angle, where any two layers have reciprocal lattice vectors which add to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (Patterns where some layers are aligned, such as alternating-twisted trilayer[46, 47] and twisted double bilayer graphene[48– 51], often have patterns that arise from only two mis- aligned sets of layers, rather than more than two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' more- over, such patterns are always singular in themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') An example of an intrinsically trilayer moir´e pattern is three square lattices twisted near 120◦, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The sum of the n = (1, 0) lattice vectors from each layer vanishes, so at 120◦ there is a singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Notice that this singular structure is not commensurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' in fact, it is a twelvefold-symmetric quasicrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In gen- eral, the singular structures will be quasicrystalline, but not necessarily with higher rotational symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 6: Reciprocal space of two square lattices stacked at a commensurate 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9◦ twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Red(blue) open circles indicate the reciprocal lattice vectors of the top(bottom) layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' black filled circles indicate shared reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Thick lines shows that the (1,2) mode of the blue layer coincides with the (2,1) mode of the red layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Light gray indicates the reciprocal commensurate lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' k1 k2 kM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 7: Lowest frequency modes of two square lattices at a small relative twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Red and blue circles indicate reciprocal lattice vectors of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The small difference between the lowest modes k1 − k2 gives the moir´e wavevector kM, from which Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (21) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' What is a singular structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Since the notion of a “singular structure” is not a stan- dard notion of the physics literature (although it has ap- peared in the mathematical literature on moir´e patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 65), it is worth spending a moment highlighting both how it is different from a commensurate structure and how it is different from a general twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' First, a multilayer system is commensurate if the com- bined system has exact translation symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In other words, there must exist lattice vectors a1,2 for the multi- layer system such that, for each layer i with lattice vec- tors a(i) 1,2, the vectors a1,2 are integer linear combinations of a(i) 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' As shown in Appendix D, this definition of com- mensurate is equivalent to every layer being individually commensurate with the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, in an N layer system with threefold or fourfold rotational symme- try, commensurability imposes 2N − 2 scalar constraints (from N − 1 vector constraints) on the size and orienta- tion of the lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' By contrast, consider the singularity condition � i ki,ni = 0, where ki,ni are each reciprocal lattice vec- tors of layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This imposes only two scalar constraints (one vector constraint) on the orientations of layers, re- gardless of the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For a bilayer system, the singularity condition is equivalent to commensurabil- ity, but with more than two layers, commensurability is a strictly stronger condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Now contrast that situation with generic twist an- gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Singular structures have a property unusual among twisted systems: the average, long-distance properties of the system are sensitive to relative translations of the layers, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Given a system with a local property f(x), the av- erage value of that property over an area A is given by 1 |A| � A f(x)d2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' If that area becomes very large, un- der appropriate convergence conditions on f, the average value converges to the Fourier transform of f at the ori- gin, ˆf(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Suppose now that f(x) can be written as a product of functions of each layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', for a trilayer system, f(x) = f1(x)f2(x)f3(x), where fi(x) is periodic with the periodicity of layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Notice the transmission function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (15) has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The zeroth Fourier mode of f is determined by Fourier modes ˆfi(ki) of each layer such that � i ki = 0, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' If the layers are not stacked in a singular structure, the only solution to � i ki = 0 is when ki = 0 in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, the average value of f in the multilayer is a product of the average values of f in each individual layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' relative translations of the layers have no impact on this zeroth Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' By contrast, for a singular structure, there exists a nontrivial combination of Fourier modes in each layer that contribute to the average value of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For instance, consider a trilayer system with reciprocal lattice vectors ki in each layer such that � i ki = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Further suppose 9 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −10 −5 0 5 10 −10 −5 0 5 10 15 20 25 30 35 15 20 25 30 35 G2 G3 G1 G2 G3 G1 GM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 8: A moir´e lattice of three unit square lattices at a relative twist of 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='3◦, resulting in a moir´e unit cell of side length 47 (drawn in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Local structures are shown at right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Top right illustrates the reciprocal lattice vectors at exactly 120◦ (left) and after the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='7◦ deviation from the singular structure (right, deviation exaggerated for illustration purposes), resulting in the moir´e reciprocal lattice vector GM shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A larger unannotated picture of the moir´e pattern is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' S2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' fi = c0,i + 2c1,i cos(ki · x), for some coefficients c0,i, c1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (19), the zeroth Fourier mode of f is ˆf(0) = c0,1c0,2c0,3 + 2c1,1c1,2c1,3 (23) where the factor of 2 derives from the positive and neg- ative contributions of the cosine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (If fi had a rotation symmetry instead of being a 1D cosine, the factor of 2 would turn into a 4 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') Now translating each layer i by ai transforms the zeroth Fourier mode into ˆf(0) = c0,1c0,2c0,3 + 2c1,1c1,2c1,3 cos �� ki · ai � , (24) which is different for generic choices of ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Thus, the physical consequence of a singular structure is that local properties of the multilayer are sensitive to relative translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This is also true for commensurate structures, but is not true for a general non-singular or non-commensurate stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, notice that for a fixed set of ki, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (24) is invariant under the special set of translations ai which satisfy � kiai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' These special translations will be important in developing our notion of configuration space for multilayer systems in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III, the condition that the physical quantity of interest is a product of properties in each layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', f = f1f2f3 for a trilayer system, simplifies the discussion, but can also be relaxed significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The more general description is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Labelling singular structures We now provide a convenient labelling schema for sin- gular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Since a singular structure is specified by a combination of reciprocal lattice vectors that adds up to zero, it can be conveniently labelled by the integer indices of the reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Let bi,1 and bi,2 be the basis of reciprocal lattice vectors 10 in layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Then a singular structure will be specified by a set of ni,j that satisfy the singularity condition � i,j ni,jbi,j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (25) For a trilayer system, the singular structure given by ni,j is labelled as (n1,1, n1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' n2,1, n2,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' n3,1, n3,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This description can be generalized to any number of layers, including bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Note that the labelling depends on the choice of reciprocal lattice vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' thus, a set of ni,j combined with knowledge of the reciprocal lattice vectors in each layer determines the singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The ni,j for an N-layer system naturally live in Z2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The singularity condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (25) defines a 1D sub- lattice in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Assuming rotational symmetry, one choice of ni,j yields another linearly-independent ni,j af- ter rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Thus, combined there is a 2D sublattice in Z2N satisfying the singularity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' It is also possi- ble for the sublattice to have a higher even dimension, as we will show for trilayer graphene in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Regard- less of dimension, we call the ni,j that satisfy the singu- larity condition the zero mode lattice, because they cor- respond to combinations of Fourier modes in each layer that contribute to the k = 0 Fourier mode of the sin- gular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Under the assumption that the sublat- tice is 2D and that the degree of rotational symmetry is known, each singular structure can be labelled by a single set of ni,j that defines one of the basis vectors of the zero mode lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' the other basis vector follows from rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' As a few concrete examples: the standard near-zero moir´e pattern of two layers is the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −1, 0) moir´e pat- tern because b1,1 − b2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The near-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='8◦ struc- ture shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 4 and the near-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='9◦ structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 5 are both (1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −2, −1) moir´e patterns because b1,1 + 2b1,2 − 2b2,1 − b2,2 = 0 in both cases, despite their different rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Finally, the intrin- sically trilayer pattern illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 8 would be the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0) moir´e, assuming the first basis vector of the three layers are chosen 120 degrees apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Degeneracy of singular structures We now consider how singular structures arise in the manifold of possible twists and lattice mismatches be- tween the layers, which we call deformation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (More generally, we could also include strains that break rota- tional symmetries in our deformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' we call this gener- alization anisotropic deformation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' However, since such deformations can result in 1D instead of 2D moir´e patterns, we neglect such transformations here and sim- plify our discussion by referring to our space of isotropic deformations by the shorter term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=') Commensurate structures of bilayer systems are spe- cial among singular structures because they are zero- dimensional manifolds in deformation space: no small G1 G2 G3 G1 G2 G3 G1 G2 G3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 9: Starting from a particular singular structure, a small twist away combined with a corresponding strain results in another singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' These transformations yield a manifold of singular structures rather than an isolated point, as occurs for bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' deformation of a bilayer singular structure yields the same singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For instance, in the simple case of aligned layers (corresponding to the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −1, 0) com- mensurate structure), no combination of small relative mismatch or twist of the two layers will yield another (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −1, 0) commensurate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This is not, however, the case for singular struc- tures with more than two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' With N layers there are 2N − 2 possible isotropic deformations (twists and isotropic strains) of the layers relative to each other: each layer beyond the first adds two additional parameters (namely, strain and mismatch with respect to the first layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The singular structure then adds two constraints (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (25) and its rotated counterpart) on this deforma- tion space, meaning that it forms a (2N −4)-dimensional manifold in this space of deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Intuitively, this is because there is a continuum of ways to change the sides of the triangle that keep it a trian- gle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For example, given a triangle formed by reciprocal lattice vectors, one can deform two of the lattices by a combination of twists and (isotropic) strains while leav- ing the third fixed and still have a triangle, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In contrast, the only way to deform the layers and preserve a singular digon formed by the reciprocal lattice vectors of a bilayer is to perform an overall twist or isotropic stretch of both layers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' These singularity-preserving deformations are at the crux of understanding what the na¨ıve configuration space description in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II D fails to see about intrinsically tri- layer moir´e patterns, namely, why the effective param- eter space seems to be periodically spanned by the two dimensional moir´e pattern even though the na¨ıve param- eter space is four-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The connection between these pictures will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The doubly-singular structure of twisted trilayer graphene We now examine twisted trilayer graphene from the perspective of singular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Twisted trilayer graphene arises at the intersection of two singular struc- tures: the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 0, 0) singular structure and the (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −1, 0) singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In this sense, it is “doubly-singular”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' therefore, with four singularity con- straints instead of the two considered in the previous 11 θ12 θ23 (1,0,-1,0,0,0) (0,0,1,0,-1,0) (1,0,0,0,-1,0) (a) δ12 = δ32 = 0 δ12 = δ32 θ12 = θ23 (1, 0, −2, 0, 1, 0) (b) θ12 = θ23 and δ12 = δ32 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 10: Several singular structures of TTLG along two specific slices of the four-dimensional parameter space (θ12, θ23, δ12, δ32) indicated by solid colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The dashed green line represents the constraint of helically-twisted trilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The bilayer singular structures shown in the left figure deviate from helically-twisted trilayer graphene to order θ, but the trilayer singular structure shown in the right figure only deviates to order θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Hence, the green singular structure produces a moir´e pattern at 1/θ2-scale, whereas the bilayer singular structures plotted in blue/red/purple produce (competing) moir´e pattern at 1/θ scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' section, the combination of singular structures is zero- dimensional, not 2D like the intrinsically trilayer pattern (the dimension is 2N −6 instead of 2N −4, where N = 3 for three layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Twisting relative to the singular structure in this case be understood as generating multiple moir´e patterns si- multaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Without a fine-tuned combination of twist and mismatch, the overlapping structure of the multiple moir´e patterns yields complex and unclear-scale patterns, as illustrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In the special case where the twist angles of the first and third layers are equal and opposite, however, some- thing special happens: at 1 θ2 length scales, a single regu- lar moir´e pattern is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This pattern is referred to as a “moir´e of moir´e,” since it arises from a moir´e pattern induced by the two competing 1 θ-scale moir´e patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This 1 θ2 -order pattern can be understood as the pat- tern arising from the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' −2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0) singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Specifically, defining k0 to be a smallest reciprocal lattice vector of graphene, the trilayer structure where the first and third layers are twisted a small amount in opposite directions away from the middle layer can be described by k1 = R(θ)k0, k2 = −2k0, and k3 = R(−θ)k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (20), the moir´e wave vector is given by kM = [R(θ) + R(−θ) − 2I]k0 = 2(cos(θ) − 1)Ik0, (26) which is of order θ2 for small θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Hence, the moir´e wave- length is of order 1 θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Moreover, since the order-θ2 deviation is only from this particular singular structure, and not from the “doubly- singular” structure, it exhibits a single 2D moir´e pattern rather than complex overlapping structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The rele- vant singular structures are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' CONFIGURATION SPACE OF INTRINSICALLY TRILAYER MOIR´E PATTERNS There is an apparent contradiction between the na¨ıve configuration space described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II D, which indi- cates that trilayers have complex moir´e patterns that cannot possibly fit on a lattice, and the intrinsically tri- layer moir´e patterns presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III, which very clearly do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We seek to resolve this contradiction by a more nuanced description of the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The missing ingredient from the na¨ıve configuration space given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (14) is a collection of “nontrivial triv- ial transformations,” which are nontrivial in that they do not correspond to overall translations, but trivial in that they do not change the local moir´e structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The correct configuration space of the moir´e pattern is the set of translations of each layer modulo overall translations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', simultaneous translations of all layers by the same amount) and these new transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We now describe how to find these additional transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We do so in a way that naturally derives not only the dimensionality of the true configuration space, but also explains why it is toroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The intuition of the argument derives from the charac- terization of singular structures provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III B 1: singular structures are those structures for which cer- tain relative translations of the layers change the aver- age value of local quantities by providing phases between different contributions to the zeroth Fourier mode of the quantity of interest, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A moir´e heterostruc- ture can be viewed as resulting from these different possi- ble phases: different regions in the moir´e heterostructure correspond to different relative translations of the singu- lar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The nontrivial trivial transformations we seek to find derive from the converse of that identification: any rela- tive translation which does not result in a phase will make no impact on average properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Such relative transla- tions that do not result in phases, therefore, are precisely the nontrivial trivial transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We find the nontrivial trivial transformations formally using in the frequency picture described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For simplicity, we take as a concrete example the (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 1, 0)-moir´e on the square lattice (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The Fourier modes are indexed by Z6, but the moir´e modes arise from the zero mode lattice described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' III B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' In this specific case, the zero mode lattice is spanned by the vectors (1, 0, 1, 0, 1, 0) and (0, 1, 0, 1, 0, 1), which we call n(1) and n(2) (each of which also have in- dices, n(1,2) i,j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A translation of layer i by ai (not necessarily a lattice vector) will multiply the Fourier mode with indices ni,j by a phase exp �� i,j ni,jbi,j · ai � , which follows from the discrete Fourier transform in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' For the relative translations which preserve the moir´e lattice, this phase vanishes when evaluated on the zero mode lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Clearly, translating each layer by the same amount, 12 ai = a, results in this phase vanishing on the zero-mode lattice, where � n(k) i,j bi,j = 0 for both k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This imposes two constraints on the six-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' The additional constraints are found by setting a1 = 0, at which point the constraint is b2,i · a2 = −b3,i · a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' the simplest two basis solutions are {a2 = b3,1, a3 = −b2,1} and {a2 = b3,2, a3 = −b2,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' These extra translations are most of the “nontrivial trivial transformations” we were searching for, and suffice to reduce the dimension- ality of the configuration space from four to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Note that this two-dimensional space is periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', a torus rather than a plane, because the sum � i,j ni,jbi,j · ai need not vanish identically for the phase to vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' in- stead, it can be a multiple of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This periodicity ensures that the final phase space is indeed a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, our final and most general characterization of the phase space is as the collection of relative transla- tions of the layers modulo those which act trivially on the zero mode lattice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=', on the combinations of modes that contribute to the zero mode in the singular structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Note that in multiply-singular structures, such as TTLG, the moir´e-generating lattice is greater than two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Therefore, there is at least a four- dimensional manifold defining the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Consequently, one cannot regard this configuration space as being periodically fully explored in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' This explains the difference between the complex patterns in TTLG and the periodic moir´e in intrinsically trilayer sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Configuration space and lattice relaxation To illustrate the usefulness of configuration space, we consider lattice relaxation in intrinsically trilayer moir´e systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Lattice relaxation is usually computed by tak- ing an average energy density of any particular stacking configuration, then enlarging regions of low-energy stack- ing while shrinking regions of high-energy stackings [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We claim that the average energy density of a singular structure on long wavelengths does not change under a nontrivial trivial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' That is to say, struc- tures in the na¨ıve configuration space (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' II D) that dif- fer by a nontrivial trivial transformation have the same energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' We now justify this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfz_5Y/content/2301.01777v1.pdf'} +page_content=' Consider the energy den- sity of a singular structure, ρ(x), and consider the energy density over some large region of radius R, 1 πR2 � |x| +pre-training +fine-tuning +Object +Encoder + name: cup + size: small + filled: no +Figure 1: Original PIGLeT Physical Dynamics Model (Zellers et al., 2021). During pre-training the model +receives as input the full symbolic representation of two objects (o0 +pre and o1 +pre) before the action is taken and +the symbolic representation of the action itself (a) and is tasked with predicting the attributes of the objects after +the action (o0 +post and o1 +post). During fine-tuning, the action encoder is replaced by an LLM to process a natural +language description of the action being taken and with what objects. +formal symbolic representations. +Our contributions are three-fold. First, we show +that it is possible to predict the physical effects of +actions from visual data. Second, we show that it +is possible to learn the task on training data where +formal symbolic representations, which are unob- +servable in real-world settings, are replaced with +NL descriptions (which can be observed through +natural interaction). Third, we evaluate all our mod- +els in a stricter zero-shot setup to promote ways +to train agents that generalize. Overall our work +paves the way for multi-modal models that learn +the effects of actions in realistic environments. +2 +Related Work +Commonsense reasoning has been highlighted as a +potential weak point of LLMs in recent years (Shen +and Kejriwal, 2021; Forbes et al., 2019; Bisk et al., +2020). Datasets such as PIGPeN (Zellers et al., +2021), commonsenseQA (Talmor et al., 2019), +VCR (Zellers et al., 2019) and GD-VCR (Yin et al., +2021) help evaluate different aspects of common- +sense reasoning in modern LLMs. In this paper, we +focus on physical commonsense reasoning, which +involves understanding the (often) unexpressed +rules of the physical world. +Forbes et al. (2019) reported that neural repre- +sentations found it challenging to infer the link +between actions and what they imply about the +attributes of objects. Accordingly, Zellers et al. +(2019) introduced the Visual Commonsense Rea- +soning (VCR) task to test how images can inform +question answering models that tackle common- +sense information. Bisk et al. (2020) designed the +PIQA benchmark to evaluate physical common- +sense reasoning in LLMs through question answer- +ing. Sampat et al. (2021) proposed an extension to +the CLEVR dataset, where an agent must reason +and answer questions about a scene after a hypo- +thetical action is taken. +Multiple approaches can improve the capabil- +ities of LLMs in commonsense reasoning, such +as using handcrafted knowledge graphs (Hwang +et al., 2021) or leveraging simulated environments +(Zellers et al., 2021). PIGLeT, in particular, com- +bines a traditional LLM and a “Physical Dynamics” +model to ground an LLM (Zellers et al., 2021). The +Physical Dynamics model enhances the common- +sense knowledge of an LLM by fine-tuning it, using +trajectories sampled from a realistic environment +(see Figure 1). Trajectories are an action and a +pair of environment states (before and after the ac- +tion) expressed in a formal symbolic representation. +Zellers et al. (2021) found that fine-tuning LLMs +with symbolic data from the simulated environment +helped them outperform other models in physical +commonsense reasoning tasks: in particular, pre- +dicting the effects of an action when executed in a +particular state. +Image inputs offer a way to ground an LLM, as +they only require general alignment with a text or +symbolic input and do not require the comprehen- +sive environment ground-truth labels that PIGLeT +uses. Gao et al. (2018) used multi-modal web +data to learn actions and their effects from images + +Action +Apply +Object +Decoder +Action +Encoder + +"The robot empties +the cup." +1 +2 +observed +latent +1 +2 +... +... +... + name: cup + size: small + filled: no +Figure 2: PIGLeT-Vis. We introduce PIGLeT-Vis, where we modify the PIGLeT architecture to replace its Sym- +bolic Object Encoder with a vision component that makes use of images of the environment before and after +an action is taken to predict the symbolic representation of objects post-action. We use an attention mechanism +over the extracted bounding boxes to obtain a visual hidden representation of an object given its name. The only +remaining symbolic inputs during pre-training are the action description and object names. +and corresponding text descriptions. Zellers et al. +(2019) used an off-the-shelf ResNet50 model (He +et al., 2016) to augment an existing BERT language +model (Devlin et al., 2019) with vision capabilities. +Transformer models such as UNITER (Chen et al., +2020), ERNIE-ViL (Yu et al., 2021), VisualBERT +(Li et al., 2020), and ViLBert (Lu et al., 2019) +have been applied to visual commonsense reason- +ing. These models use a joint transformer backbone +for images and text and vary their pre-training ob- +jectives. However, most of these models are trained +on static text-image pairs: they aren’t designed to +capture the dynamics of an environment, partic- +ularly how object attributes change with actions. +Notably, recent work by Hanna et al. (2022) uses +CLIP (Radford et al., 2021) and MOCA (Singh +et al., 2021) embeddings to predict a post-action +image given a set of possible images. In contrast, +we focus on adapting an LLM with a vision-based +component to predict the consequences of actions +on the environment. +3 +Method +We propose PIGLeT-Vis (Figure 2) for learning the +effects of actions on objects from images. We use a +pre-trained vision backbone, DETR (Carion et al., +2020), as a Vision Object Encoder and combine +it with a RoBERTa LLM (Liu et al., 2019) as an +Action Encoder. We experiment with different con- +figurations of inputs to measure the impact of the +various components of our architecture. In partic- +ular, we test a variation in which we remove the +formal symbolic labels even in training, replacing +them with NL text labels. To evaluate our models, +we use the PIGPeN dataset (Zellers et al., 2021), +which consists of a symbolic and visual representa- +tion of an environment before and after an action is +taken. However, we filter PIGPeN to create a viable +testing ground for visual grounding of physical ac- +tions and more accurately measure generalization +capabilities of models. +3.1 +Architecture +PIGLeT-Vis (shown in Figure 2) consists of sepa- +rate components, which can combine multi-modal +inputs in different ways. Through this modular +approach, we can turn off specific components to +evaluate how different inputs and model structures +affect performance on the task. We test models +with and without symbolic inputs and image inputs. +For all components, we use a dropout of p = 0.1 +in between layers and a default hidden layer size +of h = 64. +3.1.1 +Object Encoder +We reproduce Zellers et al. (2021), where all ac- +tions are assumed to involve two objects, o0 and +o1, and the symbolic representation of objects are +encoded in an Object Encoder model. The sym- +bolic representation of an object before the action +is represented by opre. Both objects (o0 +pre and o1 +pre) +in the environment are described by a vector of +38 attributes, chosen on the basis that they are the +kinds of physical attributes that are influenced by +actions. They describe an object as small/large, +cold/hot, empty/full, etc. +We first embed these symbolic object attributes + +using an embedding layer Ee×h, where e = 329 is +the total number of unique attributes and h is our +hidden size. For an object k: +ˆok +pre = E(ok +pre) +(1) +The Object Encoder Oencoder takes in the embed- +ded object attributes through a set of multi-head +attention layers to encode the symbolic representa- +tion of each object. We use the default Pytorch im- +plementation of the Transformer Encoder (Paszke +et al., 2019) with three layers and 4 heads. The first +encoded output of each object sequence is used for +representing the entire object. +hk +pre = Oencoder(ˆok +pre) +(2) +3.1.2 +Action Encoder +Actions are encoded either as a symbolic triplet +⟨action, action object, action receptacle⟩ or as an +annotated text describing an action being taken +(e.g., “robot empties the cup”). +During +pre-training, +the +Action +Encoder +Apretrain uses an action embedding layer E′ to +embed the first dimension of the action, and re- +uses the object embedding layer E to embed the +action object name ao and action receptacle name +ar. The action embedding layer E′ has dimension- +ality 10 × h for the 10 distinct actions. The three +embedded representations are summed and passed +to the Action Encoder’s linear layers to produce ha +(see equation 3). Similarly to Zellers et al. (2021), +a tanh activation is applied after each linear layer. +ha = Apretrain(E′(a) + E(ao) + E(ar))) +(3) +When fine-tuning on the annotated dataset, the +action input is text and therefore we switch out the +Action Encoder Apretrain for Afinetune—our text- +based Action Encoder. Afinetune uses a RoBERTa- +base1 model (Liu et al., 2019) to process a tok- +enized version of the text input at. The first token +([CLS]) of the RoBERTa output layer is used to rep- +resent the action sequence and then passed through +a linear layer to map the dimensionality of the hid- +den states from 256 to h. +ha = Afinetune(at) +(4) +1Implementation and pre-trained model weights are taken +from the Huggingface library (Wolf et al., 2019). +3.1.3 +Vision Object Encoder +The Vision Object Encoder takes in images (ipre +and ipost) to provide a visual representation of each +object k before and after (hk +pre and hk +post). We +use the DETR1 (Carion et al., 2020) model as a +backbone to predict N bounding boxes in a pair +of images (pre- and post-action). As DETR is pre- +trained on the COCO object detection dataset (Lin +et al., 2014), its predicted object labels do not align +with those in PIGPeN. Therefore, we instead learn +a mapping between the predicted bounding box +representations and the PIGPeN objects. For each +image, we obtain a hidden representation hb of +dimensionality N × 256 where N = 100. +We use an attention mechanism over the bound- +ing boxes’ hidden representation, conditioned on +the object names. For a given object ok, its condi- +tional representation hk +c is the encoded name of the +object: E(ok +name). We can therefore obtain the at- +tention score of a given object ok and image im by +calculating the alignment between the conditional +representation hk +c and the hidden representations of +bounding boxes hbm: +hbm = DETR(im) +(5) +αk +m = Softmax +� h +� +i=1 +(hk +chbm)i +� +(6) +We obtain the final representation for a given ob- +ject and image by multiplying our attention scores +α with the extracted output representation from +DETR and summing along the bounding box axis: +hk +om = W +� +� +b +� +j=1 +(αk +mhbm)j +� +� +(7) +We use a final output layer W to decrease the di- +mensionality of ho from the DETR dimensionality +of 256 to h. +Through the Vision Object Encoder, we replace +the previously symbolic inputs with images and can +extract [h0 +preh1 +pre] and [h0 +posth1 +post] from ipre and +ipost respectively. Note that we make the implicit +assumption that ipre and ipost contain the informa- +tion necessary to predict object attributes of the +objects post-action. +3.1.4 +Action Apply +The Action Apply Model β is a simple fuse op- +eration (concatenation in the hidden dimension) +followed by three linear layers, which combine + +the action representation ha and an object repre- +sentation of the scene pre-action hk +pre. The model +outputs an object’s representation hk +a, containing +information conditioned all inputs: +hk +a = β(ha, hk +pre) +(8) +3.2 +Object Decoder +Finally, the Object Decoder is a transformer mod- +ule that maps the object representations ho from +the pre-action state back to 38 symbolic attributes. +It uses a default three layer Transformer Decoder +(Paszke et al., 2019) that takes the hidden repre- +sentation from the Action Apply hk +a as an encoded +memory state and hk +pre as the source sequence to +predicts a label for each attribute. +˙ok +post = Odecoder(hk +a, hk +pre) +(9) +When we use image inputs, we also have access +to the post-action visual representation and can +therefore use hk +pre + hk +post instead of hk +pre. +The output has post-action object states ˙ok +post +which are compared to the ground truth ok +post to +calculate cross-entropy. As an additional loss, we +also use the cross-entropy between ˙ok +pre and ok +pre by +passing an empty hk +a to force the Object Decoder +to recreate the attributes in the pre-action state. We +weight both losses equally. +3.3 +Evaluation Metrics +Since our task involves predicting 38 attributes +for two different objects per example, we follow +Zellers et al. (2021) and report different types of +accuracy metrics on the test set (after fine-tuning). +We measure the overall accuracy by scoring how +many objects have all attributes correctly predicted +(exact match). Note that this is a high bar for a +model where the symbolic representations are la- +tent: to predict an object correctly, our model must +first estimate its attributes before the action and +then estimate whether and how these change given +an action. So we also measure the attribute-level +and action-level accuracies of each model, so as +to explore which attributes and actions are more +difficult to predict than others. +3.4 +PIGPeN-Vis Dataset Split +To evaluate physical commonsense reasoning using +PIGLeT-Vis, we filter PIGPeN (Zellers et al., 2021) +to create a subset (PIGPeN-Vis) which we use for +all our experiments. We motivate PIGPeN-Vis as +a way to isolate the effects of adding our vision +component, because while PIGPeN already has +images, these images were not used in PIGLeT. +The PIGPeN dataset consists of trajectories of +an environment before (pre) and after (post) an +action is taken. Each trajectory contains repre- +sentations of two distinct objects before and af- +ter. One of the objects is usually targeted by the +action, while the other acts as a distractor. In ad- +dition, image pairs (ipre, ipost) for each trajectory +are provided, where each image is snapshot of the +simulated photo-realistic 3D environment which +contains the objects in view (see Appendix B for +an example). Each image is an RGB image of +dimensions 640 × 385. +The original dataset is separated into two distinct +sets: +1. A pre-training set of 278, 009 trajectories, +which includes the symbolic representations +of objects o before and after a symbolic action +a is taken. A separate validation set of 33, 042 +examples is also included. +2. A fine-tuning set of 1, 000 trajectories which +has been annotated to replace the symbolic ac- +tion a with a textual representation at describ- +ing the action. Separate validation and test +sets of 500 examples each are also included. +All metrics are reported on the test set. +In PIGPeN, the object states opre and opost con- +tained 40 different attributes and 13 different ac- +tions a. Attributes range from intrinsic such as +name or moveable to stateful such as distance or +isCooked. In forming PIGPeN-Vis, we remove +two attributes and three actions from the dataset +to obtain 38 attributes and 10 possible actions (see +Appendix B for more details). +3.4.1 +Viewpoint and Action Filtering +Since the PIGPeN images were not generated with +the goal of being used as input data, we identified +several issues with the quality of certain scenes. +A notable difficulty is that in some cases, the be- +fore and after images are not captured from the +same camera angle or they have different light- +ing conditions. Changing orientations and lighting +conditions makes it difficult to use an image pair +(ipre, ipost) to isolate the outcome of an action. Con- +versely, image pairs with too few perceivable differ- +ences also break our assumption that the changes in + +the environment are perceivable. Therefore, we fil- +ter the dataset using pixel statistics to remove image +pairs that have either large perceivable differences +(likely due to changes in viewpoint) or small per- +ceivable differences (where the action’s results are +not visually salient enough) (see Appendix B.2). +We exclude 15.4% of the total dataset through +visual filtering of the original dataset. +3.4.2 +Zero-Shot Filtering +To evaluate the generalization capabilities gained +from a vision component, we further filter the +dataset to exclude a subset of training examples. +Unlike the original PIGPeN dataset which only +tested for zero-shot generalization at the level of +the fine-tuning data, we remove all instances with +selected specific objects or action-object pairs from +all training and validation sets. To minimize the +effect of removing examples from the dataset, we +pick objects and action-object pairs with an already +low number of samples in the training sets. In total, +we exclude 14 objects and 27 action-object pairs, +which amounts to less than 3% (6, 816 samples) +of the remaining training sets (see Appendix B.3). +These zero-shot examples comprise around 10% of +the test set. +After both filtering stages, PIGPeN-Vis contains +a pre-training dataset of 232, 625 trajectories with a +validation set of 26, 823, and a fine-tuning training +set of 750 examples with a validation set of 367 +examples and a test set of 398 examples. +3.5 +Training Configurations +We evaluate the impact of the vision component on +PIGPeN-Vis through five different setups: +• base: We implement a baseline model with- +out symbolic object inputs. Our implemen- +tation removes the Object Encoder entirely, +such that the model must predict the attributes +of objects solely from knowing the action and +the object names that it relates to. This model +acts as a lower bound on the capabilities of the +vision model: its performance would match +the vision model if images are irrelevant to +solving the task. +• base+symbolic: This is our implementation +of the original Zellers et al. (2021) PIGLeT +model, shown in Figure 1. This model acts as +an upper bound on the capabilities of the vi- +sion model since it observes the true symbolic +representations of objects before the action +(which the vision model must estimate). +• base+images: This is our proposed PIGLeT- +Vis, shown in Figure 2, where the Vision Ob- +ject Encoder replaces the previously symbolic +Object Encoder. This model leverages the +before and after images of the environment +as well as the name of the objects to extract +representations of the object attributes. +• base+symbolic+images: We sum the hid- +den symbolic representations of objects with +their visual representations in a unified model. +Through this setup, we evaluate whether im- +ages can provide additional information to the +already comprehensive symbolic representa- +tions. +• base+images+text-labels: We convert the +symbolic representations of the labels for the +object names and actions to their text label and +encode them using a frozen LLM during pre- +training. We use the same LLM to encode the +text labels that we later use in the fine-tuning +stage. This setup replaces all symbolic inputs +from the pre-training stage to only language +and image inputs. +Note that there are a few differences between the +original Zellers et al. (2021) model and our im- +plementation of base+symbolic. +For instance, +for simplicity, we opted to use an off-the-shelf +RoBERTa-base (Liu et al., 2019) model instead +of training our own custom GPT2 (Radford et al., +2019). Additionally, we also reduce the dimen- +sionality of the PIGLeT layers from h = 256 to +h = 64. We found that not only does this allow +faster training times as it shrinks the Physical Dy- +namics model from 11.9 million parameters to 2 +million parameters, it also improves the overall +accuracy by a small margin (+1.51%). +We train each model for 80 epochs with a batch +size of 256 using the Pytorch implementation of +the Adam optimizer (Kingma and Ba, 2014) and a +learning rate of 10−3 during pre-training and 10−5 +during fine-tuning. We run each setup over 10 +different seeds and report the average and standard +deviation for each metric (see Appendix C.1 for +more details). + +Accuracy (% ± σ) +Overall +Zero-Shot +base +21.23 ± 0.72 +5.34 ± 2.77 +base+symbolic (PIGLeT) +85.03 ± 0.45 +39.04 ± 3.37 +base+symbolic+images +86.01 ± 0.89 +35.89 ± 3.47 +base+images (PIGLeT-Vis) +45.47 ± 1.50 +7.53 ± 2.60 +base+images+text-labels +47.55 ± 2.10 +8.90 ± 3.24 +Table 1: Overall and zero-shot accuracies (PIGPeN- +Vis) +4 +Results and Discussion +We evaluate all models on our PIGPeN-Vis split +and report the overall (exact match), zero-shot, +action-level, and attribute-level accuracy results +for all setups in Tables 1 and 2. For completeness, +we also evaluate models on the original PIGPeN to +contrast the effects of our filtering operations (see +§3.4 and Appendix D) and find PIGPeN-Vis is a +more challenging subset for all models. +The base model provides a low bar estimate of +what is achievable using only the action encoder +inputs. Unsurprisingly, the base model performs +worst on overall accuracy, which demands an ex- +act match of all attributes. It does relatively well +on (individual) attribute-level accuracy, primarily +because it predicts the most common attribute for +each object. Some actions are also easier than +others—for instance, the model reaches 27.38% ac- +curacy on ToggleOn from only knowing the action +and object names. This is likely because ToggleOn +is constrained to a small set of objects and effects. +Our base+symbolic model obtains similar re- +sults to the original implementation by Zellers et al. +(2021), with an overall accuracy of 85.03%. How- +ever, it performs much worse on the zero-shot split +(39.04%) than the original PIGLeT model reported +(80.2%) (Zellers et al., 2021). This disparity can +be explained by the fact that the original zero-shot +PIGPeN dataset was not a true zero-shot dataset, be- +cause the Physical Dynamics model was exposed +to the “unseen” objects in its pre-training. The +base+symbolic model provides a high bar esti- +mate of what could be achievable if: (i) ipre and +ipost capture the symbolic environment; and (ii) the +Vision Object Encoder can subsequently extract +these features. However, as we will argue in Sec- +tion 6, both (i) and (ii) are unrealistic given the +constraints of both the dataset and the model. +Our base+images (PIGLeT-Vis) model scores +45.28% in overall accuracy but only 7.53% on the +zero-shot set. +Nevertheless, it outperforms the +base model in overall accuracy (p < 0.0001) and +in zero-shot accuracy (p = 0.08), which demon- +strates that the images improve the prediction of +the effects of actions. The base+images model +also performs significantly better than base on dif- +ficult attribute-level accuracies such as distance +(p < 0.0001). However, as before, accuracy on +individual attributes benefits from the skewed dis- +tributions of their values and does not necessar- +ily translate to high scores on predicting all 38 +attributes correctly. +Utilizing both images and symbolic representa- +tions as inputs helps the base+symbolic+images +model outperform purely symbolic inputs in over- +all accuracy, from 85.03% to 86.01% (p < 0.01). +However, image inputs also decrease the model’s +zero-shot performance from 39.04% to 35.89%, al- +though this isn’t statistically significant (p = 0.05) +due to high variance. We suspect that this high +variance is caused by an increase in noise in the +model resulting from adding images to the sym- +bolic model. However, the overall picture is more +complicated, as images can also provide gains on +certain actions (e.g., PickUp accuracy increases +from 80.48% to 86.14%) even though it causes a +decrease in many other cases (e.g., ToggleOn). +Finally, when we utilize NL descriptions to re- +place the formal symbolic inputs (action name +and object names), base+images+text-labels +improves overall accuracy when compared to +base+images from 45.47% to 47.55% (p = 0.02). +Text inputs appear to improve zero-shot accuracy, +but not by a statistically significant margin (p = +0.31). Accuracy also improves in most actions, +for instance the Slice accuracy improves from +41.64% to 45.57% (p = 0.03). So the NL descrip- +tions inform the task in a beneficial way, over and +above the raw images. But encoding the labels as +text rather than formal symbolic representations +also adds noise. +Nevertheless, text labels improve accuracy on +actions where the semantic information contained +in the label provides a richer context to help gen- +eralize to similar objects. For instance, a “cup” +and a “mug” are semantically close, and thus learn- +ing the effects of actions on a “cup” might help +the model predict the same effects on a “mug” +even if the word forms are different. +In con- +trast, the formal symbolic representations treat +the predicate symbols cup and mug as unrelated, +and so don’t benefit from the lexical relationships +that the LLM captures. Fully removing the sym- + +Action Accuracy (%) +Attribute Accuracy (%) +Open +Pickup ToggleOn Slice +size +distance temperature +base +8.33 +10.96 +27.38 +22.13 +73.78 +51.01 +95.91 +base+symbolic (PIGLeT) +85.73 +80.48 +96.90 +75.41 +94.98 +95.13 +99.85 +base+symbolic+images +88.75 86.14 +92.86 +81.31 +96.35 +96.13 +99.59 +base+images (PIGLeT-Vis) +20.83 +33.49 +70.24 +41.64 +87.03 +76.62 +96.10 +base+images+text-labels +22.92 +40.12 +67.14 +45.57 +87.89 +78.06 +96.72 +Table 2: Action and attribute specific accuracies for a subset of actions and attributes; for a comprehensive table +with standard deviations see Appendix D. size and distance each have eight possible classes while temperature +has three. +Apple +Apple +Pot +CounterTop +Before +After +SliceObject(Apple) on (CounterTop,Apple) +PutObject(Apple, Pot) on (Apple,Pot) +Figure 3: We visualize the attention of the Vision Object Encoder from a trained base+images model on two +different actions and environments. The left grid focuses on the effect of Slice(Apple) on CounterTop and +Apple, while the right grid focuses on the effects of Slice(Apple) on Apple and Pot objects. +bolic representations allows us to adapt our model +to any possible unseen object during test time. +base+images+text-labels is adaptable to gen- +eral settings without knowing the symbolic map- +ping of objects and actions in the environment. +The results of both base+symbolic+images and +base+images+text-labels make the case multi- +modal modeling of commonsense reasoning, as +both language and images are complementary to +generalize to unseen settings. +4.1 +Qualitative Attention Maps +Visualizing attention is another benefit of a vision +component, as we can see what the model focuses +on and partially explain its predictions. Figure 3 +shows two separate examples and corresponding at- +tention maps. In the left example, base+images is +tasked with predicting the attributes of CounterTop +and Apple after the Slice action is applied on the +Apple. In the right example, the Put action is ap- +plied on the Apple, and the model must predict +the attributes of the Apple and the distractor object +Pot. The two rows are the before and after images +(ipre and ipost), and the two columns are the two +objects used to condition the attention. The atten- +tion maps display the strength of the attention for +each bounding box given an object name. +Both examples in Figure 3 show that the Vision +Object Encoder can map known objects to relevant +bounding boxes. The model successfully tracks the +Apple in both cases by placing the most weight on +the bounding box targeting the Apple. However, +these examples also show the difficulty of this task— +the environments are realistic and can be filled with +more than one instance of an object. +5 +Conclusion +In this paper, we tackle the task of predicting the +effects of actions on objects’ physical attributes. In +contrast to (Zellers et al., 2021), our model does +not treat the formal symbolic representation of the +images as observed. Instead, PIGLeT-Vis supports +inference when the inputs are images alone or im- +ages plus NL descriptions and a phrase denoting +the action (e.g., “the robot empties the cup”). While +PIGPeN offers challenges for applying a multi- +modal approach, our model can extract useful in- +formation from images, opening the door for gen- +eralizing learning physical commonsense to real- +world data. Importantly, our PIGPeN-Vis split can +be used to evaluate the zero-shot capabilities of +different model configurations. Moreover, while +base+symbolic still outperforms base+images, it + +CounterTop +Apple +Before +AfterApple +Pot +Before +Afterdoes so without estimating the attributes of ob- +jects and thus solves a much easier but unrealistic +task. Through base+images+text-labels, we +show that, when replacing symbolic inputs, the +best solution is to complement image inputs with +NL descriptions to leverage information from both +modalities. Finally, our results show the need to +improve the generalization capabilities of multi- +modal models such that they can learn and adapt to +unseen situations. +6 +Limitations +There are several limitations to our approach that +result directly from the inherent limitations of PIG- +PeN and our proposed Vision Object Encoder re- +spectively. +PIGPeN was not originally designed for test- +ing commonsense reasoning using images and con- +tains numerous inconsistencies which cannot all be +solved with the PIGPeN-Vis split obtained from +filtering (Section 3.4.1). Given the presence of non- +physically salient attributes such as temperature, +images are not guaranteed to fully capture their +symbolic representations. PIGPeN includes certain +attributes which are not discernible from images, +e.g., even humans would be unable to tell a hot +plate from a cold plate from vision alone. The im- +ages in PIGPeN can also contain more than one +object (e.g., more than one cup) without ever speci- +fying which one the symbolic representation refers +to. This causes difficulty for our approach because +judging specific attributes such as distance is im- +possible if there are two cups at different distances +from the viewpoint. Additionally, PIGPeN also +discretizes continuous variables such as distance +into categories which can be hard to disambiguate. +To approach the accuracy of base+symbolic +with our vision component, we also need a vision +representation from which to correctly estimate +all latent attributes. Even if images are assumed +to be perfect representations of the symbolic en- +vironment, the model still has to extract each of +the 38 attributes correctly for both objects using +only two images. It is possible (and likely) for +the vision detection backbone to miss the target +object entirely because it is not trained to detect +the specific object in question. We see this effect +in Figure 3, where the model falls back to using a +bounding box around the sink area to describe the +CounterTop object. The DETR vision model used +to extract bounding boxes was pre-trained on the +COCO dataset (Lin et al., 2014) which does not +contain CounterTop as an object. PIGLeT-Vis is +therefore ultimately limited by the capabilities of +its vision backbone. +Ethics Statement +While this work does not introduce new data or +involve human participants, we use the PIGPeN +dataset which contains human-labelled data. The +fine-tuning portion of the dataset was annotated +through MTurk by Zellers et al. (2021) and they re- +port following best practices (paying decent wages, +providing feedback and using a qualification test) +in their data collection. We filter and use a subset of +PIGPeN and introduce methods to learn the effects +of actions in a multimodal setting. We, therefore, +believe that our work does not raise any ethical +concerns. +Acknowledgements +This work was supported in part by the UKRI Cen- +tre for Doctoral Training in Natural Language Pro- +cessing, funded by the UKRI (grant EP/S022481/1) +and the University of Edinburgh, School of Infor- +matics and School of Philosophy, Psychology & +Language Sciences. +References +Yonatan Bisk, Rowan Zellers, Ronan Le Bras, Jian- +feng Gao, and Yejin Choi. 2020. Piqa: Reasoning +about physical commonsense in natural language. 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In Proceedings of the 59th Annual Meet- +ing of the Association for Computational Linguis- +tics. +A +GPT-3 Example of Physical Reasoning +B +PIGPeN-Vis +We select an example from PIGPeN to display in +Figure 5 and Table 3. +From +the +original +dataset, +we +re- +move +two +attributes +(isUsedUp +and +salientMaterials_Organic) +because +they +are unchanged in all examples. We also remove +The weight of the potato is 150 grams. +The robot then slices the potato into thin slices. +The weight of the potato is now 75 grams. +Figure 4: Example of incorrect physical commonsense +by an LLM. When predicting what comes after the in- +put text, the large 175 billion parameter GPT-3 (Brown +et al., 2020) predicts that the weight of the potato +halves after a slicing action is taken. +EmptyLiquidFromObject +object=Cup +Action +Annotated Action +"The robot empties the +cup into the sink." +Figure 5: Image pair and actions for a selected PIGPeN +example. +3 actions (ThrowObject10, +ThrowObject100 +and ThrowObject1000) which are all related +to throwing an object across a certain distance. +These actions account for only a small subset of +the dataset and create inconsistent image pairs +due to the agent’s momentum being captured in +the images. +The angle of the camera changes +as a result of ThrowObject and this breaks our +assumption that the difference between ipre and +ipost solely reflects the effects of the action on the +environment (and not on the viewer). We therefore +reduce the total number of symbolic attributes per +object to 38 and the number of possible actions to +10. +B.1 +Attributes +The following 38 symbolic attributes are used to +describe an object in PIGPeN: +ObjectName, +parentReceptacles, +receptacleObjectIds, +distance, +mass,size, +ObjectTemperature, +breakable, +cookable, +dirtyable, +isBroken, +isCooked, +isDirty, +isFilledWithLiquid, +isOpen, +isPickedUp, +isSliced, +isToggled, +moveable, +openable, +pickupable, receptacle, salientMaterials_Ceramic, +salientMaterials_Fabric, +salientMaterials_Food, +salientMaterials_Glass, salientMaterials_Leather, +salientMaterials_Metal, +salientMaterials_Paper, +salientMaterials_Plastic, +salientMaterials_Rubber, +salientMaterials_Soap, + +pre +post +ocup +pre +ofaucet +pre +ocup +post +ofaucet +post +ObjectName +Cup +Faucet +Cup +Faucet +Contained Objects +Is contained in... +Mass +1 to 2lb +Massless +1 to 2lb +Massless +Size +small +medium +small +medium +Temperature +RoomTemp +RoomTemp +RoomTemp +RoomTemp +Distance +1 to 2ft +3 to 4 ft +1 to 2ft +3 to 4 ft +Breakable +Yes +No +Yes +No +Cookable +No +No +No +No +CanBecomeDirty +Yes +No +Yes +No +IsBroken +No +No +No +No +IsCooked +No +No +No +No +IsDirty +No +No +No +No +IsFilledWithLiquid +Yes +No +No +No +IsOpen +No +No +No +No +IsPickedUp +Yes +No +Yes +No +IsSliced +No +No +No +No +IsToggled +No +No +No +No +Moveable +No +No +No +No +Openable +No +No +No +No +Pickupable +Yes +No +Yes +No +CanHoldItems +Yes +No +Yes +No +Sliceable +No +No +No +No +Toggleable +No +Yes +No +Yes +Materials +Ceramic +Ceramic +Table 3: Attributes for a selected PIGPeN example. +The total number of attributes is 38 as the Materials +attribute is a multi-hot encoding. +salientMaterials_Sponge, +salientMaterials_Stone, +salientMaterials_Wax, +salientMaterials_Wood, +sliceable, toggleable +B.2 +Filtering Statistics +We initially filter the PIGPeN dataset using two +main strategies to remove images with too much or +too little change between the pre and post images. +In both cases, the goal is to remove pairs of images +in which it would be impossible for a vision model +to predict what has changed. +Images with too many changes are often images +taken from different viewpoints or with different +lighting conditions. We filter these images by look- +ing at the number of pixels changed between ipre +and ipost. We show the distribution of the num- +ber of pixels changed per image over the training +dataset in Figure 6. Using this visualization we +can clearly see a small peak at the extreme - where +almost all the pixels in ipost are different from ipre. +Note that since each image is an RGB image of +dimensions 640 × 385, the max number of change +is 640 × 385 × 3 = 739, 200 (we also compare +pixels across color channels). We opt to remove +all images with more than 400, 000 changes, which +corresponds to around 6.2% of the training dataset. +Images with too little change could be exam- +ples of where the action has no visual outcome and +ipre and ipost are indistinguishable. To filter these +Figure 6: Distribution of the number of pixels changed +per image in the PIGPeN dataset. +Figure 7: Distribution of the maximum pixel value +changed per image in the PIGPeN dataset. + +12000 +10000 +8000 +Count +6000 +4000 +2000 +0 +0 +100000200000300000400000500000600000700000 +num_change5000 +4000 +ur +3000 +8 +2000 +1000 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +max_changeimages we measure the maximum magnitude of +change in each pixel and each color channel be- +tween the pairs of images. We visualize the max +change across the training dataset in Figure 7. Here +a low values implies almost no salient change, and +as max change approaches zero - it becomes un- +likely that a human would be able to perceive the +difference between the pair of images. We opt for +to keep images with a max change greater than +0.2 which corresponds to excluding 7.8% of the +training dataset. +Filtering on the number of changed pixels lead +to the exclusion of around 13.89% of the training +dataset. +B.3 +Zero-shot Filtering +We remove the following 14 objects from both the +train and validation (3, 401 examples total): +HandTowel, Towel, Plunger, Watch, CD, SoapBottle, +Pen, +RemoteControl, +SoapBar, +Box, +Bottle, +CreditCard, Statue, KeyChain +We remove the following 27 action-object pairs +from both the train and validation (3, 278 examples +total): +(CloseObject,Toilet), +(DirtyObject,Pan), +(DirtyObject,Pot), +(EmptyLiquidFromObject,Bottle), +(EmptyLiquidFromObject,Pot), (OpenObject,Toilet), +(PickupObject,Box), +(PickupObject,CellPhone), +(PickupObject,CreditCard), +(PickupObject,KeyChain), +(PutObject,CD), +(PutObject,CreditCard), +(PutObject,HandTowel), +(PutObject,Laptop), +(PutObject,Lettuce), +(PutObject,Pen), +(PutObject,Plunger), +(PutObject,Pot), +(PutObject,RemoteControl), +(PutObject,SoapBar), +(PutObject,SoapBottle), +(PutObject,Statue), +(PutObject,ToiletPaper), +(PutObject,Towel), +(PutObject,Watch), +(ToggleOff,CellPhone), (ToggleOff,Television) +C +Code Release and Training +Our full code, models, and PIGPeN-Vis split can +be found at github.com/gautierdag/piglet-vis. +C.1 +Additional Training Details +As previously mentioned, there are a few differ- +ences between the original Zellers et al. (2021) +model and our implementation of base+symbolic. +We use an off-the-shelf RoBERTa-base (Liu et al., +2019) model instead of a custom GPT2 (Radford +et al., 2019). Additionally, we also reduce the di- +mensionality of the PIGLeT layers from h = 256 +to h = 64. This shrinks the overall model (ex- +cluding the LLM) from 11.9 million parameters to +less than 2 million parameters during pre-training +and improves the overall accuracy by a small mar- +gin (+1.51%). We do not run any other hyper- +parameter search throughout our experiments and +wherever possible use the same hyper-parameters +as PIGLeT. We also reduce the batch size from +1024 to 256 because we use a mix of NVIDIA +GTX 1080 and NVIDIA A100 GPUs and wish to +keep batch size constant. +The +images models use the extracted represen- +tations from a frozen off-the-shelf DETR model +(41.3 million parameters), however it is ran only +once over all images as we cache its predictions. +We do not use the “NO OBJECT” predictions from +DETR, and simply pass all 100 bounding boxes +representations to the attention mechanism. Since +we do not have access to the true bounding boxes +in PIGPeN, we do not fine-tune DETR and there- +fore ignore its prediction heads which have also +been trained on COCO and mismatch our possible +objects. +The +symbolic models use the Symbolic Object +Encoder which is an additional 800, 000 parame- +ters on its own. During fine-tuning all models use a +RoBERTA-base model (+120 million parameters) +in the Action Encoder. The +text-label model +also uses the RoBERTA-base model during pre- +training, but again this is frozen and its outputs are +cached for the full dataset. +We pre-train each model for 80 epochs and fine- +tune for 60 epochs. For all setups, pre-training +takes between 1 to 2 hours and fine-tuning takes +less than 1 hour on an NVIDIA A100 GPU. We use +the Pytorch implementation of the Adam optimizer +(Kingma and Ba, 2014) and a learning rate of 10−3 +during pre-training and 10−5 during fine-tuning. +We use early stopping on the validation loss with +a patience of 10 epochs. We run each setup over +10 different seeds (s ∈ [1, 2, ..., 10] and report the +average and standard deviation for each metric. +D +Accuracy Results +D.1 +Comparing PIGPeN and PIGPeN-Vis +Table 4 compares the overall accuracy on the orig- +inal PIGPeN dataset with our proposed PIGPeN- +Vis split. We find that our PIGPeN-Vis split is +consistently harder to solve than the original PIG- +PeN dataset. We explain the increased accuracy in + +Overall Accuracy (% ± σ) +PIGPeN +PIGPeN-Vis +∆ +base +29.18 ± 0.34 +21.23 ± 0.72 +−7.95% +base+symbolic (PIGLeT) +86.39 ± 0.79 +85.03 ± 0.45 +−1.36% +base+symbolic+images +87.45 ± 0.66 +86.01 ± 0.89 +−1.44% +base+images (PIGLet-Vis) +49.13 ± 1.53 +45.47 ± 1.50 +−3.66% +base+images+text-labels +51.28 ± 1.68 +47.55 ± 2.10 +−3.73% +Table 4: Overall Accuracies comparing full PIGPeN +with the PIGPeN-Vis split across 10 seeds. +Overall Accuracy (% ± σ) +validation +test +base +23.85 ± 0.95 +21.23 ± 0.72 +base+symbolic (PIGLeT) +88.08 ± 0.50 +85.03 ± 0.45 +base+symbolic+images +89.49 ± 0.82 +86.01 ± 0.89 +base+images +50.73 ± 2.97 +45.47 ± 1.50 +base+images+text-labels +53.33 ± 3.15 +47.55 ± 2.10 +Table 5: Validation and test overall accuracies. Note +the zero-shot accuracy is not calculated on the valida- +tion set since there are no unseen examples in the vali- +dation set to prevent leakage. +the original dataset with the fact that some of the +filtered out actions (see Appendix B) are easy to +solve from knowing the object name and action: +e.g., most of the images we exclude due to little +salient changes are appliances like stoves being +turned on or off. However, it is easy for a model +to predict the post-condition attributes of a stove, +which are mostly static, across all examples given +an action such as ToggleOn, which always has the +same effect. +D.2 +Complete Accuracy Results on +PIGPeN-Vis +Table 5 shows the overall accuracies for both the +test and validation sets. The full accuracy results +for all actions in Table 6 and for all attributes in +Table 7. +E +Additional Attention Maps +We +plot +additional +attention +visual- +izations +for +all +three +image +models +base+images, +base+symbolic+images, +and +base+images+text-labels in Figures 8, Fig- +ures 9, and Figures 11. Since the DETR object +detector remains frozen, all models have access +to the same bounding boxes and bounding +box representations. +Qualitatively, we find +that the attention weights of base+images and +base+images+text-labels both learn to map +to globally relevant bounding boxes given an +objects. +We also find the attention maps in +base+images+text-labels to be less confident +overall than base+images, likely due to the noise +introduced by the semantic text inputs. As a result, +base+images+text-labels makes less mistakes +by not focusing too much attention to the wrong +bounding box. +On the other hand, base+symbolic+images +focuses on seemingly random bounding boxes. +Since base+symbolic+images already receives +the full representation of each objects, it does +not learn to complement the object’s represen- +tation with accurate visual information. +While +base+symbolic+images extracts 1% of additional +overall accuracy from image inputs when compared +to base+symbolic, it does so by falling back to +vision for visually salient actions such as Pickup. +base+symbolic+images focuses only a narrow set +bounding boxes with overconfidence with no re- +gard for whether or not the bounding box relates +to the object. We posit that the model might use +vision to better estimate more difficult attributes to +predict such as distance in some contexts. Note +Pickup is a salient action because when the agent +in the environment picks an object up, the object is +placed directly in the middle of its field of vision +(as if the agent were holding the object in front of +it). + +(a) base+images +(b) base+symbolic+images +(c) base+images+text-labels +Figure 8: Attention maps for the effects of the EmptyLiquid action on Bowl with objects Fridge and Bowl. The top +row of each grid maps to the before environment and the bottom row maps to the after environment. The columns +map to each respective object. The Fridge object appears in the lower left of the image, and is only correctly +identified by base+images+text-labels, even though the model does place more weight to the bounding box of +the stove (lower right). +(a) base+images +(b) base+symbolic+images +(c) base+images+text-labels +Figure 9: Attention maps for the effects of the Slice action on Apple with objects CounterTop and Apple. The +top row of each grid maps to the before environment and the bottom row maps to the after environment. The +columns map to each respective object. +(a) base+images +(b) base+symbolic+images +(c) base+images+text-labels +Figure 10: Attention maps for the effects of the Dirty action on Bowl with objects Bowl and None. The top row of +each grid maps to the before environment and the bottom row maps to the after environment. The columns map to +each respective object. None can be an object in PIGPeN, but we do not predict its attributes and exclude it in all +model predictions. +(a) base+images +(b) base+symbolic+images +(c) base+images+text-labels +Figure 11: Attention maps for the effects of the Open action on Toilet with objects Toilet and ToiletPaper. +The top row of each grid maps to the before environment and the bottom row maps to the after environment. The +columns map to each respective object. This particular example is an unseen combination of action and object that +has been excluded from the training and validation set. + +CounterTop +Apple +Before +AfterCounterTop +Apple +Before +AfterCounterTop +Apple +BeforeBowl +None +Before +aBowl +None +Before +AfterBowl +None +BeforeToilet +ToiletPaper +BeforeToilet +ToiletPaper +Before +AfterToilet +ToiletPaper +BeforeFridge +Bowl +Before +AfterFridge +Bowl +Before +AfterFridge +Bowl +Before +AfterAction Accuracy (% ± σ) +Close +Dirty +EmptyLiquid +HeatUpPan +Open +base +13.20 ± 1.06 +17.71 ± 1.20 +24.75 ± 5.75 +36.33 ± 4.14 +8.33 ± 1.84 +base+symbolic +85.98 ± 1.77 +94.00 ± 3.42 +99.34 ± 1.15 +100.00 ± 0.00 +85.73 ± 0.99 +base+symbolic+images +86.80 ± 3.29 +90.29 ± 5.90 +99.02 ± 2.07 +99.17 ± 1.62 +88.75 ± 3.02 +base+images +27.42 ± 3.71 +58.57 ± 2.78 +69.34 ± 4.17 +68.67 ± 3.75 +20.83 ± 4.63 +base+images+text-labels +28.87 ± 3.19 +57.71 ± 3.24 +70.16 ± 3.17 +74.00 ± 3.16 +22.92 ± 5.79 +Pickup +Put +Slice +ToggleOff +ToggleOn +base +10.96 ± 1.92 +27.95 ± 1.19 +22.13 ± 0.86 +30.83 ± 3.39 +27.38 ± 2.57 +base+symbolic +80.48 ± 2.88 +58.39 ± 1.94 +75.41 ± 1.89 +99.40 ± 0.84 +96.90 ± 1.61 +base+symbolic+images +86.14 ± 2.56 +57.59 ± 2.31 +81.31 ± 3.96 +99.05 ± 0.75 +92.86 ± 5.14 +base+images +33.49 ± 3.45 +34.91 ± 2.43 +41.64 ± 3.80 +71.43 ± 2.75 +70.24 ± 16.00 +base+images+text-labels +40.12 ± 2.61 +38.30 ± 3.11 +45.57 ± 3.85 +69.05 ± 5.81 +67.14 ± 16.53 +Table 6: Full accuracy results table including the standard deviation over 10 seeds for all actions and setups. +Attribute Accuracy (% ± σ) +Name +Temperature +attribute +breakable +cookable +dirtyable +distance +isBroken +isCooked +isDirty +base +99.66 ± 0.07 +95.91 ± 0.41 +96.12 ± 0.07 +91.46 ± 0.36 +99.95 ± 0.07 +99.95 ± 0.10 +51.01 ± 0.93 +99.86 ± 0.00 +98.60 ± 0.06 +97.93 ± 0.19 +base+symbolic +99.64 ± 0.12 +99.85 ± 0.04 +99.48 ± 0.03 +99.84 ± 0.09 +100.00 ± 0.00 +100.00 ± 0.00 +95.13 ± 0.35 +100.00 ± 0.00 +99.85 ± 0.04 +99.71 ± 0.14 +base+symbolic+images +99.62 ± 0.09 +99.59 ± 0.27 +99.48 ± 0.04 +99.78 ± 0.10 +100.00 ± 0.00 +99.97 ± 0.09 +96.13 ± 0.40 +100.00 ± 0.00 +99.85 ± 0.04 +99.52 ± 0.32 +base+images +97.34 ± 0.65 +96.28 ± 0.74 +97.25 ± 0.13 +92.63 ± 0.75 +99.91 ± 0.10 +99.62 ± 0.20 +76.90 ± 1.05 +99.85 ± 0.05 +98.68 ± 0.19 +97.87 ± 0.34 +base+images+text-labels +98.44 ± 0.35 +96.05 ± 1.23 +97.46 ± 0.13 +93.19 ± 0.31 +99.96 ± 0.09 +99.93 ± 0.10 +78.56 ± 1.16 +99.84 ± 0.09 +98.19 ± 0.84 +97.78 ± 0.24 +isFilledWithLiquid +isOpen +isPickedUp +isSliced +isToggled +mass +moveable +openable +parentReceptacles +pickupable +base +96.79 ± 0.50 +98.84 ± 0.23 +94.83 ± 0.82 +97.99 ± 0.09 +98.36 ± 0.23 +96.51 ± 0.15 +99.90 ± 0.09 +99.97 ± 0.06 +87.44 ± 0.42 +99.84 ± 0.09 +base+symbolic +99.93 ± 0.12 +98.95 ± 0.09 +99.27 ± 0.31 +100.00 ± 0.00 +99.88 ± 0.12 +99.33 ± 0.14 +99.99 ± 0.04 +99.97 ± 0.06 +97.78 ± 0.47 +99.90 ± 0.11 +base+symbolic+images +99.84 ± 0.19 +98.67 ± 0.38 +98.96 ± 0.31 +99.97 ± 0.06 +99.74 ± 0.30 +99.59 ± 0.09 +100.00 ± 0.00 +99.99 ± 0.04 +97.26 ± 0.44 +99.88 ± 0.10 +base+images +96.88 ± 0.55 +98.81 ± 0.97 +97.43 ± 0.37 +98.28 ± 0.30 +97.92 ± 0.83 +96.41 ± 0.41 +99.79 ± 0.21 +99.74 ± 0.20 +91.05 ± 0.77 +99.59 ± 0.17 +base+images+text-labels +97.25 ± 0.45 +98.11 ± 1.14 +97.54 ± 0.53 +98.34 ± 0.29 +98.06 ± 0.55 +96.74 ± 0.24 +99.89 ± 0.09 +99.95 ± 0.10 +92.49 ± 0.69 +99.70 ± 0.09 +receptacleIds +receptacle +Ceramic +Fabric +Food +Glass +Leather +Metal +Paper +Plastic +base +84.20 ± 0.61 +99.85 ± 0.10 +98.26 ± 0.17 +99.55 ± 0.07 +99.99 ± 0.04 +98.91 ± 0.13 +99.89 ± 0.06 +98.69 ± 0.15 +99.73 ± 0.00 +98.30 ± 0.10 +base+symbolic +96.36 ± 0.18 +99.90 ± 0.09 +100.00 ± 0.00 +99.96 ± 0.07 +100.00 ± 0.00 +99.99 ± 0.04 +100.00 ± 0.00 +99.99 ± 0.04 +100.00 ± 0.00 +99.97 ± 0.06 +base+symbolic+images +96.13 ± 0.30 +99.92 ± 0.10 +99.99 ± 0.04 +99.85 ± 0.10 +99.99 ± 0.04 +99.97 ± 0.06 +100.00 ± 0.00 +100.00 ± 0.00 +100.00 ± 0.00 +99.96 ± 0.07 +base+images +82.87 ± 0.55 +99.47 ± 0.21 +99.03 ± 0.22 +99.50 ± 0.19 +99.92 ± 0.10 +99.16 ± 0.21 +99.97 ± 0.06 +98.31 ± 0.37 +99.67 ± 0.21 +98.83 ± 0.31 +base+images+text-labels +83.91 ± 0.56 +99.69 ± 0.11 +99.36 ± 0.19 +99.44 ± 0.12 +99.96 ± 0.09 +99.37 ± 0.24 +99.95 ± 0.10 +98.69 ± 0.30 +99.56 ± 0.19 +99.08 ± 0.20 +Rubber +Soap +Sponge +Stone +Wax +Wood +size +sliceable +toggleable +base +100.00 ± 0.00 +99.99 ± 0.04 +100.00 ± 0.00 +99.34 ± 0.09 +100.00 ± 0.00 +99.51 ± 0.16 +73.78 ± 0.29 +98.02 ± 0.12 +99.95 ± 0.07 +base+symbolic +100.00 ± 0.00 +100.00 ± 0.00 +100.00 ± 0.00 +99.99 ± 0.04 +100.00 ± 0.00 +99.99 ± 0.04 +94.98 ± 0.19 +100.00 ± 0.00 +99.99 ± 0.04 +base+symbolic+images +99.97 ± 0.06 +99.99 ± 0.04 +100.00 ± 0.00 +99.99 ± 0.04 +100.00 ± 0.00 +100.00 ± 0.00 +96.35 ± 0.20 +99.99 ± 0.04 +99.96 ± 0.09 +base+images +99.88 ± 0.08 +99.89 ± 0.11 +99.92 ± 0.10 +99.48 ± 0.14 +99.92 ± 0.10 +99.25 ± 0.22 +87.03 ± 1.15 +98.32 ± 0.32 +99.81 ± 0.17 +base+images+text-labels +99.85 ± 0.08 +99.92 ± 0.10 +99.88 ± 0.14 +99.60 ± 0.19 +99.95 ± 0.07 +99.37 ± 0.22 +87.89 ± 1.11 +98.32 ± 0.36 +99.95 ± 0.07 +Table 7: Full accuracy results table including the standard deviation over 10 seeds for all attributes and setups. + diff --git a/SNFKT4oBgHgl3EQfjS7T/content/tmp_files/load_file.txt b/SNFKT4oBgHgl3EQfjS7T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7c4960061a47dafc570f3a79743bf740463c594 --- /dev/null +++ b/SNFKT4oBgHgl3EQfjS7T/content/tmp_files/load_file.txt @@ -0,0 +1,1466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf,len=1465 +page_content='Learning the Effects of Physical Actions in a Multi-modal Environment Gautier Dagan, Frank Keller, Alex Lascarides School of Informatics University of Edinburgh, UK gautier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='dagan@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='uk, {keller, alex}@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='uk Abstract Large Language Models (LLMs) handle phys- ical commonsense information inadequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' As a result of being trained in a disembod- ied setting, LLMs often fail to predict an ac- tion’s outcome in a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' How- ever, predicting the effects of an action before it is executed is crucial in planning, where co- herent sequences of actions are often needed to achieve a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Therefore, we introduce the multi-modal task of predicting the outcomes of actions solely from realistic sensory inputs (images and text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Next, we extend an LLM to model latent representations of objects to bet- ter predict action outcomes in an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We show that multi-modal models can capture physical commonsense when augmented with visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Finally, we evaluate our model’s performance on novel actions and ob- jects and find that combining modalities help models to generalize and learn physical com- monsense reasoning better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 1 Introduction Large Language Models (LLMs) are trained on large corpora of disembodied texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' They are typi- cally pre-trained on a masked language modeling task: the model must predict a masked word in a text given its context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' LLMs have achieved state- of-the-art performance on many NLP tasks (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020), but they can also fail on seemingly easy and obvious tasks and in un- predictable ways (McCoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Bommasani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Commonsense knowledge is shared knowledge and is often so obvious that it is absent from the LLMs’ training data: people don’t men- tion what is already known to their interlocutors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This includes physical commonsense information, including how executed actions affect the physical attributes of objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', shape and weight (Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Humans may learn such knowledge from their embodied environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' But LLMs, be- ing trained on disembodied text, can make incorrect predictions about physical attributes and how these change when actions occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For instance, when asked what the weight of a 150 grams potato after it is sliced, GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020) incorrectly answers 75 grams (see Appendix A for the exact prompt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' GPT-3 is an LLM with 175 billion param- eters, and nonetheless its disembodied existence limits its physical commonsense estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) inject physical common- sense information into LLMs via their model PIGLeT—a modified LLM that is trained on their PIGPeN simulated 3D environment dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' PIGLeT estimates how an environment changes as a result of specific actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In training and testing, the model uses ground-truth symbolic representa- tions of the environment but not the images: it ignores visual sensory observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' These sym- bolic representations of objects in an environment are chosen to capture the possible effects of ac- tions, and include attributes like weight, size and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, in an embodied situation, an agent needs to use visual perception to estimate its interpretation of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Therefore, the sym- bolic representations should be treated as latent rather than observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We propose an alternative to the PIGLeT model, PIGLeT-Vis, which uses images directly as input into a multi-modal LLM to ground the model to its physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We compare our approach to the original PIGLeT model and evaluate the gen- eralization capabilities gained from using image inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' At test time, our model foregoes symbolic labels: only the images and the name of the action are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Thus our model tackles a more chal- lenging task than the original PIGLeT model in that it must not only predict the effect of actions but also (indirectly) estimate the symbolic representations of objects in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We also evaluate a model for predicting the effects of actions that trains on PIGPeN’s images and their associated natural lan- guage (NL) descriptions, eliminating the need for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='11845v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='CL] 27 Jan 2023 Action Apply Object Decoder name: cup size: small filled: yes observed latent "The robot empties the cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='" 2 1 Action Encoder pre-training fine-tuning Object Encoder name: cup size: small filled: no Figure 1: Original PIGLeT Physical Dynamics Model (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' During pre-training the model receives as input the full symbolic representation of two objects (o0 pre and o1 pre) before the action is taken and the symbolic representation of the action itself (a) and is tasked with predicting the attributes of the objects after the action (o0 post and o1 post).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' During fine-tuning, the action encoder is replaced by an LLM to process a natural language description of the action being taken and with what objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' formal symbolic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Our contributions are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' First, we show that it is possible to predict the physical effects of actions from visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Second, we show that it is possible to learn the task on training data where formal symbolic representations, which are unob- servable in real-world settings, are replaced with NL descriptions (which can be observed through natural interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Third, we evaluate all our mod- els in a stricter zero-shot setup to promote ways to train agents that generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Overall our work paves the way for multi-modal models that learn the effects of actions in realistic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2 Related Work Commonsense reasoning has been highlighted as a potential weak point of LLMs in recent years (Shen and Kejriwal, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Bisk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Datasets such as PIGPeN (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021), commonsenseQA (Talmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019), VCR (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) and GD-VCR (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021) help evaluate different aspects of common- sense reasoning in modern LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In this paper, we focus on physical commonsense reasoning, which involves understanding the (often) unexpressed rules of the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2019) reported that neural repre- sentations found it challenging to infer the link between actions and what they imply about the attributes of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Accordingly, Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2019) introduced the Visual Commonsense Rea- soning (VCR) task to test how images can inform question answering models that tackle common- sense information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Bisk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2020) designed the PIQA benchmark to evaluate physical common- sense reasoning in LLMs through question answer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Sampat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) proposed an extension to the CLEVR dataset, where an agent must reason and answer questions about a scene after a hypo- thetical action is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Multiple approaches can improve the capabil- ities of LLMs in commonsense reasoning, such as using handcrafted knowledge graphs (Hwang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021) or leveraging simulated environments (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' PIGLeT, in particular, com- bines a traditional LLM and a “Physical Dynamics” model to ground an LLM (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The Physical Dynamics model enhances the common- sense knowledge of an LLM by fine-tuning it, using trajectories sampled from a realistic environment (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Trajectories are an action and a pair of environment states (before and after the ac- tion) expressed in a formal symbolic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) found that fine-tuning LLMs with symbolic data from the simulated environment helped them outperform other models in physical commonsense reasoning tasks: in particular, pre- dicting the effects of an action when executed in a particular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Image inputs offer a way to ground an LLM, as they only require general alignment with a text or symbolic input and do not require the comprehen- sive environment ground-truth labels that PIGLeT uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2018) used multi-modal web data to learn actions and their effects from images Action Apply Object Decoder Action Encoder "The robot empties the cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='" 1 2 observed latent 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' name: cup size: small filled: no Figure 2: PIGLeT-Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We introduce PIGLeT-Vis, where we modify the PIGLeT architecture to replace its Sym- bolic Object Encoder with a vision component that makes use of images of the environment before and after an action is taken to predict the symbolic representation of objects post-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use an attention mechanism over the extracted bounding boxes to obtain a visual hidden representation of an object given its name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The only remaining symbolic inputs during pre-training are the action description and object names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' and corresponding text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2019) used an off-the-shelf ResNet50 model (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2016) to augment an existing BERT language model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) with vision capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Transformer models such as UNITER (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020), ERNIE-ViL (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021), VisualBERT (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020), and ViLBert (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) have been applied to visual commonsense reason- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' These models use a joint transformer backbone for images and text and vary their pre-training ob- jectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, most of these models are trained on static text-image pairs: they aren’t designed to capture the dynamics of an environment, partic- ularly how object attributes change with actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Notably, recent work by Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2022) uses CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021) and MOCA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021) embeddings to predict a post-action image given a set of possible images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In contrast, we focus on adapting an LLM with a vision-based component to predict the consequences of actions on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3 Method We propose PIGLeT-Vis (Figure 2) for learning the effects of actions on objects from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use a pre-trained vision backbone, DETR (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020), as a Vision Object Encoder and combine it with a RoBERTa LLM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) as an Action Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We experiment with different con- figurations of inputs to measure the impact of the various components of our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In partic- ular, we test a variation in which we remove the formal symbolic labels even in training, replacing them with NL text labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' To evaluate our models, we use the PIGPeN dataset (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021), which consists of a symbolic and visual representa- tion of an environment before and after an action is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, we filter PIGPeN to create a viable testing ground for visual grounding of physical ac- tions and more accurately measure generalization capabilities of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Architecture PIGLeT-Vis (shown in Figure 2) consists of sepa- rate components, which can combine multi-modal inputs in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Through this modular approach, we can turn off specific components to evaluate how different inputs and model structures affect performance on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We test models with and without symbolic inputs and image inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For all components, we use a dropout of p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 in between layers and a default hidden layer size of h = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Object Encoder We reproduce Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021), where all ac- tions are assumed to involve two objects, o0 and o1, and the symbolic representation of objects are encoded in an Object Encoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The sym- bolic representation of an object before the action is represented by opre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Both objects (o0 pre and o1 pre) in the environment are described by a vector of 38 attributes, chosen on the basis that they are the kinds of physical attributes that are influenced by actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' They describe an object as small/large, cold/hot, empty/full, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We first embed these symbolic object attributes using an embedding layer Ee×h, where e = 329 is the total number of unique attributes and h is our hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For an object k: ˆok pre = E(ok pre) (1) The Object Encoder Oencoder takes in the embed- ded object attributes through a set of multi-head attention layers to encode the symbolic representa- tion of each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use the default Pytorch im- plementation of the Transformer Encoder (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) with three layers and 4 heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The first encoded output of each object sequence is used for representing the entire object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' hk pre = Oencoder(ˆok pre) (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 Action Encoder Actions are encoded either as a symbolic triplet ⟨action, action object, action receptacle⟩ or as an annotated text describing an action being taken (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', “robot empties the cup”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' During pre-training, the Action Encoder Apretrain uses an action embedding layer E′ to embed the first dimension of the action, and re- uses the object embedding layer E to embed the action object name ao and action receptacle name ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The action embedding layer E′ has dimension- ality 10 × h for the 10 distinct actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The three embedded representations are summed and passed to the Action Encoder’s linear layers to produce ha (see equation 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Similarly to Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021), a tanh activation is applied after each linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' ha = Apretrain(E′(a) + E(ao) + E(ar))) (3) When fine-tuning on the annotated dataset, the action input is text and therefore we switch out the Action Encoder Apretrain for Afinetune—our text- based Action Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Afinetune uses a RoBERTa- base1 model (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) to process a tok- enized version of the text input at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The first token ([CLS]) of the RoBERTa output layer is used to rep- resent the action sequence and then passed through a linear layer to map the dimensionality of the hid- den states from 256 to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' ha = Afinetune(at) (4) 1Implementation and pre-trained model weights are taken from the Huggingface library (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='3 Vision Object Encoder The Vision Object Encoder takes in images (ipre and ipost) to provide a visual representation of each object k before and after (hk pre and hk post).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use the DETR1 (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020) model as a backbone to predict N bounding boxes in a pair of images (pre- and post-action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' As DETR is pre- trained on the COCO object detection dataset (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2014), its predicted object labels do not align with those in PIGPeN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Therefore, we instead learn a mapping between the predicted bounding box representations and the PIGPeN objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For each image, we obtain a hidden representation hb of dimensionality N × 256 where N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use an attention mechanism over the bound- ing boxes’ hidden representation, conditioned on the object names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For a given object ok, its condi- tional representation hk c is the encoded name of the object: E(ok name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We can therefore obtain the at- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='tention score of a given object ok and image im by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='calculating the alignment between the conditional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='representation hk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='c and the hidden representations of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='bounding boxes hbm: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='hbm = DETR(im) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='αk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='m = Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='(hk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='chbm)i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='We obtain the final representation for a given ob- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='ject and image by multiplying our attention scores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='α with the extracted output representation from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='DETR and summing along the bounding box axis: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='hk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='om = W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='(αk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='mhbm)j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='We use a final output layer W to decrease the di- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='mensionality of ho from the DETR dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='of 256 to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Through the Vision Object Encoder, we replace the previously symbolic inputs with images and can extract [h0 preh1 pre] and [h0 posth1 post] from ipre and ipost respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note that we make the implicit assumption that ipre and ipost contain the informa- tion necessary to predict object attributes of the objects post-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4 Action Apply The Action Apply Model β is a simple fuse op- eration (concatenation in the hidden dimension) followed by three linear layers, which combine the action representation ha and an object repre- sentation of the scene pre-action hk pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The model outputs an object’s representation hk a, containing information conditioned all inputs: hk a = β(ha, hk pre) (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 Object Decoder Finally, the Object Decoder is a transformer mod- ule that maps the object representations ho from the pre-action state back to 38 symbolic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' It uses a default three layer Transformer Decoder (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) that takes the hidden repre- sentation from the Action Apply hk a as an encoded memory state and hk pre as the source sequence to predicts a label for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' ˙ok post = Odecoder(hk a, hk pre) (9) When we use image inputs, we also have access to the post-action visual representation and can therefore use hk pre + hk post instead of hk pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The output has post-action object states ˙ok post which are compared to the ground truth ok post to calculate cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' As an additional loss, we also use the cross-entropy between ˙ok pre and ok pre by passing an empty hk a to force the Object Decoder to recreate the attributes in the pre-action state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We weight both losses equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='3 Evaluation Metrics Since our task involves predicting 38 attributes for two different objects per example, we follow Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) and report different types of accuracy metrics on the test set (after fine-tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We measure the overall accuracy by scoring how many objects have all attributes correctly predicted (exact match).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note that this is a high bar for a model where the symbolic representations are la- tent: to predict an object correctly, our model must first estimate its attributes before the action and then estimate whether and how these change given an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' So we also measure the attribute-level and action-level accuracies of each model, so as to explore which attributes and actions are more difficult to predict than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4 PIGPeN-Vis Dataset Split To evaluate physical commonsense reasoning using PIGLeT-Vis, we filter PIGPeN (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021) to create a subset (PIGPeN-Vis) which we use for all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We motivate PIGPeN-Vis as a way to isolate the effects of adding our vision component, because while PIGPeN already has images, these images were not used in PIGLeT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The PIGPeN dataset consists of trajectories of an environment before (pre) and after (post) an action is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Each trajectory contains repre- sentations of two distinct objects before and af- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' One of the objects is usually targeted by the action, while the other acts as a distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In ad- dition, image pairs (ipre, ipost) for each trajectory are provided, where each image is snapshot of the simulated photo-realistic 3D environment which contains the objects in view (see Appendix B for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Each image is an RGB image of dimensions 640 × 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The original dataset is separated into two distinct sets: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' A pre-training set of 278, 009 trajectories, which includes the symbolic representations of objects o before and after a symbolic action a is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' A separate validation set of 33, 042 examples is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' A fine-tuning set of 1, 000 trajectories which has been annotated to replace the symbolic ac- tion a with a textual representation at describ- ing the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Separate validation and test sets of 500 examples each are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' All metrics are reported on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In PIGPeN, the object states opre and opost con- tained 40 different attributes and 13 different ac- tions a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Attributes range from intrinsic such as name or moveable to stateful such as distance or isCooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In forming PIGPeN-Vis, we remove two attributes and three actions from the dataset to obtain 38 attributes and 10 possible actions (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Viewpoint and Action Filtering Since the PIGPeN images were not generated with the goal of being used as input data, we identified several issues with the quality of certain scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' A notable difficulty is that in some cases, the be- fore and after images are not captured from the same camera angle or they have different light- ing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Changing orientations and lighting conditions makes it difficult to use an image pair (ipre, ipost) to isolate the outcome of an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Con- versely, image pairs with too few perceivable differ- ences also break our assumption that the changes in the environment are perceivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Therefore, we fil- ter the dataset using pixel statistics to remove image pairs that have either large perceivable differences (likely due to changes in viewpoint) or small per- ceivable differences (where the action’s results are not visually salient enough) (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We exclude 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4% of the total dataset through visual filtering of the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 Zero-Shot Filtering To evaluate the generalization capabilities gained from a vision component, we further filter the dataset to exclude a subset of training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Unlike the original PIGPeN dataset which only tested for zero-shot generalization at the level of the fine-tuning data, we remove all instances with selected specific objects or action-object pairs from all training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' To minimize the effect of removing examples from the dataset, we pick objects and action-object pairs with an already low number of samples in the training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In total, we exclude 14 objects and 27 action-object pairs, which amounts to less than 3% (6, 816 samples) of the remaining training sets (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' These zero-shot examples comprise around 10% of the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' After both filtering stages, PIGPeN-Vis contains a pre-training dataset of 232, 625 trajectories with a validation set of 26, 823, and a fine-tuning training set of 750 examples with a validation set of 367 examples and a test set of 398 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='5 Training Configurations We evaluate the impact of the vision component on PIGPeN-Vis through five different setups: base: We implement a baseline model with- out symbolic object inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Our implemen- tation removes the Object Encoder entirely, such that the model must predict the attributes of objects solely from knowing the action and the object names that it relates to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This model acts as a lower bound on the capabilities of the vision model: its performance would match the vision model if images are irrelevant to solving the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+symbolic: This is our implementation of the original Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) PIGLeT model, shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This model acts as an upper bound on the capabilities of the vi- sion model since it observes the true symbolic representations of objects before the action (which the vision model must estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+images: This is our proposed PIGLeT- Vis, shown in Figure 2, where the Vision Ob- ject Encoder replaces the previously symbolic Object Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This model leverages the before and after images of the environment as well as the name of the objects to extract representations of the object attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+symbolic+images: We sum the hid- den symbolic representations of objects with their visual representations in a unified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Through this setup, we evaluate whether im- ages can provide additional information to the already comprehensive symbolic representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+images+text-labels: We convert the symbolic representations of the labels for the object names and actions to their text label and encode them using a frozen LLM during pre- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use the same LLM to encode the text labels that we later use in the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This setup replaces all symbolic inputs from the pre-training stage to only language and image inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note that there are a few differences between the original Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) model and our im- plementation of base+symbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For instance, for simplicity, we opted to use an off-the-shelf RoBERTa-base (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) model instead of training our own custom GPT2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Additionally, we also reduce the dimen- sionality of the PIGLeT layers from h = 256 to h = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We found that not only does this allow faster training times as it shrinks the Physical Dy- namics model from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='9 million parameters to 2 million parameters, it also improves the overall accuracy by a small margin (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='51%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We train each model for 80 epochs with a batch size of 256 using the Pytorch implementation of the Adam optimizer (Kingma and Ba, 2014) and a learning rate of 10−3 during pre-training and 10−5 during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We run each setup over 10 different seeds and report the average and standard deviation for each metric (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Accuracy (% ± σ) Overall Zero-Shot base 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='34 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='77 base+symbolic (PIGLeT) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='45 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='04 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='37 base+symbolic+images 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='47 base+images (PIGLeT-Vis) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='53 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='60 base+images+text-labels 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='90 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='24 Table 1: Overall and zero-shot accuracies (PIGPeN- Vis) 4 Results and Discussion We evaluate all models on our PIGPeN-Vis split and report the overall (exact match), zero-shot, action-level, and attribute-level accuracy results for all setups in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For completeness, we also evaluate models on the original PIGPeN to contrast the effects of our filtering operations (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4 and Appendix D) and find PIGPeN-Vis is a more challenging subset for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The base model provides a low bar estimate of what is achievable using only the action encoder inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Unsurprisingly, the base model performs worst on overall accuracy, which demands an ex- act match of all attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' It does relatively well on (individual) attribute-level accuracy, primarily because it predicts the most common attribute for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Some actions are also easier than others—for instance, the model reaches 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='38% ac- curacy on ToggleOn from only knowing the action and object names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This is likely because ToggleOn is constrained to a small set of objects and effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Our base+symbolic model obtains similar re- sults to the original implementation by Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021), with an overall accuracy of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' How- ever, it performs much worse on the zero-shot split (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='04%) than the original PIGLeT model reported (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2%) (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This disparity can be explained by the fact that the original zero-shot PIGPeN dataset was not a true zero-shot dataset, be- cause the Physical Dynamics model was exposed to the “unseen” objects in its pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The base+symbolic model provides a high bar esti- mate of what could be achievable if: (i) ipre and ipost capture the symbolic environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' and (ii) the Vision Object Encoder can subsequently extract these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, as we will argue in Sec- tion 6, both (i) and (ii) are unrealistic given the constraints of both the dataset and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Our base+images (PIGLeT-Vis) model scores 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='28% in overall accuracy but only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='53% on the zero-shot set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Nevertheless, it outperforms the base model in overall accuracy (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='0001) and in zero-shot accuracy (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='08), which demon- strates that the images improve the prediction of the effects of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The base+images model also performs significantly better than base on dif- ficult attribute-level accuracies such as distance (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, as before, accuracy on individual attributes benefits from the skewed dis- tributions of their values and does not necessar- ily translate to high scores on predicting all 38 attributes correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Utilizing both images and symbolic representa- tions as inputs helps the base+symbolic+images model outperform purely symbolic inputs in over- all accuracy, from 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03% to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, image inputs also decrease the model’s zero-shot performance from 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='04% to 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89%, al- though this isn’t statistically significant (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='05) due to high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We suspect that this high variance is caused by an increase in noise in the model resulting from adding images to the sym- bolic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, the overall picture is more complicated, as images can also provide gains on certain actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', PickUp accuracy increases from 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='48% to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14%) even though it causes a decrease in many other cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', ToggleOn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Finally, when we utilize NL descriptions to re- place the formal symbolic inputs (action name and object names), base+images+text-labels improves overall accuracy when compared to base+images from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='47% to 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='55% (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Text inputs appear to improve zero-shot accuracy, but not by a statistically significant margin (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Accuracy also improves in most actions, for instance the Slice accuracy improves from 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='64% to 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='57% (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' So the NL descrip- tions inform the task in a beneficial way, over and above the raw images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' But encoding the labels as text rather than formal symbolic representations also adds noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Nevertheless, text labels improve accuracy on actions where the semantic information contained in the label provides a richer context to help gen- eralize to similar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For instance, a “cup” and a “mug” are semantically close, and thus learn- ing the effects of actions on a “cup” might help the model predict the same effects on a “mug” even if the word forms are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In con- trast, the formal symbolic representations treat the predicate symbols cup and mug as unrelated, and so don’t benefit from the lexical relationships that the LLM captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Fully removing the sym- Action Accuracy (%) Attribute Accuracy (%) Open Pickup ToggleOn Slice size distance temperature base 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='96 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='38 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='13 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='78 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='91 base+symbolic (PIGLeT) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='73 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='48 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='90 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='41 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='98 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='85 base+symbolic+images 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='86 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='31 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='35 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='59 base+images (PIGLeT-Vis) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='83 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='49 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='24 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='64 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='62 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='10 base+images+text-labels 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='92 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='12 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='57 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='06 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='72 Table 2: Action and attribute specific accuracies for a subset of actions and attributes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' for a comprehensive table with standard deviations see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' size and distance each have eight possible classes while temperature has three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Apple Apple Pot CounterTop Before After SliceObject(Apple) on (CounterTop,Apple) PutObject(Apple, Pot) on (Apple,Pot) Figure 3: We visualize the attention of the Vision Object Encoder from a trained base+images model on two different actions and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The left grid focuses on the effect of Slice(Apple) on CounterTop and Apple, while the right grid focuses on the effects of Slice(Apple) on Apple and Pot objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' bolic representations allows us to adapt our model to any possible unseen object during test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+images+text-labels is adaptable to gen- eral settings without knowing the symbolic map- ping of objects and actions in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The results of both base+symbolic+images and base+images+text-labels make the case multi- modal modeling of commonsense reasoning, as both language and images are complementary to generalize to unseen settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Qualitative Attention Maps Visualizing attention is another benefit of a vision component, as we can see what the model focuses on and partially explain its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Figure 3 shows two separate examples and corresponding at- tention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In the left example, base+images is tasked with predicting the attributes of CounterTop and Apple after the Slice action is applied on the Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In the right example, the Put action is ap- plied on the Apple, and the model must predict the attributes of the Apple and the distractor object Pot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The two rows are the before and after images (ipre and ipost), and the two columns are the two objects used to condition the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The atten- tion maps display the strength of the attention for each bounding box given an object name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Both examples in Figure 3 show that the Vision Object Encoder can map known objects to relevant bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The model successfully tracks the Apple in both cases by placing the most weight on the bounding box targeting the Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, these examples also show the difficulty of this task— the environments are realistic and can be filled with more than one instance of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 5 Conclusion In this paper, we tackle the task of predicting the effects of actions on objects’ physical attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In contrast to (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2021), our model does not treat the formal symbolic representation of the images as observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Instead, PIGLeT-Vis supports inference when the inputs are images alone or im- ages plus NL descriptions and a phrase denoting the action (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', “the robot empties the cup”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' While PIGPeN offers challenges for applying a multi- modal approach, our model can extract useful in- formation from images, opening the door for gen- eralizing learning physical commonsense to real- world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Importantly, our PIGPeN-Vis split can be used to evaluate the zero-shot capabilities of different model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Moreover, while base+symbolic still outperforms base+images, it CounterTop Apple Before AfterApple Pot Before Afterdoes so without estimating the attributes of ob- jects and thus solves a much easier but unrealistic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Through base+images+text-labels, we show that, when replacing symbolic inputs, the best solution is to complement image inputs with NL descriptions to leverage information from both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Finally, our results show the need to improve the generalization capabilities of multi- modal models such that they can learn and adapt to unseen situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 6 Limitations There are several limitations to our approach that result directly from the inherent limitations of PIG- PeN and our proposed Vision Object Encoder re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' PIGPeN was not originally designed for test- ing commonsense reasoning using images and con- tains numerous inconsistencies which cannot all be solved with the PIGPeN-Vis split obtained from filtering (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Given the presence of non- physically salient attributes such as temperature, images are not guaranteed to fully capture their symbolic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' PIGPeN includes certain attributes which are not discernible from images, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', even humans would be unable to tell a hot plate from a cold plate from vision alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The im- ages in PIGPeN can also contain more than one object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', more than one cup) without ever speci- fying which one the symbolic representation refers to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This causes difficulty for our approach because judging specific attributes such as distance is im- possible if there are two cups at different distances from the viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Additionally, PIGPeN also discretizes continuous variables such as distance into categories which can be hard to disambiguate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' To approach the accuracy of base+symbolic with our vision component, we also need a vision representation from which to correctly estimate all latent attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Even if images are assumed to be perfect representations of the symbolic en- vironment, the model still has to extract each of the 38 attributes correctly for both objects using only two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' It is possible (and likely) for the vision detection backbone to miss the target object entirely because it is not trained to detect the specific object in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We see this effect in Figure 3, where the model falls back to using a bounding box around the sink area to describe the CounterTop object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The DETR vision model used to extract bounding boxes was pre-trained on the COCO dataset (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2014) which does not contain CounterTop as an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' PIGLeT-Vis is therefore ultimately limited by the capabilities of its vision backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Ethics Statement While this work does not introduce new data or involve human participants, we use the PIGPeN dataset which contains human-labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The fine-tuning portion of the dataset was annotated through MTurk by Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) and they re- port following best practices (paying decent wages, providing feedback and using a qualification test) in their data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We filter and use a subset of PIGPeN and introduce methods to learn the effects of actions in a multimodal setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We, therefore, believe that our work does not raise any ethical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Acknowledgements This work was supported in part by the UKRI Cen- tre for Doctoral Training in Natural Language Pro- cessing, funded by the UKRI (grant EP/S022481/1) and the University of Edinburgh, School of Infor- matics and School of Philosophy, Psychology & Language Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' References Yonatan Bisk, Rowan Zellers, Ronan Le Bras, Jian- feng Gao, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 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+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Ke Shen and Mayank Kejriwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' On the gener- alization abilities of fine-tuned commonsense lan- guage representation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In Artificial Intelli- gence XXXVIII, page 3–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Kunal Pratap Singh, Suvaansh Bhambri, Byeonghwi Kim, Roozbeh Mottaghi, and Jonghyun Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Factorizing perception and policy for interactive instruction following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1888–1897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Alon Talmor, Jonathan Herzig, Nicholas Lourie, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' CommonsenseQA: A ques- tion answering challenge targeting commonsense knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4149–4158, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pier- ric Cistac, Tim Rault, Rémi Louf, Morgan Funtow- icz, and Jamie Brew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Huggingface’s trans- formers: State-of-the-art natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' CoRR, abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Da Yin, Liunian Harold Li, Ziniu Hu, Nanyun Peng, and Kai-Wei Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Broaden the Vision: Geo-Diverse Visual Commonsense Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Fei Yu, Jiji Tang, Weichong Yin, Yu Sun, Hao Tian, Hua Wu, and Haifeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Ernie-vil: Knowledge enhanced vision-language representa- tions through scene graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Rowan Zellers, Yonatan Bisk, Ali Farhadi, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' From recognition to cognition: Visual commonsense reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In 2019 IEEE/CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), page 6713–6724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Rowan Zellers, Ari Holtzman, Matthew Peters, Roozbeh Mottaghi, Aniruddha Kembhavi, Ali Farhadi, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Piglet: Language grounding through neuro-symbolic interaction in a 3d world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meet- ing of the Association for Computational Linguis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' A GPT-3 Example of Physical Reasoning B PIGPeN-Vis We select an example from PIGPeN to display in Figure 5 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' From the original dataset, we re- move two attributes (isUsedUp and salientMaterials_Organic) because they are unchanged in all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We also remove The weight of the potato is 150 grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The robot then slices the potato into thin slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The weight of the potato is now 75 grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Figure 4: Example of incorrect physical commonsense by an LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' When predicting what comes after the in- put text, the large 175 billion parameter GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2020) predicts that the weight of the potato halves after a slicing action is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' EmptyLiquidFromObject object=Cup Action Annotated Action "The robot empties the cup into the sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='" Figure 5: Image pair and actions for a selected PIGPeN example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 3 actions (ThrowObject10, ThrowObject100 and ThrowObject1000) which are all related to throwing an object across a certain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' These actions account for only a small subset of the dataset and create inconsistent image pairs due to the agent’s momentum being captured in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The angle of the camera changes as a result of ThrowObject and this breaks our assumption that the difference between ipre and ipost solely reflects the effects of the action on the environment (and not on the viewer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We therefore reduce the total number of symbolic attributes per object to 38 and the number of possible actions to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Attributes The following 38 symbolic attributes are used to describe an object in PIGPeN: ObjectName,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' parentReceptacles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' receptacleObjectIds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' ObjectTemperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' breakable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' cookable,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Toggleable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Materials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Ceramic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Ceramic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Table 3: Attributes for a selected PIGPeN example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The total number of attributes is 38 as the Materials attribute is a multi-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' salientMaterials_Sponge, salientMaterials_Stone, salientMaterials_Wax, salientMaterials_Wood, sliceable, toggleable B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 Filtering Statistics We initially filter the PIGPeN dataset using two main strategies to remove images with too much or too little change between the pre and post images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' In both cases, the goal is to remove pairs of images in which it would be impossible for a vision model to predict what has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Images with too many changes are often images taken from different viewpoints or with different lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We filter these images by look- ing at the number of pixels changed between ipre and ipost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We show the distribution of the num- ber of pixels changed per image over the training dataset in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Using this visualization we can clearly see a small peak at the extreme - where almost all the pixels in ipost are different from ipre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note that since each image is an RGB image of dimensions 640 × 385, the max number of change is 640 × 385 × 3 = 739, 200 (we also compare pixels across color channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We opt to remove all images with more than 400, 000 changes, which corresponds to around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2% of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Images with too little change could be exam- ples of where the action has no visual outcome and ipre and ipost are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' To filter these Figure 6: Distribution of the number of pixels changed per image in the PIGPeN dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Figure 7: Distribution of the maximum pixel value changed per image in the PIGPeN dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 12000 10000 8000 Count 6000 4000 2000 0 0 100000200000300000400000500000600000700000 num_change5000 4000 ur 3000 8 2000 1000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='0 max_changeimages we measure the maximum magnitude of change in each pixel and each color channel be- tween the pairs of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We visualize the max change across the training dataset in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Here a low values implies almost no salient change, and as max change approaches zero - it becomes un- likely that a human would be able to perceive the difference between the pair of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We opt for to keep images with a max change greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 which corresponds to excluding 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='8% of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Filtering on the number of changed pixels lead to the exclusion of around 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89% of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='3 Zero-shot Filtering We remove the following 14 objects from both the train and validation (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 401 examples total): HandTowel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Towel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Plunger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Watch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' CD,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Statue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' KeyChain We remove the following 27 action-object pairs from both the train and validation (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' 278 examples total): (CloseObject,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Toilet),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (DirtyObject,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Pan),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (PutObject,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='ToiletPaper),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (PutObject,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Towel),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (PutObject,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Watch),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (ToggleOff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='CellPhone),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (ToggleOff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='Television) C Code Release and Training Our full code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' and PIGPeN-Vis split can be found at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='com/gautierdag/piglet-vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Additional Training Details As previously mentioned, there are a few differ- ences between the original Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (2021) model and our implementation of base+symbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use an off-the-shelf RoBERTa-base (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019) model instead of a custom GPT2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Additionally, we also reduce the di- mensionality of the PIGLeT layers from h = 256 to h = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This shrinks the overall model (ex- cluding the LLM) from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='9 million parameters to less than 2 million parameters during pre-training and improves the overall accuracy by a small mar- gin (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='51%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We do not run any other hyper- parameter search throughout our experiments and wherever possible use the same hyper-parameters as PIGLeT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We also reduce the batch size from 1024 to 256 because we use a mix of NVIDIA GTX 1080 and NVIDIA A100 GPUs and wish to keep batch size constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The +images models use the extracted represen- tations from a frozen off-the-shelf DETR model (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='3 million parameters), however it is ran only once over all images as we cache its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We do not use the “NO OBJECT” predictions from DETR, and simply pass all 100 bounding boxes representations to the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Since we do not have access to the true bounding boxes in PIGPeN, we do not fine-tune DETR and there- fore ignore its prediction heads which have also been trained on COCO and mismatch our possible objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The +symbolic models use the Symbolic Object Encoder which is an additional 800, 000 parame- ters on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' During fine-tuning all models use a RoBERTA-base model (+120 million parameters) in the Action Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The +text-label model also uses the RoBERTA-base model during pre- training, but again this is frozen and its outputs are cached for the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We pre-train each model for 80 epochs and fine- tune for 60 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' For all setups, pre-training takes between 1 to 2 hours and fine-tuning takes less than 1 hour on an NVIDIA A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use the Pytorch implementation of the Adam optimizer (Kingma and Ba, 2014) and a learning rate of 10−3 during pre-training and 10−5 during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We use early stopping on the validation loss with a patience of 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We run each setup over 10 different seeds (s ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', 10] and report the average and standard deviation for each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' D Accuracy Results D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='1 Comparing PIGPeN and PIGPeN-Vis Table 4 compares the overall accuracy on the orig- inal PIGPeN dataset with our proposed PIGPeN- Vis split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We find that our PIGPeN-Vis split is consistently harder to solve than the original PIG- PeN dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We explain the increased accuracy in Overall Accuracy (% ± σ) PIGPeN PIGPeN-Vis ∆ base 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='72 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95% base+symbolic (PIGLeT) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='79 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='45 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='36% base+symbolic+images 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='66 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='44% base+images (PIGLet-Vis) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='13 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='53 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='50 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='66% base+images+text-labels 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='28 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='68 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='10 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='73% Table 4: Overall Accuracies comparing full PIGPeN with the PIGPeN-Vis split across 10 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Overall Accuracy (% ± σ) validation test base 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='72 base+symbolic (PIGLeT) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='50 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='45 base+symbolic+images 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='82 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 base+images 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='73 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='97 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='50 base+images+text-labels 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='33 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='15 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='10 Table 5: Validation and test overall accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note the zero-shot accuracy is not calculated on the valida- tion set since there are no unseen examples in the vali- dation set to prevent leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' the original dataset with the fact that some of the filtered out actions (see Appendix B) are easy to solve from knowing the object name and action: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=', most of the images we exclude due to little salient changes are appliances like stoves being turned on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' However, it is easy for a model to predict the post-condition attributes of a stove, which are mostly static, across all examples given an action such as ToggleOn, which always has the same effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='2 Complete Accuracy Results on PIGPeN-Vis Table 5 shows the overall accuracies for both the test and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The full accuracy results for all actions in Table 6 and for all attributes in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' E Additional Attention Maps We plot additional attention visual- izations for all three image models base+images, base+symbolic+images, and base+images+text-labels in Figures 8, Fig- ures 9, and Figures 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Since the DETR object detector remains frozen, all models have access to the same bounding boxes and bounding box representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Qualitatively, we find that the attention weights of base+images and base+images+text-labels both learn to map to globally relevant bounding boxes given an objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We also find the attention maps in base+images+text-labels to be less confident overall than base+images, likely due to the noise introduced by the semantic text inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' As a result, base+images+text-labels makes less mistakes by not focusing too much attention to the wrong bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' On the other hand, base+symbolic+images focuses on seemingly random bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Since base+symbolic+images already receives the full representation of each objects, it does not learn to complement the object’s represen- tation with accurate visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' While base+symbolic+images extracts 1% of additional overall accuracy from image inputs when compared to base+symbolic, it does so by falling back to vision for visually salient actions such as Pickup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' base+symbolic+images focuses only a narrow set bounding boxes with overconfidence with no re- gard for whether or not the bounding box relates to the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' We posit that the model might use vision to better estimate more difficult attributes to predict such as distance in some contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Note Pickup is a salient action because when the agent in the environment picks an object up, the object is placed directly in the middle of its field of vision (as if the agent were holding the object in front of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (a) base+images (b) base+symbolic+images (c) base+images+text-labels Figure 8: Attention maps for the effects of the EmptyLiquid action on Bowl with objects Fridge and Bowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The top row of each grid maps to the before environment and the bottom row maps to the after environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The columns map to each respective object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The Fridge object appears in the lower left of the image, and is only correctly identified by base+images+text-labels, even though the model does place more weight to the bounding box of the stove (lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (a) base+images (b) base+symbolic+images (c) base+images+text-labels Figure 9: Attention maps for the effects of the Slice action on Apple with objects CounterTop and Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The top row of each grid maps to the before environment and the bottom row maps to the after environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The columns map to each respective object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (a) base+images (b) base+symbolic+images (c) base+images+text-labels Figure 10: Attention maps for the effects of the Dirty action on Bowl with objects Bowl and None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The top row of each grid maps to the before environment and the bottom row maps to the after environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The columns map to each respective object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' None can be an object in PIGPeN, but we do not predict its attributes and exclude it in all model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' (a) base+images (b) base+symbolic+images (c) base+images+text-labels Figure 11: Attention maps for the effects of the Open action on Toilet with objects Toilet and ToiletPaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The top row of each grid maps to the before environment and the bottom row maps to the after environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' The columns map to each respective object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' This particular example is an unseen combination of action and object that has been excluded from the training and validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' CounterTop Apple Before AfterCounterTop Apple Before AfterCounterTop Apple BeforeBowl None Before aBowl None Before AfterBowl None BeforeToilet ToiletPaper BeforeToilet ToiletPaper Before AfterToilet ToiletPaper BeforeFridge Bowl Before AfterFridge Bowl Before AfterFridge Bowl Before AfterAction Accuracy (% ± σ) Close Dirty EmptyLiquid HeatUpPan Open base 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='20 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='06 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='71 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='20 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='33 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='84 base+symbolic 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='77 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='42 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='34 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='15 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='00 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='99 base+symbolic+images 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='80 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='29 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='29 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='90 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='02 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='07 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='62 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='02 base+images 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='42 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='71 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='78 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='34 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='17 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='67 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='83 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='63 base+images+text-labels 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='87 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='19 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='71 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='24 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='16 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='17 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='16 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='92 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='79 Pickup Put Slice ToggleOff ToggleOn base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='96 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='92 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='19 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='86 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='83 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='39 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='38 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='57 base+symbolic 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='48 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='88 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='39 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='94 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='89 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='84 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='90 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='61 base+symbolic+images 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='56 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='59 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='31 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='31 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='86 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 base+images 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='49 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='45 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='91 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='43 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='64 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='80 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='43 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='75 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='24 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='00 base+images+text-labels 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='12 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='61 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='30 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='11 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='57 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='85 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='05 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='81 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='14 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='53 Table 6: Full accuracy results table including the standard deviation over 10 seeds for all actions and setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content=' Attribute Accuracy (% ± σ) Name Temperature attribute breakable cookable dirtyable distance isBroken isCooked isDirty base 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='07 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='41 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='07 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='36 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='07 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='10 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='01 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} +page_content='07 Table 7: Full accuracy results table including the standard deviation over 10 seeds for all attributes and setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFKT4oBgHgl3EQfjS7T/content/2301.11845v1.pdf'} diff --git a/SdFAT4oBgHgl3EQf1h7S/content/2301.08710v1.pdf b/SdFAT4oBgHgl3EQf1h7S/content/2301.08710v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..109d864d279112499b4efb2ad11e29ea8631aa44 --- /dev/null +++ b/SdFAT4oBgHgl3EQf1h7S/content/2301.08710v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c01ce1a2f9a88a2e5390c894c8bb7461343e86c2ada614c0c68764d8f2e5d1b5 +size 15450368 diff --git a/T9E1T4oBgHgl3EQfugV-/content/tmp_files/2301.03389v1.pdf.txt b/T9E1T4oBgHgl3EQfugV-/content/tmp_files/2301.03389v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f634f64dab0edb62ad6ebbcc4c5024f5388140a0 --- /dev/null +++ b/T9E1T4oBgHgl3EQfugV-/content/tmp_files/2301.03389v1.pdf.txt @@ -0,0 +1,959 @@ +On the α-index of minimally 2-connected graphs with given +order or size ∗ +Jiayu Loua,b, Ligong Wanga,b,†, Ming Yuana +aSchool of Mathematics and Statistics, Northwestern Polytechnical University, +Xi’an, Shaanxi 710129, P.R. China. +b Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, +Xi’an, Shaanxi 710129, P.R. China. +E-mail: jyloumath@163.com, lgwangmath@163.com, ym19980508@mail.nwpu.edu.cn +Abstract +For any real α ∈ [0, 1], Nikiforov defined the Aα-matrix of a graph G as +Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency ma- +trix and the diagonal matrix of vertex degrees of G, respectively. The largest +eigenvalue of Aα(G) is called the α-index or the Aα-spectral radius of G. A +graph is minimally k-connected if it is k-connected and deleting any arbitrary +chosen edge always leaves a graph which is not k-connected. In this paper, +we characterize the extremal graphs with the maximum α-index for α ∈ [ 1 +2, 1) +among all minimally 2-connected graphs with given order or size, respectively. +Key Words: minimally 2-connected graph, α-index, extremal graph +AMS Subject Classification (2020): 05C50, 05C40, 05C35. +1 +Introduction +Let G = (V (G), E(G)) be a simple undirected graph with vertex set V (G) and edge set E(G). +Let n = |V (G)| and m = |E(G)| denote the order and size of G, respectively. For a vertex +v ∈ V (G), its neighbor set is denoted by NG(v) (or, N(v) for short) and its closed neighbor +set is defined as NG[v] = NG(v) ∪ {v} (or, N[v] for short). The degree of vertex v is denoted +by dG(v) = |NG(v)| (or, d(v) for short). Let ∆(G) and δ(G) be the maximum degree and +minimum degree of G, respectively. For a vertex set S ⊆ V (G), let G[S] be the subgraph of +G induced by S. For A, B ⊂ V (G), we denote by e(A) the number of edges in G[A] and by +e(A, B) the number of edges with one endpoint in A and one endpoint in B. Let G−v denote +the graph obtained from G by deleting the vertex v together with all the edges incident with +v. Similarly, Let G − uv (resp. G + uv) denote the graph obtained from G by deleting (resp. +adding) the edge uv ∈ E(G) (resp. uv /∈ E(G)). Let Ks,t be a complete bipartite graph +with bipartition (X, Y ), where |X| = s and |Y | = t. For an odd number m, SK2, m−1 +2 +(see +Fig. 1) denotes the graph obtained from the complete bipartite graph K2, m−1 +2 +by subdividing +∗Supported by the National Natural Science Foundation of China (No. 12271439). +†Corresponding author. +1 +arXiv:2301.03389v1 [math.CO] 2 Nov 2022 + +one edge. A cycle C of G is said to have a chord if there is an edge of G that joins a pair of +non-adjacent vertices from C. +The adjacency matrix A(G) of G is an n × n matrix (aij)n×n, where aij = 1 if vivj ∈ E(G) +and 0 otherwise. +Let D(G) be the diagonal matrix of vertex degrees of G. +The signless +Laplacian matrix of G is defined as Q(G) = D(G) + A(G). The largest eigenvalue of A(G) is +called the index or the spectral radius of G, and the largest eigenvalue of Q(G) is called the +Q-index or the signless Laplacian spectral radius of G. For any real α ∈ [0, 1], Nikiforov [21] +proposed to study the convex linear combinations Aα(G) of A(G) and D(G) defined by +Aα(G) = αD(G) + (1 − α)A(G). +It is easy to see that A(G) = A0(G), D(G) = A1(G) and Q(G) = 2A 1 +2 (G). The largest +eigenvalue of Aα(G), denoted by ρα(G), is called the α-index or the Aα-spectral radius of G. +For a connected graph G, Aα(G) is irreducible. By the Perron-Frobenius Theorem, ρα(G) is +positive, and there exists a unique positive unit eigenvector corresponding to ρα(G), which is +called the α-Perron vector of G. +Let G be a set of graphs. For the work on extremal spectral problems, one of the most +important problems is to find the upper or lower bounds for some spectral parameter (index, +Q-index or α-index, etc.) in G and characterize the extremal graphs. There are two classic +problems related to this problem. One is the Brualdi-Soheild problem [5]: find an upper +bound for the indices in G of order n and characterize the extremal graphs, and the other +is the Brualdi-Hoffman problem [4]: find an upper bound for the indices in G of size m and +characterize the extremal graphs. For related researches, one may refer to [2, 8, 24–27]. +It is interesting to consider the above two problems under the restrictions of other pa- +rameters or special classes of graphs. A graph is said to be H-free if it does not contain a +subgraph isomorphic to H. Berman and Zhang [1] characterized the graphs with maximum +index among all connected graphs with order n and cut vertices k. Liu, Lu and Tian [17] +determined the graphs with the maximum index among all the unicyclic graphs with order +n and diameter d. Zhai, Lin and Shu [30] characterized the graphs with the maximum in- +dex among the K2,r+1-free (resp. {C+ +3 , C+ +4 }-free) graphs with size m. For the Q-index and +α-index counterparts of the above problems, many researchers also have some corresponding +results. Zhai, Xue and Lou [31] determined the graph with the maximum Q-index among all +graphs with size m and clique number ω (resp. chromatic number χ). Lin, Huang and Xue +[16] characterized the graph with the maximum α-index among all connected graphs with +order n and cut vertices k. Guo and Zhang [14] determined the graphs with the maximum +α-index for α ∈ [ 1 +2, 1) among the C4-free (resp. Halin) graphs with order n. For more results, +one can refer to [11, 12, 18, 28, 32]. In recent years, the relationship between the spectral +parameter and forbidden subgraphs has been a hot research topic. We refer the interested +reader to the surveys [6, 15, 20] for more results. +A graph is k-connected (resp. k-edge-connected) if removing fewer than k vertices (resp. +edges) always leaves the remaining graph connected, and is minimally k-(edge)-connected if +it is k-connected (resp. +k-edge-connected) and deleting any arbitrary chosen edge always +leaves a graph which is not k-connected (resp. k-edge-connected). In recent works, some +researchers restrict G to (minimally) k-(edge)-connected graphs of order n or size m. A graph +is minimally 1-(edge)-connected if and only if it is a tree. It is natural to ask which graphs +have the maximal indices among all minimally k-(edge)-connected graphs for k ≥ 2. Fan, +Goryainov and Lin [10] asked the following question for k ≥ 2. +2 + +Fig. 1. The graphs SK2, m−1 +2 +and G(a, b) +Problem 1.1. What is the maximum (Q-)index and what are the corresponding extremal +graphs among minimally k-(edge)-connected graph for k ≥ 2? +Chen and Guo [7] and Lou, Min and Huang [19] characterized the extremal graphs with +the maximum index among all minimally 2-(edge)-connected graphs with given order or size, +respectively. Fan, Goryainov and Lin [10] determined the extremal graphs with the maxi- +mum Q-index among all minimally 2-(edge)-connected graphs with given order, meanwhile, +they characterized the extremal graphs with the maximum (Q-)index among all minimally +3-connected graphs with given order. Guo and Zhang [13, 33] characterized the extremal +graphs with the maximum Q-index among all (minimally) 2-connected graphs with given +size. Analogously, we ask the following question with respect to α-index. +Problem 1.2. What is the maximum α-index and what are the corresponding extremal graphs +among minimally k-(edge)-connected graph for k ≥ 2? +In this paper, we characterize the extremal graphs with the minimum α-index for α ∈ [ 1 +2, 1) +among all minimally 2-connected graphs with given order or size, respectively. +Theorem 1.3. Let G be a minimally 2-connected graph with order n ≥ 5. If α ∈ [ 1 +2, 1), then +ρα(G) ≤ ρα(K2,n−2), with equality if and only if G ∼= K2,n−2. +Theorem 1.4. Let G be a minimally 2-connected graph with size m and α ∈ [ 1 +2, 1). +(i) If m ≥ 6 is an even number, then ρα(G) ≤ ρα(K2, m +2 ), with equality if and only if +G ∼= K2, m +2 . +(ii) If m ≥ 9 is an odd number, then ρα(G) ≤ ρα(SK2, m−1 +2 ), where ρα(SK2, m−1 +2 ) is the +largest root of x3−( m+5 +2 α+1)x2+( m+5 +2 α2+ 5(m−1) +2 +α+2−m)x−2mα2−(m−5)α+m−3 = +0, with equality if and only if G ∼= SK2, m−1 +2 . +The rest of this paper is organized as follows. In Section 2, we recall some notions and +lemmas that will be used later, and prove some new lemmas. In Sections 3 and 4, we give +the proof of Theorems 1.3 and 1.4, respectively. +3 + +4 +a +b +m+5 +2 +G(a, b) +22 +Preliminaries +In this section, we introduce some preliminary results that are used in the proof of our main +results. +Lemma 2.1. ([21]) If G is a graph with no isolated vertices, then +ρα(G) ≤ max +u∈V (G) +� +� +�αd(u) + 1 − α +d(u) +� +uv∈E(G) +d(v) +� +� +� . +If α ∈ ( 1 +2, 1) and G is connected, with equality if and only if G is regular. +Lemma 2.2. ([21]) Let G be a graph with ∆(G) = ∆. If α ∈ [0, 1 +2], then +ρα(G) ≥ α(∆ + 1). +If α ∈ [ 1 +2, 1), then +ρα(G) ≥ α∆ + (1 − α)2 +α +. +Lemma 2.3. ([3]) If G is a minimally 2-(edge)-connected graph, then δ(G) = 2. +Lemma 2.4. ([9]) A minimally 2-connected graph with more than three vertices contains no +triangles. +Lemma 2.5. ([9]) A minimally 2-connected graph with n ≥ 4 has at most 2n − 4 edges, with +equality if and only if G ∼= K2,n−2. +Lemma 2.6. ([23]) A 2-connected graph G is minimally 2-connected if and only if no cycle +of G has a chord. +Lemma 2.7. ([22, 29]) Let G be a connected graph with α ∈ [0, 1). +For u, v ∈ V (G), +suppose N ⊆ N(v)\(N(u) ∪ {u}). Let G′ = G − {vw : w ∈ N} + {uw : w ∈ N}. Let +X = (x1, x2, . . . , xn)T be the α-Perron vector of G corresponding to ρα(G). If N ̸= ∅ and +xu ≥ xv, then ρα (G′) > ρα(G). +We say that u and v are equivalent in G if there exists an automorphism p : G → G such +that p(u) = v. +Lemma 2.8. ([21]) Let G be a connected graph of order n, and let u and v be equivalent +vertices in G. If X = (x1, x2, . . . , xn)T is an eigenvector to ρα(G), then xu = xv. +Lemma 2.9. ([21]) Let a ≥ b ≥ 1. If α ∈ [0, 1], then the largest eigenvalue of Aα (Ka,b) is +ρα(Ka,b) = 1 +2(α(a + b) + +� +α2(a + b)2 + 4ab(1 − 2α)). +4 + +Lemma 2.10. Let m be an odd integer. Then ρα(SK2, m−1 +2 ) is the largest root of the following +equation: +x3 − (m + 5 +2 +α + 1)x2 + (m + 5 +2 +α2 + 5(m − 1) +2 +α + 2 − m)x − 2mα2 − (m − 5) α + m − 3 = 0. +Proof. +Let V (SK2, m−1 +2 ) = {v1, v2, ..., v m+5 +2 } (see Fig. 1) and X = (x1, x2, ..., x m+5 +2 )T be +the α-Perron vector of SK2, m−1 +2 , where xi denotes the coordinate corresponding to vi for +1 ≤ i ≤ m+5 +2 . By Lemma 2.8, we have +x1 = x2, x3 = x4, x5 = ... = x m+5 +2 . +Let ρα = ρα(SK2, m−1 +2 ). Since Aα(SK2, m−1 +2 )X = ραX, then +� +� +� +� +� +� +� +� +� +(ρα − (α + 1)) x1 = (1 − α)x3, +(ρα − (m − 1)α +2 +) x3 = (1 − α)x1 + (m − 3)(1 − α) +2 +x5, +(ρα − 2α) x5 = 2(1 − α)x3. +Since X = (x1, x2, . . . , x m+5 +2 )T is an eigenvector corresponding to ρα, it follows that X ̸= 0. +This implies that +������ +ρα − (α + 1) +−(1 − α) +0 +−(1 − α) +ρα − (m−1)α +2 +− (m−3)(1−α) +2 +0 +−2(1 − α) +ρα − 2α +������ += 0. +Hence ρα is the largest root of the following equation +������ +x − (α + 1) +−(1 − α) +0 +−(1 − α) +x − (m−1)α +2 +− (m−3)(1−α) +2 +0 +−2(1 − α) +x − 2α +������ += 0. +By computation, we conclude that ρα is the largest root of the following equation: +x3 − (m + 5 +2 +α + 1)x2 + (m + 5 +2 +α2 + 5(m − 1) +2 +α + 2 − m)x − 2mα2 − (m − 5)α + m − 3 = 0. +This completes the proof. +Lemma 2.11. Let m ≥ 9 and α ∈ [ 1 +2, 1). Then +f(α, m) > 0 and g(α, m) < 0, +where f(α, m) = 2(5α3 − 6α2 + 2α)m5 − 2(123α3 − 156α2 + 60α − 4)m4 + 4(517α3 − 650α2 ++ 254α − 20)m3 − 4(1783α3 − 2008α2 + 664α − 16)m2 + 2(5377α3 − 5014α2 ++ 1202α + 136)m − 2(2951α3 − 2052α2 + 228α + 196) +and g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 +− 135α2 + 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16). +5 + +Proof. The first, second and third order partial derivatives of function f(α, m) with respect +to m are as follows: +fm(α, m) = 10(5α3 − 6α2 + 2α)m4 − 8(123α3 − 156α2 + 60α − 4)m3 + 12(517α3 − 650α2 ++ 254α − 20)m2 − 8(1783α3 − 2008α2 + 664α − 16)m + 2(5377α3 − 5014α2 ++ 1202α + 136), +fmm(α, m) = 40(5α3 − 6α2 + 2α)m3 − 24(123α3 − 156α2 + 60α − 4)m2 + 24(517α3 − 650α2 ++ 254α − 20)m − 8(1783α3 − 2008α2 + 664α − 16), +fmmm(α, m) = 120(5α3 − 6α2 + 2α)m2 − 48(123α3 − 156α2 + 60α − 4)m + 24(517α3 − 650α2 ++ 254α − 20). +Since α ∈ [ 1 +2, 1), then 5α3 − 6α2 + 2α > 0. Let m0 be the minimum point of fmmm(α, m). +Then we have +m0 = 48(123α3 − 156α2 + 60α − 4) +240 (5α3 + 6α2 − 2α) += 123α3 − 156α2 + 60α − 4 +5(5α3 − 6α2 + 2α) +< 9. +It follows that fmmm(α, m) is increasing for m ≥ 9 and +fmmm(α, m) ≥ fmmm(α, 9) = 96(82α3 − 68α2 − 4α + 13). +It is easy to verify that 82α3 − 68α2 − 4α + 13 > 0 for α ∈ [ 1 +2, 1), that is, fmmm(α, m) > 0. +It follows that fmm(α, m) is increasing for m ≥ 9 and +fmm(α, m) ≥ fmm(α, 9) = 64(64α3 + 62α2 − 137α + 56). +Note that 64α3 + 62α2 − 137α + 56 > 0 for α ∈ [ 1 +2, 1), that is, fmm(α, m) > 0. It follows that +fm(α, m) is increasing for m ≥ 9 and +fm(α, m) ≥ fm(α, 9) = −32(137α3 − 590α2 + 538α − 166). +It is obvious that 137α3 − 590α2 + 538α − 166 < 0 for α ∈ [ 1 +2, 1), that is, fm(α, m) > 0. It +follows that f(α, m) is increasing for m ≥ 9 and +f(α, m) ≥ f(α, 9) = −64(43α3 − 117α2 + 69α − 22) > 0. +Similarly, we have +g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 − 135α2 ++ 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16) < 0. +for m ≥ 9 and α ∈ [ 1 +2, 1). +This completes the proof. +3 +Proof of Theorem 1.3 +In this section, we give the proof of Theorem 1.3. +Proof. Let G be a minimally 2-connected graph with order n ≥ 5, then ∆(G) ≤ n − 2 by +6 + +Lemmas 2.3 and 2.4. Notice that K2,n−2 is a minimally 2-connected graph. By Lemma 2.9, +we have +ρα(K2,n−2) = 1 +2(αn + +� +α2n2 + 8(n − 2)(1 − 2α)). +When ∆(G) = n − 2, it is easy to see that G ∼= K2,n−2 by Lemmas 2.3 and 2.4. By Lemma +2.2, we have +ρα(K2,n−2) ≥ α∆(K2,n−2) + (1 − α)2 +α += α(n − 2) + (1 − α)2 +α +(1) +for α ∈ [ 1 +2, 1). Thus we assume that ∆(G) ≤ n − 3. +Let w be a vertex of G such that +αd(w) + 1 − α +d(w) +� +wv∈E(G) +d(v) = max +u∈V (G) +� +� +�αd(u) + 1 − α +d(u) +� +uv∈E(G) +d(v) +� +� +� . +By Lemma 2.1, we have +ρα(G) ≤ αd(w) + 1 − α +d(w) +� +wv∈E(G) +d(v). +(2) +By Lemma 2.3, we have 2 ≤ d(w) ≤ ∆(G) ≤ n − 3. By Lemma 2.4, we know that N(w) is +an independent set, that is, e(N(w)) = 0. Then +� +wv∈E(G) +d(v) = 2e(N(w)) + e(N(w), V (G)\N(w)) = e(N(w), V (G)\N(w)). +(3) +Next we prove that ρα(G) ≤ ρα(K2,n−2) for 2 ≤ d(w) ≤ n − 3. We consider the following +two cases. +Case 1. d(w) = 2. +If e(V (G)\N[w]) = 0, then G ∼= K2,n−2 by Lemmas 2.3 and 2.4. +If e(V (G)\N[w]) ̸= 0. We assume that there exists an edge v1v2 ∈ E(G[V (G)\N[w]]). By +Lemma 2.4, we have NN(w)(v1) ∩ NN(w)(v1) = ∅. Let B = V (G)\(N[w] ∪ {v1, v2}). Then +e(N(w), V (G)\N[w]) ≤ dN(w)(v1) + dN(w)(v2) + e(N(w), B) = d(w) + d(w)|B| = 2n − 8. +and so � +wv∈E(G) d(v) ≤ 2n − 6. Combining this with (2), we have +ρα(G) ≤ 2α + 1 − α +2 +� +wv∈E(G) +d(v) ≤ 2α + (1 − α)(n − 3). +Noting that +α(n − 2) + (1 − α)2 +α +− (2α + (1 − α)(n − 3)) = (2α2 − α)n − 6α2 + α + 1 +α +≥ 0 +for n ≥ 5 and α ∈ [ 1 +2, 1), we have ρα(G) ≤ α(n − 2) + (1−α)2 +α +. Combining this with (1), we +have +ρα(G) ≤ α(n − 2) + (1 − α)2 +α +≤ ρα (K2,n−2) +7 + +for n ≥ 5 and α ∈ [ 1 +2, 1). +Case 2. 3 ≤ d(w) ≤ n − 3. +In order to prove ρα(G) ≤ ρα(K2,n−2), it is enough to prove +ρα(G) ≤ 1 +2(αn + +� +α2n2 + 8(n − 2)(1 − 2α)), +that is, to prove ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) ≤ 0. For convenience, we denote +Aα(G) = Aα, A(G) = A and D(G) = D. Let +B = (bij)n×n = A2 +α − αnAα + 2(2α − 1)(n − 2)In, +where In is the n × n unit matrix. Let cu(B) be the sum of all elements in the u-th column +of B. Then we have the following claim. +Claim 2.1. cu(B) ≤ 0 for n ≥ 5 and α ∈ [ 1 +2, 1). +Proof. Since Aα = αD + (1 − α)A, then +B =(αD + (1 − α)A)2 − αn(αD + (1 − α)A) + 2(2α − 1)(n − 2)In +=α2D2 + (1 − α)2A2 + α(1 − α)DA + α(1 − α)AD − α2nD − +� +αn − α2n +� +A ++ 2(2α − 1)(n − 2)In. +It is easy to see that cu(A) = cu(D) = d(u), cu(A2) = cu(DA) = � +uv∈E(G) d(v) and +cu(AD) = d2(u). Combining (3) with Lemma 2.5, we have +� +wv∈E(G) +d(v) = e(N(w), V (G)\N(w)) ≤ |E(G)| ≤ 2n − 4, +with equality if and only if G ∼= K2,n−2. It follows that +cu(B) =α2d2(u) + (1 − α)2 +� +uv∈E(G) +d(v) + α(1 − α) +� +uv∈E(G) +d(v) + α(1 − α)d2(u) − α2nd(u) +− +� +αn − α2n +� +d(u) + 2(2α − 1)(n − 2) +=α2d2(u) + (1 − α) +� +uv∈E(G) +d(v) − αnd(u) + 2(2α − 1)(n − 2) +≤α2d2(u) + (1 − α)(2n − 4) − αnd(u) + 2(2α − 1)(n − 2) +=α +� +d2(u) − nd(u) + 2n − 4 +� +≤ max +� +α(9 − 3n + 2n − 4), α +� +(n − 3)2 − n(n − 3) + 2n − 4 +�� +=α(−n + 5) ≤ 0 +for n ≥ 5 and α ∈ [ 1 +2, 1), with equality if and only if G ∼= K2,n−2. +This completes the proof of the claim. +Let X = (x1, x2, . . . , xn)T be the α-Perron vector of G corresponding to ρα(G) satisfying +�n +i=1 xi = 1. Then +BX = +� +ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) +� +X. +8 + +Hence we have +ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) += +n +� +i=1 +� +ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) +� +xi += +n +� +i=1 +(BX)i = +n +� +i=1 +( +n +� +j=1 +bijxj) = +n +� +j=1 +( +n +� +i=1 +bij)xj = +n +� +j=1 +cj(B)xj ≤ 0. +Combining the above arguments, we have ρα(G) ≤ ρα(K2,n−2) for n ≥ 5 and α ∈ [ 1 +2, 1), +with equality if and only if G ∼= K2,n−2. +These complete the proof. +4 +Proof of Theorem 1.4 +In this section, we give the proof of Theorem 1.4. +Proof. Let G be a minimally 2-connected graph with size m. For any v ∈ V (G), it is easy +to see that G − v is connected and |E(G − v)| = m − d(v), then we have +d(v) < |V (G − v)| ≤ m − d(v) + 1, +with equality in the right inequality if and only if G − v is a tree. It follows that d(v) < m+1 +2 . +Combining this with Lemma 2.3, we have 2 ≤ d(v) < m+1 +2 . +Let w be a vertex of G such that +αd(w) + 1 − α +d(w) +� +wv∈E(G) +d(v) = max +u∈V (G) +� +� +�αd(u) + 1 − α +d(u) +� +uv∈E(G) +d(v) +� +� +� . +By Lemma 2.1, we have +ρα(G) ≤ αd(w) + 1 − α +d(w) +� +wv∈E(G) +d(v). +(4) +By Lemma 2.4, we know that N(w) is an independent set, that is, e(N(w)) = 0. Then +� +wv∈E(G) +d(v) = 2e(N(w)) + e(N(w), V (G)\N(w)) = e(N(w), V (G)\N(w)). +(5) +Combining (4) and (5), we have +ρα(G) ≤ αd(w) + 1 − α +d(w) e(N(w), V (G)\N(w)). +(6) +(i) Let m ≥ 6 be an even number. Then we have 2 ≤ d(w) ≤ m +2 . Notice that K2, m +2 is a +minimally 2-connected graph. By Lemma 2.9, we have +ρα(K2, m +2 ) = 1 +4((m + 4)α + +� +(m + 4)2α2 + 16m(1 − 2α)). +9 + +If d(w) = m +2 , then d(v) = 2 for any v ∈ N(w) and e(N(w), V (G)\N[w]) = m +2 by Lemma 2.3. +Since G − w is connected, then G ∼= K2, m +2 . +Next we prove that ρα(G) ≤ ρα(K2, m +2 ) for 2 ≤ d(w) ≤ m−2 +2 . We consider the following two +cases. +Case 1. d(w) = 2. +If e(V (G)\N[w]) = 0, then G ∼= K2, m +2 by Lemmas 2.3 and 2.4. +If e(V (G)\N[w]) ̸= 0, then e(N(w), V (G)\N(w)) ≤ m − 1 by Lemma 2.4. Combining this +with (6), we have +ρα(G) ≤ 2α + 1 − α +2 +(m − 1) ≤ m + 3 +4 +< m + 4 +4 +for m ≥ 6 and α ∈ [ 1 +2, 1). In order to prove ρα(G) < ρα(K2, m +2 ), it is enough to prove +m + 4 +4 +≤ 1 +4((m + 4)α + +� +(m + 4)2α2 + 16m(1 − 2α)), +that is, to prove (1 − 2α)m2 − 8(1 − 2α)m + 16(1 − 2α) ≤ 0 for m ≥ 6 and α ∈ [ 1 +2, 1). It is +easy to check that this is true. Hence we have ρα(G) < ρα(K2, m +2 ) for m ≥ 6 and α ∈ [ 1 +2, 1). +Case 2. 3 ≤ d(w) ≤ m−2 +2 . +In order to prove ρα(G) ≤ ρα(K2, m +2 ), it is enough to prove +ρα(G) ≤ 1 +4((m + 4)α + +� +(m + 4)2α2 + 16m(1 − 2α)), +that is, to prove 2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m < 0. For convenience, we denote +Aα(G) = Aα, A(G) = A and D(G) = D. Let +B = (bij)n×n = 2A2 +α − (m + 4)αAα + 2(2α − 1)mIn, +where In is the n × n unit matrix. Let cu(B) be the sum of all elements in the u-th column +of matrix B. Then we have the following claim. +Claim 2.1. cu(B) ≤ 0 for m ≥ 6 and α ∈ [ 1 +2, 1). +Proof. Since Aα = αD + (1 − α)A, then +B =2(αD + (1 − α)A)2 − (m + 4)α(αD + (1 − α)A) + 2(2α − 1)mIn +=2α2D2 + 2(1 − α)2A2 + 2α(1 − α)DA + 2α(1 − α)AD − (m + 4)α2D +− (m + 4)α(1 − α)A + 2(2α − 1)mIn. +It is easy to see that cu(A) = cu(D) = d(u), cu(A2) = cu(DA) = � +uv∈E(G) d(v) and cu(AD) = +d2(u). Since e(N(w), V (G)\N(w)) ≤ |E(G)| = m, then � +wv∈E(G) d(v) ≤ m by (5). It follows +that +cu(B) =2α2d2(u) + 2(1 − α)2 +� +wv∈E(G) +d(v) + 2α(1 − α) +� +wv∈E(G) +d(v) + 2α(1 − α)d2(u) +− (m + 4)α2d(u) − (m + 4)α(1 − α)d(u) + 2(2α − 1)m +=2α2d2(u) + 2(1 − α) +� +wv∈E(G) +d(v) − (m + 4)αd(u) + 2(2α − 1)m +≤2αd2(u) − (m + 4)αd(u) + 2mα +10 + +=α(2d2(u) − (m + 4)d(u) + 2m) +≤ max +� +α(18 − 3(m + 4) + 2m), α +�(m − 2)2 +2 +− (m + 4)(m − 2) +2 ++ 2m +�� +=α(−m + 6) ≤ 0 +for m ≥ 6 and α ∈ [ 1 +2, 1), with equality if and only if G ∼= K2, m +2 . +This completes the proof of the claim. +Let X = (x1, x2, . . . , xn)T be the α-Perron vector of G corresponding to ρα(G) satisfying +�n +i=1 xi = 1. Then +BX = +� +2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m +� +X. +Hence we have +2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m += +n +� +i=1 +� +2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m +� +xi += +n +� +i=1 +(BX)i = +n +� +i=1 +( +n +� +j=1 +bijxj) = +n +� +j=1 +( +n +� +i=1 +bij)xj = +n +� +j=1 +cj(B)xj ≤ 0. +Combining the above arguments, we have ρα(G) ≤ ρα(K2, m +2 ) for m ≥ 6 and α ∈ [ 1 +2, 1), +with equality if and only if G ∼= K2, m +2 . We complete the proof of (i). +(ii) Let m ≥ 9 be an odd number. Then we have 2 ≤ d(w) ≤ m−1 +2 . Notice that SK2, m +2 is +a minimally 2-connected graph. Next we complete the proof with three facts. +Fact 1. e(N(w), V (G)\N(w)) ≤ m − 1. +Proof. Since e(N(w), V (G)\N(w)) ≤ |E(G)| = m, then � +wv∈E(G) d(v) ≤ m by (5). For a +contradiction, we suppose e(N(w), V (G)\N(w)) = m, then e(V (G)\N[w]) = 0. We consider +the following two cases. +Case 1.1. d(w) = 2. +In this case, we have d(v) = 2 for any v ∈ V (G)\N(w) by Lemma 2.3. This implies that +G is a complete bipartite graph K2,b and m(K2,b) = 2b, which contradicts the fact that m is +odd. +Case 1.2. 3 ≤ d(w) ≤ m−1 +2 . +Let v1, v2 ∈ V (G)\N[w] be any two vertices. If NN(w)(v1)∩NN(w)(v2) = ∅ , then d(wi) = 2 +for any wi ∈ N(w) by Lemma 2.3. It follows that m = e(N(w), V (G)\N(w)) = 2d(w) ≤ m−1, +a contradiction. +If NN(w)(v1) ∩ NN(w)(v2) ̸= ∅ and NN(w)(v1) ̸= NN(w)(v2), we assume +w12 ∈ NN(w)(v1) ∩ NN(w)(v2). +By Lemma 2.3, there exists wi ∈ NN(w)(vi)\w12 for each +i ∈ {1, 2}. Hence, G contains a cycle ww1v1w12v2w2w with a chord ww12, which contradicts +Lemma 2.6. If NN(w)(v1) = NN(w)(v2) = N(w), then we have δ(G) ≥ 3. This contradicts with +Lemma 2.3. Hence NN(w)(v1) = NN(w)(v2) ̸= N(w). It follows that G − w is disconnected, a +contradiction. +Through the above two cases, we know that e(N(w), V (G)\N(w)) ≤ m − 1. +Fact 2. ρα(G) < ρα(SK2, m−1 +2 ) for 3 ≤ d(w) ≤ m−3 +2 . +Proof. Combining Fact 1 and (6), we have +ρα(G) ≤ αd(w) + 1 − α +d(w) (m − 1). +11 + +Let q(x) = αx + 1−α +x (m − 1). Since α ∈ [ 1 +2, 1), it is easy to see that the function q(x) is +convex for x > 0 and its maximum in any closed internal is attained at one of the ends of this +internal. Hence when 3 ≤ x ≤ m−3 +2 , we have +ρα(G) ≤ max +� +3α + 1 − α +3 +(m − 1), m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1) +� +. +Noting that +m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1) − (3α + 1 − α +3 +(m − 1)) += (5α − 2)m2 − (56α − 20)m + 99α − 18 +6(m − 3) +> 0 +for m ≥ 9 and α ∈ [ 1 +2, 1), we have +ρα(G) ≤ m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1) +for m ≥ 9 and α ∈ [ 1 +2, 1). By Lemma 2.10, we know ρα(SK2, m−1 +2 ) is the largest root of the +following equation: +p(x) = x3 − (m + 5 +2 +α + 1)x2 + (m + 5 +2 +α2 + 5(m − 1) +2 +α + 2 − m)x +− 2mα2 − (m − 5)α + m − 3 = 0. +By computation, we have +−8(m − 3)3 · p(m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1)) = f(α, m), +where f(α, m) = 2(5α3 − 6α2 + 2α)m5 − 2(123α3 − 156α2 + 60α − 4)m4 + 4(517α3 − 650α2 ++ 254α − 20)m3 − 4(1783α3 − 2008α2 + 664α − 16)m2 + 2(5377α3 − 5014α2 ++ 1202α + 136)m − 2(2951α3 − 2052α2 + 228α + 196). +By Lemma 2.11, we have f(α, m) > 0 for m ≥ 9 and α ∈ [ 1 +2, 1). It follows that +p(m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1)) < 0 +for α ∈ [ 1 +2, 1) and m ≥ 9. This implies that +ρα(G) ≤ m − 3 +2 +α + 2(1 − α) +m − 3 (m − 1) < ρα(SK2, m−1 +2 ) +for m ≥ 9 and α ∈ [ 1 +2, 1). +Fact 3. ρα(G) ≤ ρα(SK2, m−1 +2 ) for d(w) = 2 or d(w) = m−1 +2 , with equality if and only if +G ∼= SK2, m−1 +2 . +Proof. We consider the following two cases. +Case 3.1. d(w) = 2. +We consider the following two cases and assume N(w) = {w1, w2}. +Subcase 3.1.1. e(N(w), V (G)\N(w)) = m − 1. +12 + +Since e(N(w), V (G)\N(w)) = m − 1, then e(V (G)\N[w]) = 1. Let e(V (G)\N[w]) = v1v2. +By Lemma 2.4, we can see that w1(resp. w2) is adjacent to only one vertex of v1 and v2. +Without loss of generality, we assume that w1v1 ∈ E(G) and w2v2 ∈ E(G). By Lemma +2.3, we have NN(w)(v) = {w1, w2} for any v ∈ V (G)\(N[w] ∪ {v1, v2}). +It follows that +G ∼= SK2, m−1 +2 . +Subcase 3.1.2. e(N(w), V (G)\N(w)) ≤ m − 2. +By (6), we have +ρα(G) ≤ 2α + 1 − α +2 +(m − 2). +By Lemma 2.10, we know ρα(SK2, m−1 +2 ) is the largest root of the following equation: +p(x) = x3 − (m + 5 +2 +α + 1)x2 + (m + 5 +2 +α2 + 5(m − 1) +2 +α + 2 − m)x +− 2mα2 − (m − 5)α + m − 3 = 0. +By computation, we have +4 · p(2α + 1 − α +2 +(m − 2)) = g(α, m), +where g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 +− 135α2 + 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16). +By Lemma 2.11, we have g(α, m) < 0 for m ≥ 9 and α ∈ [ 1 +2, 1). It follows that +p(2α + 1 − α +2 +(m − 2)) < 0 +for m ≥ 9 and α ∈ [ 1 +2, 1). This implies that +ρα(G) ≤ 2α + 1 − α +2 +(m − 2) < ρα(SK2, m−1 +2 ) +for m ≥ 9 and α ∈ [ 1 +2, 1). +Case 3.2. d(w) = m−1 +2 . +By Lemma 2.3, we have e(N(w), V (G)\N(w)) ≥ m − 1. Combining this with Fact 1, we +have e(N(w), V (G)\N(w)) = m − 1, that is, e(V (G)\N[w]) = 1. It follows that d(wi) = 2 for +wi ∈ N(w) by Lemmas 2.3 and 2.4. Let e(V (G)\N[w]) = v1v2. Then V (G)\N[w] = {v1, v2}. +Otherwise G − w is disconnected, a contridiction. +This implies that G = G(a, b), where +1 ≤ a ≤ b and a + b = m−1 +2 +(see Fig. 1). We assume NN(w)(v1) = {w1, w2, . . . , wa} and +NN(w)(v2) = {wa+1, wa+2, . . . , wa+b}. +It is easy to see that G(1, m−3 +2 ) ∼= SK2, m−1 +2 . +Let +X = (x1, x2, . . . , xn)T be the α-Perron vector of G corresponding to ρα(G). Without loss of +generality, we assume xv1 ≤ xv2. Then G(1, m−3 +2 ) = G(a, b) − {v1wi : i = 1, 2, . . . , a − 1} + +{v2wi : i = 1, 2, . . . , a − 1}. By Lemma 2.7, we have +ρα(G) = ρα(G(a, b)) ≤ ρα(G(1, m − 3 +2 +)) = ρα(SK2, m−1 +2 ) +for m ≥ 9 and α ∈ [ 1 +2, 1), with equality if and only if a = 1, that is, G ∼= SK2, m−1 +2 . +Combining Facts 2 and 3, we have ρα(G) ≤ ρα(SK2, m−1 +2 ) for m ≥ 9 and α ∈ [ 1 +2, 1), with +equality if and only if G ∼= SK2, m−1 +2 . We complete the proof of (ii). +13 + +References +[1] A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. +Theory Ser. B 83 (2001) 233–240. +[2] B. Bollob´as, Extremal Graph Theory, Academic Press, London, New York, 1978. +[3] J.A. Bondy, U.S.R. Murty, Graph theory, Springer, New York, 2008. +[4] R.A. Brualdi, A.J. Hoffman, On the spectral radius of (0,1)-matrices, Linear Algebra +Appl. 65 (1985) 133–146. +[5] R.A. Brualdi, E.S. Solheid, On the spectral radius of complementary acyclic matrices of +zeros and ones, SIAM J. Algebra. Discrete Methods 7 (1986) 265–272. +[6] M.Z. Chen, X.-D. 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Zhang, Sharp upper bounds on the Q-index of (minimally) 2-connected +graphs with given size, Discrete Appl. Math. 320 (2022) 408–415. +[14] S.-G. Guo, R. Zhang, The sharp upper bounds on the Aα-spectral radius of C4-free +graphs and Halin graphs, Graphs Combin. 38 (2022) 19. +[15] Y.T. Li, W.J. Liu, L.H. Feng, A survey on spectral conditions for some extremal graph +problems, Adv. Math. 51 (2) (2022) 193–258. +[16] H.Q. Lin, X. Huang, J. Xue, A note on the Aα-spectral radius of graphs, Linear Algebra +Appl. 557 (2018) 430–437. +[17] H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, +Linear Algebra Appl. 420 (2007) 449–457. +[18] X.X. Liu, H.J. Broersma, L.G. Wang, On a conjecture of Nikiforov involving a spectral +radius condition for a graph to contain all trees, Discrete Math. 345 (2022) 113112. +14 + +[19] Z.Z. Lou, G. Min, Q.X. Huang, On the spectral radius of minimally 2-(edge)-connected +graphs with given size, https://arxiv.org/abs/2206.07872. +[20] V. Nikiforov, Some new results in extremal graph theory, in: Surveys in Combinatories +2011, London Math. Soc. Lecture Note Ser. 392 (2011) 141–181. +[21] V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 +(2017) 81–107. +[22] V. Nikiforov, O. Rojo, On the α-index of graphs with pendent paths, Linear Algebra +Appl. 550 (2018) 87–104. +[23] M. Plummer, On minimal blocks, Trans. Amer. Math. Soc. 134 (1968) 85–94. +[24] P. Rowlinson, On the maximal index of graphs with a prescribed number of edges, Linear +Algebra Appl. 110 (1988) 43–53. +[25] Z. Stani´c, Inequalities for Graph Eigenvalues, Cambridge Unoversity Press, New York, +2015. +[26] P.R. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra +Appl. 87 (1987) 267–269. +[27] D. Stevanovi´c, Spectral Radius of Graphs, Academic Press, New York, 2015. +[28] G.X. Tian, Y.X. Chen, S.Y. Cui, The extremal α-index of graphs with no 4-cycle and +5-cycle, Linear Algebra Appl. 619 (2021) 160–175. +[29] J. Xue, H.Q. Lin, S.T. Liu, J.L. Shu, On the Aα-spectral radius of a graph, Linear +Algebra Appl. 550 (2018) 105–120. +[30] M.Q. Zhai, H.Q. Lin, J.L. Shu, Spectral extrema of graphs with fixed size: cycles and +complete bipartite graphs, Electron. J. Comb. 95 (2021) 103322. +[31] M.Q. Zhai, J. Xue, Z.Z. Lou, The signless Laplacian spectral radius of graphs with a +prescribed number of edges, Linear Algebra Appl. 603 (2020) 154–165. +[32] H.H. Zhang, S.C. Li, On the Laplacian spectral radius of bipartite graphs with fixed +order and size, Discrete Appl. Math. 229 (2017) 139–147. +[33] R. Zhang, S.-G. Guo, Maxima of the Laplacian spectral radius of (minimally) 2-connected +graphs with fixed size, Linear Algebra Appl. 651 (2022) 390–406. +15 + diff --git a/T9E1T4oBgHgl3EQfugV-/content/tmp_files/load_file.txt b/T9E1T4oBgHgl3EQfugV-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f32b3ec3edc28a9a2182375971ea1c94cf2a65db --- /dev/null +++ b/T9E1T4oBgHgl3EQfugV-/content/tmp_files/load_file.txt @@ -0,0 +1,643 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf,len=642 +page_content='On the α-index of minimally 2-connected graphs with given order or size ∗ Jiayu Loua,b, Ligong Wanga,b,†, Ming Yuana aSchool of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' b Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' E-mail: jyloumath@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='com, lgwangmath@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='com, ym19980508@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='nwpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='cn Abstract For any real α ∈ [0, 1], Nikiforov defined the Aα-matrix of a graph G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency ma- trix and the diagonal matrix of vertex degrees of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The largest eigenvalue of Aα(G) is called the α-index or the Aα-spectral radius of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' A graph is minimally k-connected if it is k-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In this paper, we characterize the extremal graphs with the maximum α-index for α ∈ [ 1 2, 1) among all minimally 2-connected graphs with given order or size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Key Words: minimally 2-connected graph, α-index, extremal graph AMS Subject Classification (2020): 05C50, 05C40, 05C35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1 Introduction Let G = (V (G), E(G)) be a simple undirected graph with vertex set V (G) and edge set E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let n = |V (G)| and m = |E(G)| denote the order and size of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For a vertex v ∈ V (G), its neighbor set is denoted by NG(v) (or, N(v) for short) and its closed neighbor set is defined as NG[v] = NG(v) ∪ {v} (or, N[v] for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The degree of vertex v is denoted by dG(v) = |NG(v)| (or, d(v) for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let ∆(G) and δ(G) be the maximum degree and minimum degree of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For a vertex set S ⊆ V (G), let G[S] be the subgraph of G induced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For A, B ⊂ V (G), we denote by e(A) the number of edges in G[A] and by e(A, B) the number of edges with one endpoint in A and one endpoint in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G−v denote the graph obtained from G by deleting the vertex v together with all the edges incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Similarly, Let G − uv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' G + uv) denote the graph obtained from G by deleting (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' adding) the edge uv ∈ E(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' uv /∈ E(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let Ks,t be a complete bipartite graph with bipartition (X, Y ), where |X| = s and |Y | = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For an odd number m, SK2, m−1 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1) denotes the graph obtained from the complete bipartite graph K2, m−1 2 by subdividing ∗Supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 12271439).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='03389v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='CO] 2 Nov 2022 one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' A cycle C of G is said to have a chord if there is an edge of G that joins a pair of non-adjacent vertices from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The adjacency matrix A(G) of G is an n × n matrix (aij)n×n, where aij = 1 if vivj ∈ E(G) and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let D(G) be the diagonal matrix of vertex degrees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The largest eigenvalue of A(G) is called the index or the spectral radius of G, and the largest eigenvalue of Q(G) is called the Q-index or the signless Laplacian spectral radius of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For any real α ∈ [0, 1], Nikiforov [21] proposed to study the convex linear combinations Aα(G) of A(G) and D(G) defined by Aα(G) = αD(G) + (1 − α)A(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to see that A(G) = A0(G), D(G) = A1(G) and Q(G) = 2A 1 2 (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The largest eigenvalue of Aα(G), denoted by ρα(G), is called the α-index or the Aα-spectral radius of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For a connected graph G, Aα(G) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By the Perron-Frobenius Theorem, ρα(G) is positive, and there exists a unique positive unit eigenvector corresponding to ρα(G), which is called the α-Perron vector of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G be a set of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For the work on extremal spectral problems, one of the most important problems is to find the upper or lower bounds for some spectral parameter (index, Q-index or α-index, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=') in G and characterize the extremal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' There are two classic problems related to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' One is the Brualdi-Soheild problem [5]: find an upper bound for the indices in G of order n and characterize the extremal graphs, and the other is the Brualdi-Hoffman problem [4]: find an upper bound for the indices in G of size m and characterize the extremal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For related researches, one may refer to [2, 8, 24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is interesting to consider the above two problems under the restrictions of other pa- rameters or special classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' A graph is said to be H-free if it does not contain a subgraph isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Berman and Zhang [1] characterized the graphs with maximum index among all connected graphs with order n and cut vertices k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Liu, Lu and Tian [17] determined the graphs with the maximum index among all the unicyclic graphs with order n and diameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Zhai, Lin and Shu [30] characterized the graphs with the maximum in- dex among the K2,r+1-free (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' {C+ 3 , C+ 4 }-free) graphs with size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For the Q-index and α-index counterparts of the above problems, many researchers also have some corresponding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Zhai, Xue and Lou [31] determined the graph with the maximum Q-index among all graphs with size m and clique number ω (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' chromatic number χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lin, Huang and Xue [16] characterized the graph with the maximum α-index among all connected graphs with order n and cut vertices k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Guo and Zhang [14] determined the graphs with the maximum α-index for α ∈ [ 1 2, 1) among the C4-free (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Halin) graphs with order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For more results, one can refer to [11, 12, 18, 28, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In recent years, the relationship between the spectral parameter and forbidden subgraphs has been a hot research topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We refer the interested reader to the surveys [6, 15, 20] for more results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' A graph is k-connected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' k-edge-connected) if removing fewer than k vertices (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' edges) always leaves the remaining graph connected, and is minimally k-(edge)-connected if it is k-connected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' k-edge-connected) and deleting any arbitrary chosen edge always leaves a graph which is not k-connected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' k-edge-connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In recent works, some researchers restrict G to (minimally) k-(edge)-connected graphs of order n or size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' A graph is minimally 1-(edge)-connected if and only if it is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is natural to ask which graphs have the maximal indices among all minimally k-(edge)-connected graphs for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Fan, Goryainov and Lin [10] asked the following question for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The graphs SK2, m−1 2 and G(a, b) Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' What is the maximum (Q-)index and what are the corresponding extremal graphs among minimally k-(edge)-connected graph for k ≥ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Chen and Guo [7] and Lou, Min and Huang [19] characterized the extremal graphs with the maximum index among all minimally 2-(edge)-connected graphs with given order or size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Fan, Goryainov and Lin [10] determined the extremal graphs with the maxi- mum Q-index among all minimally 2-(edge)-connected graphs with given order, meanwhile, they characterized the extremal graphs with the maximum (Q-)index among all minimally 3-connected graphs with given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Guo and Zhang [13, 33] characterized the extremal graphs with the maximum Q-index among all (minimally) 2-connected graphs with given size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Analogously, we ask the following question with respect to α-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' What is the maximum α-index and what are the corresponding extremal graphs among minimally k-(edge)-connected graph for k ≥ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In this paper, we characterize the extremal graphs with the minimum α-index for α ∈ [ 1 2, 1) among all minimally 2-connected graphs with given order or size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G be a minimally 2-connected graph with order n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If α ∈ [ 1 2, 1), then ρα(G) ≤ ρα(K2,n−2), with equality if and only if G ∼= K2,n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G be a minimally 2-connected graph with size m and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (i) If m ≥ 6 is an even number, then ρα(G) ≤ ρα(K2, m 2 ), with equality if and only if G ∼= K2, m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (ii) If m ≥ 9 is an odd number, then ρα(G) ≤ ρα(SK2, m−1 2 ), where ρα(SK2, m−1 2 ) is the largest root of x3−( m+5 2 α+1)x2+( m+5 2 α2+ 5(m−1) 2 α+2−m)x−2mα2−(m−5)α+m−3 = 0, with equality if and only if G ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In Section 2, we recall some notions and lemmas that will be used later, and prove some new lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In Sections 3 and 4, we give the proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 3 4 a b m+5 2 G(a, b) 22 Preliminaries In this section, we introduce some preliminary results that are used in the proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([21]) If G is a graph with no isolated vertices, then ρα(G) ≤ max u∈V (G) � � �αd(u) + 1 − α d(u) � uv∈E(G) d(v) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If α ∈ ( 1 2, 1) and G is connected, with equality if and only if G is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([21]) Let G be a graph with ∆(G) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If α ∈ [0, 1 2], then ρα(G) ≥ α(∆ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If α ∈ [ 1 2, 1), then ρα(G) ≥ α∆ + (1 − α)2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([3]) If G is a minimally 2-(edge)-connected graph, then δ(G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([9]) A minimally 2-connected graph with more than three vertices contains no triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([9]) A minimally 2-connected graph with n ≥ 4 has at most 2n − 4 edges, with equality if and only if G ∼= K2,n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([23]) A 2-connected graph G is minimally 2-connected if and only if no cycle of G has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([22, 29]) Let G be a connected graph with α ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For u, v ∈ V (G), suppose N ⊆ N(v)\\(N(u) ∪ {u}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G′ = G − {vw : w ∈ N} + {uw : w ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , xn)T be the α-Perron vector of G corresponding to ρα(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If N ̸= ∅ and xu ≥ xv, then ρα (G′) > ρα(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We say that u and v are equivalent in G if there exists an automorphism p : G → G such that p(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([21]) Let G be a connected graph of order n, and let u and v be equivalent vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , xn)T is an eigenvector to ρα(G), then xu = xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ([21]) Let a ≥ b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If α ∈ [0, 1], then the largest eigenvalue of Aα (Ka,b) is ρα(Ka,b) = 1 2(α(a + b) + � α2(a + b)2 + 4ab(1 − 2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 4 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let m be an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then ρα(SK2, m−1 2 ) is the largest root of the following equation: x3 − (m + 5 2 α + 1)x2 + (m + 5 2 α2 + 5(m − 1) 2 α + 2 − m)x − 2mα2 − (m − 5) α + m − 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let V (SK2, m−1 2 ) = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=', v m+5 2 } (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1) and X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=', x m+5 2 )T be the α-Perron vector of SK2, m−1 2 , where xi denotes the coordinate corresponding to vi for 1 ≤ i ≤ m+5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='8, we have x1 = x2, x3 = x4, x5 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' = x m+5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let ρα = ρα(SK2, m−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since Aα(SK2, m−1 2 )X = ραX, then � � � � � � � � � (ρα − (α + 1)) x1 = (1 − α)x3, (ρα − (m − 1)α 2 ) x3 = (1 − α)x1 + (m − 3)(1 − α) 2 x5, (ρα − 2α) x5 = 2(1 − α)x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , x m+5 2 )T is an eigenvector corresponding to ρα, it follows that X ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This implies that ������ ρα − (α + 1) −(1 − α) 0 −(1 − α) ρα − (m−1)α 2 − (m−3)(1−α) 2 0 −2(1 − α) ρα − 2α ������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence ρα is the largest root of the following equation ������ x − (α + 1) −(1 − α) 0 −(1 − α) x − (m−1)α 2 − (m−3)(1−α) 2 0 −2(1 − α) x − 2α ������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By computation, we conclude that ρα is the largest root of the following equation: x3 − (m + 5 2 α + 1)x2 + (m + 5 2 α2 + 5(m − 1) 2 α + 2 − m)x − 2mα2 − (m − 5)α + m − 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then f(α, m) > 0 and g(α, m) < 0, where f(α, m) = 2(5α3 − 6α2 + 2α)m5 − 2(123α3 − 156α2 + 60α − 4)m4 + 4(517α3 − 650α2 + 254α − 20)m3 − 4(1783α3 − 2008α2 + 664α − 16)m2 + 2(5377α3 − 5014α2 + 1202α + 136)m − 2(2951α3 − 2052α2 + 228α + 196) and g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 − 135α2 + 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' The first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' second and third order partial derivatives of function f(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' m) with respect to m are as follows: fm(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' m) = 10(5α3 − 6α2 + 2α)m4 − 8(123α3 − 156α2 + 60α − 4)m3 + 12(517α3 − 650α2 + 254α − 20)m2 − 8(1783α3 − 2008α2 + 664α − 16)m + 2(5377α3 − 5014α2 + 1202α + 136),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' fmm(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' m) = 40(5α3 − 6α2 + 2α)m3 − 24(123α3 − 156α2 + 60α − 4)m2 + 24(517α3 − 650α2 + 254α − 20)m − 8(1783α3 − 2008α2 + 664α − 16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' fmmm(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' m) = 120(5α3 − 6α2 + 2α)m2 − 48(123α3 − 156α2 + 60α − 4)m + 24(517α3 − 650α2 + 254α − 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since α ∈ [ 1 2, 1), then 5α3 − 6α2 + 2α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let m0 be the minimum point of fmmm(α, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then we have m0 = 48(123α3 − 156α2 + 60α − 4) 240 (5α3 + 6α2 − 2α) = 123α3 − 156α2 + 60α − 4 5(5α3 − 6α2 + 2α) < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that fmmm(α, m) is increasing for m ≥ 9 and fmmm(α, m) ≥ fmmm(α, 9) = 96(82α3 − 68α2 − 4α + 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to verify that 82α3 − 68α2 − 4α + 13 > 0 for α ∈ [ 1 2, 1), that is, fmmm(α, m) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that fmm(α, m) is increasing for m ≥ 9 and fmm(α, m) ≥ fmm(α, 9) = 64(64α3 + 62α2 − 137α + 56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Note that 64α3 + 62α2 − 137α + 56 > 0 for α ∈ [ 1 2, 1), that is, fmm(α, m) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that fm(α, m) is increasing for m ≥ 9 and fm(α, m) ≥ fm(α, 9) = −32(137α3 − 590α2 + 538α − 166).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is obvious that 137α3 − 590α2 + 538α − 166 < 0 for α ∈ [ 1 2, 1), that is, fm(α, m) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that f(α, m) is increasing for m ≥ 9 and f(α, m) ≥ f(α, 9) = −64(43α3 − 117α2 + 69α − 22) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Similarly, we have g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 − 135α2 + 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 In this section, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G be a minimally 2-connected graph with order n ≥ 5, then ∆(G) ≤ n − 2 by 6 Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Notice that K2,n−2 is a minimally 2-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='9, we have ρα(K2,n−2) = 1 2(αn + � α2n2 + 8(n − 2)(1 − 2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' When ∆(G) = n − 2, it is easy to see that G ∼= K2,n−2 by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2, we have ρα(K2,n−2) ≥ α∆(K2,n−2) + (1 − α)2 α = α(n − 2) + (1 − α)2 α (1) for α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Thus we assume that ∆(G) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let w be a vertex of G such that αd(w) + 1 − α d(w) � wv∈E(G) d(v) = max u∈V (G) � � �αd(u) + 1 − α d(u) � uv∈E(G) d(v) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1, we have ρα(G) ≤ αd(w) + 1 − α d(w) � wv∈E(G) d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (2) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3, we have 2 ≤ d(w) ≤ ∆(G) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4, we know that N(w) is an independent set, that is, e(N(w)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then � wv∈E(G) d(v) = 2e(N(w)) + e(N(w), V (G)\\N(w)) = e(N(w), V (G)\\N(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (3) Next we prove that ρα(G) ≤ ρα(K2,n−2) for 2 ≤ d(w) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' d(w) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If e(V (G)\\N[w]) = 0, then G ∼= K2,n−2 by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If e(V (G)\\N[w]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We assume that there exists an edge v1v2 ∈ E(G[V (G)\\N[w]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4, we have NN(w)(v1) ∩ NN(w)(v1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let B = V (G)\\(N[w] ∪ {v1, v2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then e(N(w), V (G)\\N[w]) ≤ dN(w)(v1) + dN(w)(v2) + e(N(w), B) = d(w) + d(w)|B| = 2n − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' and so � wv∈E(G) d(v) ≤ 2n − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining this with (2), we have ρα(G) ≤ 2α + 1 − α 2 � wv∈E(G) d(v) ≤ 2α + (1 − α)(n − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Noting that α(n − 2) + (1 − α)2 α − (2α + (1 − α)(n − 3)) = (2α2 − α)n − 6α2 + α + 1 α ≥ 0 for n ≥ 5 and α ∈ [ 1 2, 1), we have ρα(G) ≤ α(n − 2) + (1−α)2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining this with (1), we have ρα(G) ≤ α(n − 2) + (1 − α)2 α ≤ ρα (K2,n−2) 7 for n ≥ 5 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 3 ≤ d(w) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In order to prove ρα(G) ≤ ρα(K2,n−2), it is enough to prove ρα(G) ≤ 1 2(αn + � α2n2 + 8(n − 2)(1 − 2α)), that is, to prove ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For convenience, we denote Aα(G) = Aα, A(G) = A and D(G) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let B = (bij)n×n = A2 α − αnAα + 2(2α − 1)(n − 2)In, where In is the n × n unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let cu(B) be the sum of all elements in the u-th column of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' cu(B) ≤ 0 for n ≥ 5 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since Aα = αD + (1 − α)A, then B =(αD + (1 − α)A)2 − αn(αD + (1 − α)A) + 2(2α − 1)(n − 2)In =α2D2 + (1 − α)2A2 + α(1 − α)DA + α(1 − α)AD − α2nD − � αn − α2n � A + 2(2α − 1)(n − 2)In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to see that cu(A) = cu(D) = d(u), cu(A2) = cu(DA) = � uv∈E(G) d(v) and cu(AD) = d2(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining (3) with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='5, we have � wv∈E(G) d(v) = e(N(w), V (G)\\N(w)) ≤ |E(G)| ≤ 2n − 4, with equality if and only if G ∼= K2,n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that cu(B) =α2d2(u) + (1 − α)2 � uv∈E(G) d(v) + α(1 − α) � uv∈E(G) d(v) + α(1 − α)d2(u) − α2nd(u) − � αn − α2n � d(u) + 2(2α − 1)(n − 2) =α2d2(u) + (1 − α) � uv∈E(G) d(v) − αnd(u) + 2(2α − 1)(n − 2) ≤α2d2(u) + (1 − α)(2n − 4) − αnd(u) + 2(2α − 1)(n − 2) =α � d2(u) − nd(u) + 2n − 4 � ≤ max � α(9 − 3n + 2n − 4), α � (n − 3)2 − n(n − 3) + 2n − 4 �� =α(−n + 5) ≤ 0 for n ≥ 5 and α ∈ [ 1 2, 1), with equality if and only if G ∼= K2,n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , xn)T be the α-Perron vector of G corresponding to ρα(G) satisfying �n i=1 xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then BX = � ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 8 Hence we have ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) = n � i=1 � ρα(G)2 − αnρα(G) + 2(2α − 1)(n − 2) � xi = n � i=1 (BX)i = n � i=1 ( n � j=1 bijxj) = n � j=1 ( n � i=1 bij)xj = n � j=1 cj(B)xj ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining the above arguments, we have ρα(G) ≤ ρα(K2,n−2) for n ≥ 5 and α ∈ [ 1 2, 1), with equality if and only if G ∼= K2,n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' These complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4 In this section, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let G be a minimally 2-connected graph with size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For any v ∈ V (G), it is easy to see that G − v is connected and |E(G − v)| = m − d(v), then we have d(v) < |V (G − v)| ≤ m − d(v) + 1, with equality in the right inequality if and only if G − v is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that d(v) < m+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining this with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3, we have 2 ≤ d(v) < m+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let w be a vertex of G such that αd(w) + 1 − α d(w) � wv∈E(G) d(v) = max u∈V (G) � � �αd(u) + 1 − α d(u) � uv∈E(G) d(v) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1, we have ρα(G) ≤ αd(w) + 1 − α d(w) � wv∈E(G) d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (4) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4, we know that N(w) is an independent set, that is, e(N(w)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then � wv∈E(G) d(v) = 2e(N(w)) + e(N(w), V (G)\\N(w)) = e(N(w), V (G)\\N(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (5) Combining (4) and (5), we have ρα(G) ≤ αd(w) + 1 − α d(w) e(N(w), V (G)\\N(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (6) (i) Let m ≥ 6 be an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then we have 2 ≤ d(w) ≤ m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Notice that K2, m 2 is a minimally 2-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='9, we have ρα(K2, m 2 ) = 1 4((m + 4)α + � (m + 4)2α2 + 16m(1 − 2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 9 If d(w) = m 2 , then d(v) = 2 for any v ∈ N(w) and e(N(w), V (G)\\N[w]) = m 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since G − w is connected, then G ∼= K2, m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Next we prove that ρα(G) ≤ ρα(K2, m 2 ) for 2 ≤ d(w) ≤ m−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' d(w) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If e(V (G)\\N[w]) = 0, then G ∼= K2, m 2 by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If e(V (G)\\N[w]) ̸= 0, then e(N(w), V (G)\\N(w)) ≤ m − 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining this with (6), we have ρα(G) ≤ 2α + 1 − α 2 (m − 1) ≤ m + 3 4 < m + 4 4 for m ≥ 6 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In order to prove ρα(G) < ρα(K2, m 2 ), it is enough to prove m + 4 4 ≤ 1 4((m + 4)α + � (m + 4)2α2 + 16m(1 − 2α)), that is, to prove (1 − 2α)m2 − 8(1 − 2α)m + 16(1 − 2α) ≤ 0 for m ≥ 6 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to check that this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence we have ρα(G) < ρα(K2, m 2 ) for m ≥ 6 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 3 ≤ d(w) ≤ m−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In order to prove ρα(G) ≤ ρα(K2, m 2 ), it is enough to prove ρα(G) ≤ 1 4((m + 4)α + � (m + 4)2α2 + 16m(1 − 2α)), that is, to prove 2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For convenience, we denote Aα(G) = Aα, A(G) = A and D(G) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let B = (bij)n×n = 2A2 α − (m + 4)αAα + 2(2α − 1)mIn, where In is the n × n unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let cu(B) be the sum of all elements in the u-th column of matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' cu(B) ≤ 0 for m ≥ 6 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since Aα = αD + (1 − α)A, then B =2(αD + (1 − α)A)2 − (m + 4)α(αD + (1 − α)A) + 2(2α − 1)mIn =2α2D2 + 2(1 − α)2A2 + 2α(1 − α)DA + 2α(1 − α)AD − (m + 4)α2D − (m + 4)α(1 − α)A + 2(2α − 1)mIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to see that cu(A) = cu(D) = d(u), cu(A2) = cu(DA) = � uv∈E(G) d(v) and cu(AD) = d2(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since e(N(w), V (G)\\N(w)) ≤ |E(G)| = m, then � wv∈E(G) d(v) ≤ m by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that cu(B) =2α2d2(u) + 2(1 − α)2 � wv∈E(G) d(v) + 2α(1 − α) � wv∈E(G) d(v) + 2α(1 − α)d2(u) − (m + 4)α2d(u) − (m + 4)α(1 − α)d(u) + 2(2α − 1)m =2α2d2(u) + 2(1 − α) � wv∈E(G) d(v) − (m + 4)αd(u) + 2(2α − 1)m ≤2αd2(u) − (m + 4)αd(u) + 2mα 10 =α(2d2(u) − (m + 4)d(u) + 2m) ≤ max � α(18 − 3(m + 4) + 2m), α �(m − 2)2 2 − (m + 4)(m − 2) 2 + 2m �� =α(−m + 6) ≤ 0 for m ≥ 6 and α ∈ [ 1 2, 1), with equality if and only if G ∼= K2, m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , xn)T be the α-Perron vector of G corresponding to ρα(G) satisfying �n i=1 xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then BX = � 2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence we have 2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m = n � i=1 � 2ρα(G)2 − (m + 4)αρα(G) + 2(2α − 1)m � xi = n � i=1 (BX)i = n � i=1 ( n � j=1 bijxj) = n � j=1 ( n � i=1 bij)xj = n � j=1 cj(B)xj ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining the above arguments, we have ρα(G) ≤ ρα(K2, m 2 ) for m ≥ 6 and α ∈ [ 1 2, 1), with equality if and only if G ∼= K2, m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We complete the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' (ii) Let m ≥ 9 be an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then we have 2 ≤ d(w) ≤ m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Notice that SK2, m 2 is a minimally 2-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Next we complete the proof with three facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' e(N(w), V (G)\\N(w)) ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since e(N(w), V (G)\\N(w)) ≤ |E(G)| = m, then � wv∈E(G) d(v) ≤ m by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' For a contradiction, we suppose e(N(w), V (G)\\N(w)) = m, then e(V (G)\\N[w]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' d(w) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' In this case, we have d(v) = 2 for any v ∈ V (G)\\N(w) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This implies that G is a complete bipartite graph K2,b and m(K2,b) = 2b, which contradicts the fact that m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 3 ≤ d(w) ≤ m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let v1, v2 ∈ V (G)\\N[w] be any two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If NN(w)(v1)∩NN(w)(v2) = ∅ , then d(wi) = 2 for any wi ∈ N(w) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that m = e(N(w), V (G)\\N(w)) = 2d(w) ≤ m−1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If NN(w)(v1) ∩ NN(w)(v2) ̸= ∅ and NN(w)(v1) ̸= NN(w)(v2), we assume w12 ∈ NN(w)(v1) ∩ NN(w)(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3, there exists wi ∈ NN(w)(vi)\\w12 for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence, G contains a cycle ww1v1w12v2w2w with a chord ww12, which contradicts Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' If NN(w)(v1) = NN(w)(v2) = N(w), then we have δ(G) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This contradicts with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence NN(w)(v1) = NN(w)(v2) ̸= N(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that G − w is disconnected, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Through the above two cases, we know that e(N(w), V (G)\\N(w)) ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ρα(G) < ρα(SK2, m−1 2 ) for 3 ≤ d(w) ≤ m−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining Fact 1 and (6), we have ρα(G) ≤ αd(w) + 1 − α d(w) (m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 11 Let q(x) = αx + 1−α x (m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Since α ∈ [ 1 2, 1), it is easy to see that the function q(x) is convex for x > 0 and its maximum in any closed internal is attained at one of the ends of this internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Hence when 3 ≤ x ≤ m−3 2 , we have ρα(G) ≤ max � 3α + 1 − α 3 (m − 1), m − 3 2 α + 2(1 − α) m − 3 (m − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Noting that m − 3 2 α + 2(1 − α) m − 3 (m − 1) − (3α + 1 − α 3 (m − 1)) = (5α − 2)m2 − (56α − 20)m + 99α − 18 6(m − 3) > 0 for m ≥ 9 and α ∈ [ 1 2, 1), we have ρα(G) ≤ m − 3 2 α + 2(1 − α) m − 3 (m − 1) for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='10, we know ρα(SK2, m−1 2 ) is the largest root of the following equation: p(x) = x3 − (m + 5 2 α + 1)x2 + (m + 5 2 α2 + 5(m − 1) 2 α + 2 − m)x − 2mα2 − (m − 5)α + m − 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By computation, we have −8(m − 3)3 · p(m − 3 2 α + 2(1 − α) m − 3 (m − 1)) = f(α, m), where f(α, m) = 2(5α3 − 6α2 + 2α)m5 − 2(123α3 − 156α2 + 60α − 4)m4 + 4(517α3 − 650α2 + 254α − 20)m3 − 4(1783α3 − 2008α2 + 664α − 16)m2 + 2(5377α3 − 5014α2 + 1202α + 136)m − 2(2951α3 − 2052α2 + 228α + 196).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='11, we have f(α, m) > 0 for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that p(m − 3 2 α + 2(1 − α) m − 3 (m − 1)) < 0 for α ∈ [ 1 2, 1) and m ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This implies that ρα(G) ≤ m − 3 2 α + 2(1 − α) m − 3 (m − 1) < ρα(SK2, m−1 2 ) for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' ρα(G) ≤ ρα(SK2, m−1 2 ) for d(w) = 2 or d(w) = m−1 2 , with equality if and only if G ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' d(w) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We consider the following two cases and assume N(w) = {w1, w2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' e(N(w), V (G)\\N(w)) = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 12 Since e(N(w), V (G)\\N(w)) = m − 1, then e(V (G)\\N[w]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let e(V (G)\\N[w]) = v1v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4, we can see that w1(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' w2) is adjacent to only one vertex of v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Without loss of generality, we assume that w1v1 ∈ E(G) and w2v2 ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3, we have NN(w)(v) = {w1, w2} for any v ∈ V (G)\\(N[w] ∪ {v1, v2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that G ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' e(N(w), V (G)\\N(w)) ≤ m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By (6), we have ρα(G) ≤ 2α + 1 − α 2 (m − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='10, we know ρα(SK2, m−1 2 ) is the largest root of the following equation: p(x) = x3 − (m + 5 2 α + 1)x2 + (m + 5 2 α2 + 5(m − 1) 2 α + 2 − m)x − 2mα2 − (m − 5)α + m − 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By computation, we have 4 · p(2α + 1 − α 2 (m − 2)) = g(α, m), where g(α, m) = (2α4 + 6α3 − 9α2 + 1)m3 + (8α4 + 9α3 − 34α2 + α)m2 − 2(35α4 + 154α3 − 135α2 + 20α + 14)m − 4(75α4 − 79α3 + 137α2 − 21α − 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='11, we have g(α, m) < 0 for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that p(2α + 1 − α 2 (m − 2)) < 0 for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This implies that ρα(G) ≤ 2α + 1 − α 2 (m − 2) < ρα(SK2, m−1 2 ) for m ≥ 9 and α ∈ [ 1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' d(w) = m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3, we have e(N(w), V (G)\\N(w)) ≥ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining this with Fact 1, we have e(N(w), V (G)\\N(w)) = m − 1, that is, e(V (G)\\N[w]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It follows that d(wi) = 2 for wi ∈ N(w) by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let e(V (G)\\N[w]) = v1v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then V (G)\\N[w] = {v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Otherwise G − w is disconnected, a contridiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' This implies that G = G(a, b), where 1 ≤ a ≤ b and a + b = m−1 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We assume NN(w)(v1) = {w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , wa} and NN(w)(v2) = {wa+1, wa+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , wa+b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' It is easy to see that G(1, m−3 2 ) ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Let X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , xn)T be the α-Perron vector of G corresponding to ρα(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Without loss of generality, we assume xv1 ≤ xv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Then G(1, m−3 2 ) = G(a, b) − {v1wi : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , a − 1} + {v2wi : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' , a − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content='7, we have ρα(G) = ρα(G(a, b)) ≤ ρα(G(1, m − 3 2 )) = ρα(SK2, m−1 2 ) for m ≥ 9 and α ∈ [ 1 2, 1), with equality if and only if a = 1, that is, G ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Combining Facts 2 and 3, we have ρα(G) ≤ ρα(SK2, m−1 2 ) for m ≥ 9 and α ∈ [ 1 2, 1), with equality if and only if G ∼= SK2, m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' We complete the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' 13 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} +page_content=' Berman, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E1T4oBgHgl3EQfugV-/content/2301.03389v1.pdf'} 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Bar and Ronen Talmon Senior Member, IEEE +Abstract—We consider the problem of estimating the direction +of arrival of desired acoustic sources in the presence of multiple +acoustic interference sources. All the sources are located in noisy +and reverberant environments and are received by a microphone +array. We propose a new approach for designing beamformers +based on the Riemannian geometry of the manifold of Hermitian +positive definite matrices. Specifically, we show theoretically that +incorporating the Riemannian mean of the spatial correlation +matrices into frequently-used beamformers gives rise to beam +patterns that reject the directions of interference sources and +result in a higher signal-to-interference ratio. We experimentally +demonstrate the advantages of our approach in designing several +beamformers in the presence of simultaneously active multiple +interference sources. +Index Terms—Array Signal Processing, Direction of Arrival +Estimation, Interference Rejection, Hermitian Positive Semidefi- +nite Matrices, Riemannian Geometry. +I. INTRODUCTION +E +STIMATION of the direction of arrival (DoA) of an +acoustic source is prevalent in signal processing; it is +an important step in many tasks, such as source localization, +beamforming, source separation, spectrum sensing, and speech +enhancement [1], to name but a few. Despite the large research +attention it has drawn in the past decades, acoustic DoA +estimation is still considered a challenging open problem. +Especially in noisy and reverberant environments and in the +presence of interference sources, it continues to be an active +research field. +Acoustic source localization, and particularly DoA estima- +tion, are often addressed using beamforming [2]. Many beam- +formers have been proposed over the years for these tasks. +One class of beamformers is based on the steered response +power of a beamformer output. For example, considering the +maximum likelihood criterion for a single source, the output +power of the beamformer from all the directions is computed, +and the DoA is identified as the direction with the maximal +power [3]–[6]. Another example is the Minimum Variance +Distortionless Response (MVDR) beamformer [7]–[9], which +was first introduced by Capon [10]. The MVDR beamformer +extracts the DoA of each of the existing sources, maintaining a +unit gain at their direction while minimizing the response from +other directions. An important generalization of the MVDR +beamformer is the Linearly Constrained Minimum Variance +(LCMV) beamformer [11], obtained by minimizing the output +A. Bar and R. Talmon are with the Viterbi Faculty of Electrical and Com- +puter Engineering, Technion—Israel Institute of Technology, Haifa 32000, +Israel (e-mail: amitayb@campus.technion.ac.il; ronen@ef.technion.ac.il). This +work was supported by the European Union’s Horizon 2020 research and +innovation programme under grant agreement No. 802735-ERC-DIFFOP. +power under multiple linear constraints, and can be used for +DoA estimation as well [12]. Another line of beamformers is +derived based on a subspace approach, i.e., by identifying the +subspace of the desired sources, which is assumed to contain +only a small portion of the noise and the interference sources. +A prominent subspace method, which is also used for DoA +estimation, is MUltiple Signal Classification (MUSIC) [13]– +[16]. +In this paper, we consider DoA estimation in a reverberant +enclosure consisting of desired sources along with interference +sources. We assume the desired sources are constantly active, +whereas the interference sources are only intermittently active. +The number of sources, their locations, and their times of +activity are all unknown. Consequently, their identification as +desired or interference is unknown as well. The power of +the different sources is also unknown, and the interference +sources could, in fact, be stronger than the desired sources +with overlapping activity periods. Our goal is to estimate +the DoA of the desired sources in the presence of possibly +simultaneously active, multiple interference sources. +This setting poses a major challenge to the common practice +in existing methods that rely on maximal power because +estimating the DoA of the strongest sources might result in +distinct beams in the direction of the interference sources +rather than the desired sources. Furthermore, these beams +could mask the beams pointing at the directions of desired +sources. +We address this challenge from a geometric standpoint. Our +approach relies on the observation that the frequently-used +beamformers implicitly consider Euclidean geometry when +processing sample correlation matrices. Therefore, since the +sample correlation matrices are Hermitian Positive Definite +(HPD) matrices, important geometric information is not fully +utilized. Instead, we propose a new approach for beamforming +design that is based on the Riemannian geometry of the +manifold of HPD matrices [17], [18]. Concretely, we analyze +the received signal in short time windows and consider the +Riemannian mean [17] of the sample correlation matrices in +these windows. Then, we leverage particular spectral proper- +ties of the Riemannian mean. In [19], it was shown that the +Riemannian mean of HPD matrices preserves shared spectral +components and attenuates unshared spectral components. +Consequently, the continual activity of the desired sources and +the intermittent activity of the interference sources enable us +to associate desired sources with shared spectral components +and interference sources with unshared spectral components. +By combining the above, we show that the incorporation of +the Riemannian mean into the beamformer design leads to + +2 +interference rejection, i.e., gives rise to beam patterns that +implicitly reject the beams pointing at the interference sources +and preserve the beams pointing at the desired sources. The +resulting beam patterns are, in turn, used for the estimation of +the DoA of the desired sources. Importantly, our approach is +applicable to a large number of beamformers used for DoA +estimation. +By incorporating our Riemannian approach, we present new +implementations of several beamformers for DoA estimation +of the desired sources: the Delay and Sum (DS) beamformer, +subspace-based beamformers, and the MVDR beamformer. +We show that the Riemannian geometry preserves better the +desired sources subspace in comparison to the Euclidean +geometry. Additionally, we analytically show that, when in- +corporated into a DS beamformer, the proposed Riemannian +approach results in a higher Signal to Interference Ratio +(SIR) compared to its Euclidean counterpart. Furthermore, +we show that the lower the noise is, the advantage of the +Riemannian approach over the Euclidean approach increases. +Experimentally, we showcase the performance of the differ- +ent beamformers in reverberant environments, including in +the presence of simultaneously active multiple interference +sources, whose locations are unknown. In particular, we focus +on demonstrating the advantages of the Riemannian approach +over the standard Euclidean approach, providing empirical +support for the analytical results. +We conclude the introduction with three remarks. First, a +similar setting to ours, consisting of desired sources accom- +panied by interference sources, was considered in [20] and +[21] but in the context of signal enhancement. In [20], a +single desired source and a single interference source were +considered, and in [21], multiple desired sources and multiple +interference sources were considered. However, in both works, +it was assumed that there is at least one segment for each +source, desired or interference, in which it is the only active +source. Furthermore, in [21], the number of the desired sources +and their activity patterns were assumed to be available. +Second, in the context of radar, the Riemannian geometry of +the Toeplitz HPD matrices was used in [22] and [23] for target +detection by comparing Riemannian distances to a threshold. +In the radar settings, [24] estimated the correlation matrix as a +linear combination of correlation matrices, with weights that +are based on the Riemannian distance. Third, in this paper, +we demonstrate the Riemannian approach for designing beam- +formers that reject interference sources for DoA estimation. +However, other applications, e.g., signal enhancement, could +also benefit from beam patterns that reject interference sources. +This paper is organized as follows. In section II, we present +a brief background on the HPD manifold. In Section III, +we formulate the problem and the setting. In Section IV we +describe the proposed approach and present the algorithm +for DoA estimation. In Section V, we provide a theoretical +analysis of the proposed approach for the DS beamformer. In +Section VI, extensions of the approach to other beamformers +are presented. Section VII shows simulation results demon- +strating our Riemannian approach. Lastly, we conclude the +work in Section VIII. +II. BACKGROUND ON THE HPD MANIFOLD +An HPD matrix, Γ ∈ Cn×n, is a Hermitian matrix, i.e. Γ = +ΓH, where ΓH is the conjugate transpose of Γ, whose real +eigenvalues are strictly positive. Associating the space of HPD +matrices with the Affine Invariant metric [25] constitutes a +Riemannian manifold, M. The distance between two matrices +Γ1 and Γ2, induced by the Affine Invariant metric, is given +by +d2 +R(Γ1, Γ2) = +��� log +� +Γ +− 1 +2 +2 +Γ1Γ +− 1 +2 +2 +� ��� +2 +F , +(1) +where ∥ · ∥F is the Frobenius norm. +The Riemannian mean, ΓR, of a set of point {Γi|Γi ∈ M} +is defined by the Fr´echet mean as follows +ΓR ≡ arg min +Γ∈M +� +i +d2 +R(Γ, Γi). +(2) +In general, there is no closed-form expression for the Rieman- +nian mean of more than two matrices, and a solution can be +found using an iterative procedure [26]. The computation of +the Riemannian mean requires two maps. The Logarithm map, +which projects an HPD matrix Γi ∈ M to the tangent space +of the HPD manifold at Γ, is given by +LogΓ(Γi) = Γ +1 +2 log(Γ− 1 +2 ΓiΓ− 1 +2 )Γ +1 +2 . +(3) +The Exponential map, which projects a vector T from the +tangent space at Γ, is given by +ExpΓ(T ) = Γ +1 +2 exp(Γ− 1 +2 T Γ− 1 +2 )Γ +1 +2 . +(4) +A discussion about the choice of the Affine Invariant metric +appears in Appendix D. +III. PROBLEM FORMULATION +We consider the problem of localizing ND desired sources +in the presence of NI interference sources. All the sources are +static and located in a reverberant environment. The signals +are received at a noisy microphone array of M microphones, +which are positioned in known, but possibly arbitrary, posi- +tions. The acoustic environment between each source and each +microphone is modeled by the Acoustic Impulse Response +(AIR). The signal at the mth microphone is given by: +zm(n) = +ND +� +j=1 +sd +j(n) ∗ hd +jm(n) + +NI +� +j=1 +si +j(n) ∗ hi +jm(n) ++ vm(n), +(5) +where sd +j(n) is the jth desired source, si +j(n) is the jth +interference source, and vm(n) is the mth microphone noise. +We denote by hd +jm(n) the AIR between the mth microphone +and the jth desired source. Similarly, we denote by hi +jm(n) +the AIR between the mth microphone and the jth interference +source. +The sources are characterized by their activation times. The +desired sources are active during the entire interval. In contrast, +the interference sources are only partially active, namely active +during segments of the interval. Additionally, we assume that +the desired sources, the interference sources, and the noise are +all uncorrelated. The noise is assumed to be spatially white. + +3 +The received signal is processed using the Short-Time +Fourier Transform (STFT). We denote by sd +j(l, k) the STFT +at the lth window and the kth frequency of sd +j(n). The +notation for si +j(l, k), hd +jm(l, k), hi +jm(l, k), and vm(l, k) follows +similarly. Then, the STFT at the lth window and the kth +frequency of the received signal is given by +zm(l, k) = +ND +� +j=1 +sd +j(l, k)hd +jm(l, k) + +NI +� +j=1 +si +j(l, k)hi +jm(l, k) ++ vm(l, k), +(6) +where we assume the length of the window is much larger +than the AIR length. +We stack the received signals, {zm(l, k)}m, from all the +microphones to obtain a column vector z(l, k) ∈ CM×1 +z(l, k) = [z1(l, k) ... zM(l, k)]⊤. +(7) +Its explicit expression is +z(l, k) = Hd(l, k)sd(l, k) + Hi(l, k)si(l, k) + v(l, k), (8) +where sd(l, k) and si(l, k) denote the stacked STFT repre- +sentations of the desired sources and the interference sources, +respectively, and are given by +sd(l, k) = [sd +1(l, k) ... sd +ND(l, k)]⊤ +si(l, k) = [si +1(l, k) ... si +NI(l, k)]⊤, +(9) +and the noise term is +v(l, k) = [v1(l, k) ... vM(l, k)]⊤. +(10) +The Acoustic Transfer Functions (ATFs) from the jth desired +source and the jth interference source to the microphone array +are +hd +j(l, k) = [hd +j1(l, k) ... hd +jM(l, k)]⊤ +j = 1, ..., ND +hi +j(l, k) = [hi +j1(l, k) ... hi +jM(l, k)]⊤ +j = 1, ..., NI, +(11) +and in a matrix form +Hd(l, k) = [hd +1(l, k) ... hd +ND(l, k)] +Hi(l, k) = [hi +1(l, k) ... hi +NI(l, k)]. +(12) +Henceforth, we focus on a single frequency bin and omit +the frequency index. Throughout the paper, we refer to z(l) as +the received signal. Since all the sources are static the ATFs +do not change over time, so in the following, we omit their +STFT window index l. +Our goal is to estimate the direction to the desired sources, +given z(l), +l = 1, . . . , LSTFT, where LSTFT is the number +of STFT windows. The main challenge is the existence of +interference sources, positioned at unknown locations with +possibly high signal power. +IV. PROPOSED APPROACH +Typically, the DoA estimation of a desired source is based +on the output of a beamformer. In this section, we present +the proposed approach applied to the Delay-and-Sum (DS) +beamformer. In Section VI, we extend the proposed approach +to other beamformers. +We consider arbitrary indexing of the microphones in the +array and designate the first microphone as the reference +microphone. Let d(θ) denote the steering vector of the array to +direction θ relative to the first (reference) microphone, which +is given by +d(θ) = [1, ejφ2(θ), ..., ejφM(θ)]⊤, +(13) +where φm(θ) is the phase of the received signal at the mth +microphone with respect to the first microphone. For example, +for a uniform linear array and the typical microphone indexing, +we have φm(θ) = 2π · m δ +λ sin θ, where λ is the wavelength +of the received signal, and δ is the distance between the +microphones. +The weights for the DS beamformer are set according to +the steering vector, and its output is given by +yDS(θ, l) = dH(θ)z(l). +(14) +We refer to the squared absolute value of the output of the +beamformer as the beam pattern. By (14), the DS beam pattern +is given by +|yDS(θ, l)|2 = |dH(θ)z(l)|2 = dH(θ)z(l)zH(l)d(θ). +(15) +Note that the rank-1 matrix z(l)zH(l) could be viewed as +the sample correlation matrix of the population correlation +matrix E[z(l)zH(l)] based on one sample. Since the desired +source is constantly active and assumed to be at a fixed +location during the entire interval, we propose to improve the +correlation estimation by averaging z(l)zH(l) over multiple +STFT windows. We divide the STFT of the signal into Ls +disjoint segments, each consisting of Lw consecutive STFT +windows, i.e., LSTFT = Ls ·Lw (for more details regarding the +partitioning see Section V-D). Then, the sample correlation +matrix over each segment is computed as follows +Γi = 1 +Lw +i·Lw +� +l=(i−1)·Lw+1 +z(l)zH(l), +(16) +where i is the segment index. To obtain a full rank correlation +matrix, the number of STFT windows is set to be larger than +the dimension of the matrix (the number of microphones), i.e., +Lw ≥ M. +The incorporation of the Riemannian geometry is realized +by viewing each matrix Γi as a point on the HPD manifold +[25] and considering their Riemannian mean, denoted as ΓR +and given by: +ΓR = arg min +Γ∈M +Ls +� +i=1 +d2 +R(Γ, Γi). +(17) +In general, there is no closed-form solution to (17) on the HPD +manifold for more than two points [26]. Therefore, Algorithm +1 proposed in [27] is used to compute the Riemannian mean +of the Ls correlation matrices. +Once ΓR is at hand, the DS beam pattern, PDS(θ; ΓR), is +computed by +PDS(θ; ΓR) = dH(θ)ΓRd(θ). +(18) +In the case of a single desired source and assuming the +direct path is dominant in the AIR, the direction to it is set as + +4 +the direction achieving the maximum value of the DS beam +pattern, i.e., +ˆθ = arg max +θ +PDS(θ; ΓR). +(19) +In the case of ND desired sources, the ND directions are set +according to the ND strongest lobes in the beam pattern. The +algorithm for a single desired source is described in Algorithm +2. +Algorithm 1 Riemannian mean for the HPD manifold [27] +Input: a set of K HPD matrices {Γj}K +j=1 +Output: the Riemannian mean ΓR +1: Compute ΓR = 1 +K +�K +j=1 Γj +2: do +1) Compute the Euclidean mean in the tangent plane: +P = 1 +K +�K +j=1 LogΓR(Γj) +2) Update ΓR = ExpΓR(P) +3) Stop if ∥P∥F < ǫ (∥ · ∥F is the Frobenius norm) +Algorithm 2 Direction estimation in the presence of multiple +interference sources +Input: The received signal in the STFT domain {z(l)}LSTFT +l=1 +Output: The estimated direction of the desired source ˆθ +1: Divide {z(l)}LSTFT +l=1 +into Ls consecutive segments +2: For each segment i, compute the correlation matrix Γi +using (16) +3: Compute ΓR of the set {Γi}LS +i=1 using Algorithm 1 +4: Compute PDS(θ; ΓR) according to (18) +5: Return ˆθ = arg maxθ PDS(θ; ΓR) +As a baseline, we consider the common practice of the +DS beam pattern computation, which is typically based on +the sample correlation matrix over the entire interval, i.e., +PDS(θ; ΓE), where +ΓE = +1 +LSTFT +LSTFT +� +l=1 +z(l)zH(l). +(20) +We observe that computing the Euclidean mean of the +correlation matrices per segment, {Γi}Ls +i=1, results in ΓE in +(20). So, ΓE in (20) is the Euclidean counterpart of ΓR, the +correlation matrix resulting from the Riemannian approach. +We will show that our Riemannian approach exploits the +assumption that the desired source is constantly active and at +a fixed location, whereas the interference sources are inter- +mittent. More specifically, we will show both theoretically in +Section V and empirically in Section VII that the Riemannian +mean attenuates the intermittent interferences while preserving +the constantly active sources. In contrast, the standard Eu- +clidean mean accumulates all the sources, and as a result, the +main lobe could deviate from the direction of a desired source, +and even focus on an interference source. +We show in Section V and Section VII that our proposed +approach results in a beam pattern that rejects the interference +sources, allowing the beamformer to extract the DoA of the +desired sources. +We remark that the proposed approach only requires that the +desired sources are the only sources active during the entire +interval. Unlike other works (e.g. [21]), we do not need to +know the activation times of each interference, nor the number +of interference sources. Furthermore, we do not assume that +there exists a segment, at which a desired source is the only +active source, namely, it could always be accompanied by +interference sources. +In terms of complexity, the Riemannian approach requires +the computation of the Riemannian mean of the correlation +matrices, which is more complex than the Euclidean mean. +However, the dimension of the correlation matrices is deter- +mined by the number of microphones in the array, which is +typically not high. Furthermore, there is an efficient estimator +for the Riemannian mean, which is updated iteratively. In +Appendix C, we present the estimator along with an imple- +mentation for the streaming data setting. +A. Performance Evaluation +To evaluate the performance of the proposed approach, we +define the output Signal to Interference Ratio (SIR) as follows: +SIRj(Γ) = P(θd; Γ) +P(θi +j; Γ) , +(21) +where P(θ; Γ) is the beam pattern computed using the corre- +lation matrix Γ, θd is the direction of a desired source, and +θi +j is the direction of the jth interference. When using the DS +beamformer, the output SIR becomes +SIRj(Γ) = dH(θd)Γd(θd) +dH(θi +j)Γd(θi +j) . +(22) +This measure of performance is used because the main +challenge in this setting is the presence of interference sources +rather than the microphone noise. +V. ANALYSIS +In this section, we analyze the proposed approach which is +based on Riemannian geometry and compare it to its Euclidean +counterpart. The proofs of the statements appear in Appendix +E. +We begin with a short derivation, demonstrating that Rie- +mannian geometry preserves better the desired source sub- +space in comparison to Euclidean geometry. Consider a single +desired source and assume its ATF, denoted by h0, is a com- +mon eigenvector of all the correlation matrices per segment +associated with the same eigenvalue (this assumption is made +formal in Assumption 1 in the sequel). In this case, according +to Lemma 2 (see Appendix D), h0 is an eigenvector of both +means and it is associated with the same eigenvalue, namely +λ0(ΓR) = λ0(ΓE), where λi(Γ) is the ith eigenvalue of Γ. +In addition, all other eigenvectors span the interference and +noise subspaces. By the following relation between the means +of the two geometries [28] +ΓR ⪯ ΓE. +(23) + +5 +we get that +λ0(ΓR) +�M−1 +i=1 λi(ΓR) +≥ +λ0(ΓE) +�M−1 +i=1 λi(ΓE) +, +(24) +due to the equality of the numerators and (23). The inequality +in (24) implies that the desired source subspace is more +dominant relative to the subspace of the interference and noise +in the Riemannian mean compared to the Euclidean mean. +In the remainder of this section, we extend this analysis and +present additional results. +A. Assumptions +To make the analysis tractable, we consider a single de- +sired source and multiple interference sources. Therefore, we +simplify the notations by omitting the superscripts (·)i and +(·)d associated with the interference sources and the desired +sources and setting the index of the desired source to 0. +For the purpose of analysis, we make the following assump- +tions: +Assumption 1. hH +0 hj = 0, +∀j = 1, ..., NI. +Assumption 2. hH +l hj = 0, +∀l ̸= j. +It follows from Assumption 1 and Assumption 2 that the +ATFs, associated with the desired source and the interference +sources are all uncorrelated. These are common assumptions, +e.g., see [21]. We note that we do not assume there exists +a segment at which only one of the sources is active (e.g., +as in [21]). In case an interference source is only partially +active during a segment, we consider it active during the entire +segment. +In the analysis, we consider the population correlation +matrix of the received signal, neglecting the estimation errors +stemming from the finite sample in a segment. The population +correlation matrix of the ith segment is given by +Γi = σ2 +0h0hH +0 + HΛiHH + σ2 +vIM×M, +(25) +where the diagonal matrix Λi captures the signal power of the +interference sources and is given by: +Λi = diag +� +σ2 +1(i) · Ii∈L1, ... , σ2 +NI(i) · Ii∈LNI +� +, +(26) +where H = Hi(l, k) due to the omission of the indices and +σ2 +j (i) = E[|sj(n)|2|n ∈ ith segment] is the expected signal +power of the jth interference source at the ith segment, Lj +is the set of segments at which the jth interference source is +active, and Ii∈Lj is an indicator function, attaining the value +of 1 when the jth interference is active during the ith segment +and 0 otherwise. We denote by τj = |Lj| +Ls +the relative number +of segments during which the jth interference source is active. +We assume the same expected power at all the segments in +the interval, i.e., σ2 +j (i) = σ2 +j +for all j = 1, . . . , NI and i = +1, . . . , Ls. +We continue with defining the Signal to Noise Ratio (SNR) +as +SNR = σ2 +0 +σ2v +, +(27) +where σ2 +0 and σ2 +v are the power of the desired source and +the power of the noise, respectively, and are assumed fixed +over the interval as well. The power of the desired source in +(27) is considered without the effect of the acoustic channel. +We note that the actual SNR at the microphones is typically +significantly lower due to the attenuation of the acoustic +channel. Since we focus on a single frequency bin, (27) is +the narrowband SNR. +To capture the correlation between the steering vectors and +the ATFs, we define +ρrs = +|⟨dr, hs⟩|2 +∥dr∥2 · ∥hs∥2 = |⟨dr, hs⟩|2 +M∥hs∥2 , +(28) +where r, s = 0, 1, 2, ..., NI, indicating the desired source or an +interference source. +We conclude the preliminaries of the analysis with two +additional assumptions. +Assumption 3. ρrr is fixed ∀r, and ρrs is fixed ∀r ̸= s. +Assumption 3 implies that the correlation between the ATFs +and the steering vectors depends only on whether they are +associated with the same source or not. Following Assumption +3, henceforth we denote κ = ρrr and ρ = ρrs for r ̸= s. +Assumption 4. κ > ρ. +Assumption 4 is typically made in the context of source +localization. It implies that the correlation between a steering +vector to a source and the ATF associated with that source +is higher than the correlation between a steering vector to a +source and the ATF associated with a different source. +B. Main Results +Our first result states that the output SIR (22) of the +Riemannian-based DS beam pattern is higher than the output +SIR of the Euclidean-based DS beam pattern. +Proposition 1. For every interference source j, the following +holds +SIRj(ΓR) > SIRj(ΓE), +(29) +for any number of microphones in the array. +Examining the dependency of the output SIR on the noise +power σ2 +v leads to the following result. +Proposition 2. If +σ2 +0∥h0∥2 ≥ σ2 +j τj∥hj∥2, ∀j, +(30) +then +∂ +∂σ2v +SIRj(ΓR) < +∂ +∂σ2v +SIRj(ΓE) < 0. +(31) +Namely, the lower the noise power is the higher the SIR is, +and the improvement in SIRj(ΓR) is greater than the improve- +ment in SIRj(ΓE). Since we established that the Riemannian +approach is better than the Euclidean one in terms of the SIR +in Proposition 1, Proposition 2 implies that increasing the +SNR further increases the gap between the two approaches. +Nevertheless, it also indicates that the performance of the +Riemannian approach in terms of the SIR is more sensitive + +6 +to noise compared to the Euclidean counterpart. Note that +this statement holds under condition (30), which implies that +the received power of the desired source is stronger than the +received power of each interference source, considering the +attenuation stemming from the activity duration. See more +details in Appendix E-C. +The proofs of Proposition 1 and Proposition 2 rely on the +following lemma, which is important in its own right. +Lemma 1. The Riemannian or the Euclidean mean of the +population correlation matrices of the segments (25) over the +entire interval can be written in the same parametric form as: +Γ = σ2 +0h0hH +0 + +NI +� +j=1 +µ2 +jhjhH +j + σ2 +vI. +(32) +The Riemannian mean ΓR is obtained by setting the parame- +ters µj to +µ2 +j = (σ2 +j ∥hj∥2 + σ2 +v)τj(σ2 +v)1−τj − σ2 +v +∥hj∥2 +, +(33) +and the Euclidean mean ΓE is obtained by setting +µ2 +j = σ2 +j τj. +(34) +We note that only assumptions 1 and 2 are necessary for +this lemma to hold. In addition, we note that if the interference +sources are always active, i.e., |Lj| = Ls, ∀j, it holds that +ΓR = ΓE. +Lemma 1 shows that both ΓR and ΓE, i.e., the population +correlation matrix of the segments in (25), can be decom- +posed into three terms associated with the desired source, +the interference sources, and the noise. By (32), the desired +source term and the noise term (the first and third terms) are +the same in both the Riemannian and the Euclidean means. +In contrast, the coefficients in (33) and (34) imply that the +amplitude of the interference sources term (the second term) +depends on the used geometry. By further inspecting the +expressions of {µ2 +j}, we see that the interference attenuation +using the Riemannian geometry in (33) is more involved than +its Euclidean counterpart in (34), depending not only on the +interference power and the duration of activity but also on the +noise power and the corresponding ATF. +Furthermore, considering µj in (34), the condition (30) +could be viewed as the dominance of the desired source after +the attenuation of the Euclidean mean. Discussion about the +condition (30) for µj in the Riemannian case in (33) appears +in Appendix E-C. +Next, we examine a family of correlation matrices that +pertain to the same parametric form as in (32) in Lemma 1, +i.e., +Γa = h0hH +0 + +NI +� +j=1 +ajhjhH +j + σ2 +vI, +(35) +for some coefficients a = [a1, a2, ..., aNI]. Without loss of +generality, we set the coefficient of h0hH +0 to 1. We note that +Γa is in accordance with (25). +For any j, we have that +Γopt ≡ arg max +Γa +SIRj(Γa) = h0hH +0 + σ2 +vI, +(36) +where a = 0. Consequently, +SIRj(Γopt) = dH +0 (h0hH +0 + σ2 +vI)d0 +dH +j (h0hH +0 + σ2vI)dj +. +(37) +Considering vanishing noise, i.e., when the noise power +approaches zero, the following result stems from Lemma 1 +by considering the limit limσ2v→0 µ2 +j = 0 using (33). +Corollary 1. +lim +σ2v→0 ΓR = Γopt. +(38) +According to Corollary 1, the Riemannian mean approaches +the optimal correlation matrix as the noise becomes negligible. +By adding a condition on the presence of the interference +sources, from Lemma 1 and (33) we also have the following. +Corollary 2. For any interference source j, if τj < 1 +2, then +lim +σ2 +j →∞,σ2v→0 ΓR = Γopt. +(39) +Additionally, if τj < 1 +2 for all j, then +lim +σ2 +j →∞∀j=1,...,NI,σ2 +v→0 ΓR = Γopt. +(40) +Corollary 2 implies that for vanishing noise, even when +all the interference sources have infinite power, the desired +source is still the dominant source in the DS beam pattern +using the Riemannian mean. Following (37) it holds that +limσ2 +j →∞,σ2v→0 SIRj(Γopt) = +κ +ρ +> 1 for all j. For the +Euclidean mean it holds that limσ2 +j →∞,σ2v→0 SIRj(ΓE) = ρ +κ < +1 < κ +ρ = limσ2 +j →∞,σ2v→0 SIRj(Γopt) for all j. Note that we +consider noise power approaching zero rather than strictly +zero, because when σ2 +v = 0, the correlation matrix is singular, +and therefore, lies outside the HPD manifold. Additionally, in +practice, noise is always present. +To illustrate the obtained expressions for the Riemannian +and the Euclidean SIR, we present the following simple +example. +Example 1. Consider an anechoic environment without at- +tenuation, for which κ = 1 and ρ = 0, and two interference +sources. Each interference source is active at a different +segment, i.e., Ls = 2, L1 = {1}, and L2 = {2}. All +the sources have the same power. In this setting, for the +Riemannian geometry, we have +SIRj(ΓR) = +� +M +σ2v ++ 1, +(41) +and for Euclidean geometry, we have: +SIR(ΓE) = 2(M + σ2 +v) +M + 2σ2v +. +(42) +Therefore, in the limit of σ2 +v → 0, or M → ∞, we have +SIR(ΓR) = ∞, whereas SIR(ΓE) ≈ 2. +We conclude this analysis with a few remarks. First, we +note that ΓE leads to the ML estimator by taking ˆθ0 = +arg maxθ PDS(θ; ΓE) for the interference-free setting. In this +case, the Riemannian mean coincides with the Euclidean + +7 +mean, i.e., the proposed method coincides with the ML es- +timator. The main advantage of the proposed method lies in +attenuating the interference sources while preserving the de- +sired source. Second, under assumptions 1 and 2, the number +of sources, both desired and interference, are limited by the +number of microphones in the array, i.e., NI + ND < M. In +Section V-C we alleviate Assumption 2, which removes this re- +striction. Third, following the same techniques in the proof of +Proposition 1 and Proposition 2, similar results are derived for +an alternative definition of the SIR: SIRtot(Γ) ≡ +dH +0 Γd0 +�NI +j=1 dH +j Γdj , +which captures the ratio between the desired source and the +sum of all interference sources. See Appendix F for more +details. +C. Relation to Signal Enhancement +For signal enhancement in reverberant environments, the +estimation of the ATF of the desired source is typically +required. In our setting, there is no segment at which the +desired source is the only active source, and therefore, the +ATF estimation is done in the presence of the interference +sources. In such a case, the following quantity could be of +interest: +SIRj(Γ) = hH +0 Γh0 +hH +j Γhj +, +(43) +which is different than (22) in the use of the ATFs instead of +the steering vectors. +Similarly to Proposition 1, the following Proposition 3 +examines the performance in terms of the SIR defined in +(43). Here, assumptions 2-4 are not required, and therefore, +the ATFs of the interference sources could be correlated, +and the number of sources is not limited by the number of +microphones in the array. +Proposition 3. Under Assumption 1, for all j we have: +SIRj(ΓR) ≥ SIRj(ΓE). +(44) +Another interesting component in signal enhancement is the +Relative Transfer Function (RTF) between different micro- +phones [29]–[31]. We compute the RTFs with respect to the +first microphone, i.e., +hj +hj(1). Since the RTFs are proportional +to the ATFs, assumption 1 holds for the RTFs, and therefore, +all the derived results apply to the RTFs as well. +D. The Segments and the Interference Sources Activity +In this section, we investigate the effect of misalignment +between the segments and the activity of the interference +sources. We consider two interference sources and two seg- +ments. We denote by α the offset between the segments and +the activity of the interference sources. For simplicity, we +consider alternately active interference sources. Suppose the +first interference source is active during α ∈ [0, 1] of the first +segment and during 1−α of the second segment, and suppose +the second interference source is active during 1 − α and α +of the first and second segments, respectively. In this case, the +correlation matrices of the two segments are given by: +Γ1(α) = σ2 +0h0hH +0 + α2σ2 +1h1hH +1 + (1 − α)2σ2 +2h2hH +2 + σ2 +vI +Γ2(α) = σ2 +0h0hH +0 + (1 − α)2σ2 +1h1hH +1 + α2σ2 +2h2hH +2 + σ2 +vI. +(45) +The correlation matrices in (45) depend on α, and as a result, +their Riemannian mean ΓR(α) and their Euclidean mean +ΓE(α) depend on α as well. +Examining the dependency of the SIR on α leads to the +following result. +Proposition 4. For any α ∈ [0, 1], we have +SIR(ΓR(α)) ≥ SIR(ΓE(α)). +(46) +Proposition 4 states that for every misalignment between +the segments and the activity of the interference sources, the +Riemannian mean leads to higher SIR in comparison to its +Euclidean counterpart. Equality in (46) is obtained for α = 1 +2, +which means 50% offset. In this case, it holds that Γ1 = Γ2, +and both means are the same. +Empirically, we found that the advantage of the Riemannian +mean over the Euclidean mean decreases as the offset between +the segments and the activity of the interference sources +increases. We leave the question of optimal partitioning of +the STFT windows into segments to future work. +VI. EXTENSION TO OTHER BEAMFORMERS +In this section, to broaden its applicability, we demonstrate +the incorporation of the Riemannian approach in other beam- +formers. Each beamformer generates a beam pattern from +which the directions to the desired sources are estimated +according to the highest peaks in the beam pattern. +As a subspace (SbSp) approach, we implement MUSIC +[13] in the following way. given ND desired sources, we take +the leading ND eigenvectors of the correlation matrix, Γ, and +construct the signal subspace matrix, U(Γ) ∈ CM×ND, whose +columns are the ND eigenvectors. Then, the SbSp beam pattern +is defined by +PSbSp(θ; Γ) = dH(θ)U(Γ)UH (Γ)d(θ). +(47) +We note that the appropriate number of eigenvectors ND (the +dimension of the subspace) needs to be estimated. +Similarly to the DS beamformer based on the DS beam +pattern in (18) and (19), the Riemannian and the Euclidean +SbSp methods are given by PSbSp(θ; ΓR) and PSbSp(θ; ΓE), +respectively, according to (47). The SbSp method could also +benefit from our Riemannian approach. Recalling assumptions +1 and 2 and the structure of the mean correlation matrices ΓR +and ΓE in (32), we see that h0 is an eigenvector of both ΓR +and ΓE, spanning the signal subspace (we assume a single +desired source for simplicity). Moreover, the vectors {hj} are +also eigenvectors of ΓR and ΓE, spanning the subspace of +the interference sources. SbSp methods typically focus on the +principal eigenvector. Following (32), the principal eigenvector +is determined according to the largest coefficient among σ0 +and {µj}NI +j=1. The parameter µj in (33) for the Riemannian +mean ΓR is smaller than µj in (34) for the Euclidean mean + +8 +ΓE, as proven in Proposition 1. As a result, the signal sub- +space is more dominant relative to the interference subspace +when considering the Riemannian mean in comparison to the +Euclidean mean, implying better results for the Riemannian +SbSp approach. For an interference source with sufficiently +high power, the leading eigenvector of ΓE could span the +interference subspace rather than the signal subspace, whereas +the leading eigenvector of ΓR spans the signal subspace. In +Section VII, we demonstrate these SbSp methods empirically +and show the advantage of the Riemannian approach. +Our approach is also applicable to the MVDR beamformer +[10], whose beam pattern is given by +PMVDR(θ; Γ) = +1 +dH(θ)Γ−1d(θ). +(48) +The typical beam pattern of the MVDR beamformer is ob- +tained by using ΓE in (48), namely PMVDR(θ; ΓE). We propose +to use the Riemannian mean by setting Γ to be ΓR in (48) to +obtain PMVDR(θ; ΓR). +VII. SIMULATION RESULTS +In this section, we demonstrate the performance of the pro- +posed approach based on Riemannian geometry, and compare +it to Euclidean geometry, implicitly considered by the common +practice. Additionally, we compare our approach to a heuristic +method, based on the intersection of subspaces. The intersec- +tion leads to the rejection of non-common components, such +as the interference sources subspace, and preserves common +components, such as the desired source subspace. We refer to +it as the intersection beamformer (see Appendix A for more +details). +We consider a reverberant enclosure of dimensions 5m × +4m × 3.5m consisting of a microphone array with M = 12 +microphones. The AIRs between the different sources and the +array are generated based on the image method [32], as im- +plemented by the simulator in [33]. The sampling frequency is +16KHz, and the length of the AIRs is set to 2048 samples.The +emitted signals are generated as white Gaussian noise. The +desired source is constantly active, whereas the interference +sources are active only intermittently. +The received signal is transformed to the time-frequency +domain using STFT with a window size of 1024 samples with +50% overlap. We test all methods using a single frequency bin, +of index 250, chosen according to the microphone spacing, +which is 4.36cm. Here, the correlation matrix estimation is +based on 16 STFT windows, which results in a segment +duration of 1.024s. +All the sources are positioned on a 140◦ arc of radius 2m +from the center of the array on the XY plane. The heights of +the interference sources vary randomly, uniformly distributed +between 0.5m and 3m. The height of the desired source is set +to 1.8m. Figure 1 presents the room layout, where the (two) +interferences are marked by green squares, the desired source +by a red star, and the microphone array by blue circles. The +leftmost microphone is positioned at (2.0436m, 1m, 2m), and +the rest are positioned 4.36cm apart along the x-axis. While +we focus on this specific configuration, we note that we tested +other configurations that yielded similar results. +0 +1 +2 +3 +4 +5 +x[m] +1 +2 +3 +y[m] +Microphone +Target +Interference +Fig. 1. The reverberant room with the microphone array (blue circles), the +desired source (red star), and the interference sources (green squares). Left: +a 3D view. Right: a 2D view. +We examine the performance of both the DS and the SbSp +methods. Algorithm 2 is used for the proposed Riemannian +DS, and the common practice is implemented by replacing +step 3 in Algorithm 2 with (20). The SbSp methods require +knowing the dimension of the signal space of the mean corre- +lation matrix. For the intersection method only, the dimension +of the signal space of each segment is also required. Since +the desired source may not be the strongest source received at +the array, we need to consider all the active sources, and not +merely the strongest one when estimating the dimension of the +signal subspace. To estimate the dimension, we implement a +heuristic algorithm, based on the spectral gap. We consider the +dimension of the signal space to be the number of eigenvalues +higher than a threshold. For all methods, the threshold for the +mean correlation matrix is the mean value plus the standard +deviation of the eigenvalues (normalized to a unit sum). For the +intersection method, the threshold for the correlation matrix of +each segment is 1.5 times the mean of the eigenvalues of the +correlation matrix. Apart from this practical implementation, +we also present results for an oracle implementation, assuming +the dimension of the signal space is perfectly known. +For quantitative evaluation, we use two metrics. The first is +the mean of the empirical output SIR with respect to all the +interference sources, which is given by: +SIR = 1 +NI +NI +� +j=1 +P(θd) +P(θi +j) , +(49) +where P is the beam pattern computed using the evaluated +method. The second metric is the directivity [34, ch.2], which +is given by +D(Γ) = +P(θd) +1 +2 +� π +0 P(θ) sin(θ)dθ . +(50) +In the first experiment, we consider two interference +sources, each active at a single, but disjoint, segment, resulting +in a signal of 2.048s. The reverberation time is set to 150ms, +and the SNR is 50dB. To remove the dependency on the +particular layout, the SNR follows the definition in (27), which +is with respect to the sources’ emitted power. We note that the +effective SNR at the microphone is significantly lower (as the +power of the source is attenuated by the AIR). Empirically, +the effective SNR at the microphones is approximately 30dB +lower. Furthermore, since we focus on a single frequency bin, +we consider the narrowband SNR, which is typically much +higher than the broadband SNR. + +Microphone +Target +Interference4 +352 +1 +0 +4 +2 +4 +3 +2 +1 +y[m] +0 +0 +x[m]9 +0 +30 +60 +90 +120 +150 +180 +-20 +-15 +-10 +-5 +0 +R (Riem) +E (Euc) +Desired +Interference +0 +30 +60 +90 +120 +150 +180 +-20 +-15 +-10 +-5 +0 +1 (1st segment) +2 (2nd segment +Desired +Interference +Fig. 2. The DS beam pattern using ΓR in solid blue and ΓE in dashed red +(top), and Γ1 and Γ2 in different shades of orange (bottom). The black solid +line indicates the direction of the desired source, and the dashed black lines +indicate the directions to the interference sources. The Input SIR is −6dB. +We start with an example of the DS beam pattern (see (18)), +computed using ΓR and ΓE, which is presented at the top +of Figure 2. The Riemannian DS beam pattern is shown in +solid blue and the Euclidean DS beam pattern is shown in +dashed red. Both beam patterns are in a dB (log) scale. The +directions to the desired source and the interference sources +are represented by a black solid line and a dashed black line, +respectively. We see that by using ΓR the main lobe is directed +towards the desired source. In contrast, the beam pattern using +ΓE is peaked at 2 different directions, none of which is the +direction to the desired source. The bottom of Figure 2 is the +same as the top, only with the beam pattern computed using +the correlation matrix of each of the two segments, presented +in different shades of orange. Even though the main lobes +of the two beam patterns are not pointing toward the desired +source, the Riemannian mean leads to a beam pattern with the +main lobe directed at the desired source, whereas the lobes +to other directions are highly attenuated. We emphasize that +viewing the Euclidean DS beam pattern in addition to the +beam pattern of each segment separately does not allow correct +identification of the desired source. +Next, we randomly generate 200 different pairs of positions +for the interference sources. For each pair, the target source +is located at 20 different equally spaced directions along the +arc (with the height of 1.8m). Thus, in total, 4000 different +scenarios are examined. +Figure 3 presents the mean output SIR (top) and the +directivity (bottom) for the DS method using the correlation +matrix estimates: ΓR (based on Riemannian geometry) in +blue and ΓE (based on Euclidean geometry) in red. The box +indicates the 25th and 75th percentiles, and the line marks +the median. We test the different methods, Riemannian or +Euclidean, in varying input SIR values, i.e. the SIR with +respect to the source’s power (excluding the AIRs and the +beamformer processing). +We see that the Riemannian DS method attains high output +SIR values, even for strong interference sources (high input +SIR). In contrast, the Euclidean DS method results in relatively +low output SIR values. The gap in the output SIR values +-15 +-10 +-5 +0 +5 +10 +15 +[dB] +Riemannian Euclidean +-6[dB] +-6[dB] +Riemannian +Euclidean +-10[dB] +-10[dB] +Riemannian +Euclidean +-20[dB] +-20[dB] +-10 +-5 +0 +5 +[dB] +Riemannian Euclidean +-6[dB] +-6[dB] +Riemannian +Euclidean +-10[dB] +-10[dB] +Riemannian +Euclidean +-20[dB] +-20[dB] +Fig. 3. +The mean output SIR (top) and the directivity (bottom) for two +interference sources, for the Riemannian and the Euclidean DS method. The +x-axis indicates the input SIR, and the y-axis indicates the output SIR. The +box indicates the 25th and 75th percentiles, and the central line marks the +median. Several input SIR values are presented. +between the Riemannian DS, and the Euclidean DS is up +to 10dB. These results coincide with Proposition 1, stating +that SIRj(ΓR) > SIRj(ΓE), for every interference source j. +We emphasize that the Euclidean mean, ΓE, is equivalent to +the common practice of using the entire signal for a single +correlation matrix estimation. Since both the mean output SIR +and the directivity present similar trends, and due to space +considerations, in the following, we only present the mean +output SIR. +Figure 4 is the same as Figure 3, but presenting the SbSp +method with the addition of the intersection method, which +appears in orange. The top subfigure presents the results +for the practical implementation that includes estimating the +dimension, whereas the bottom subfigure presents the results +for the oracle. We see that the Riemannian approach outper- +forms its Euclidean counterpart, by approximately 20dB. In +addition, the oracle SbSp method is better than the practical +SbSp. In comparison to Fig. 3(top), it can be seen that the +Riemannian SbSp method results in higher output SIRs than +the Riemannian DS method. In contrast, for the Euclidean +approach, the SbSp method yields slightly lower SIRs than +the DS method. The reason is that the Riemannian mean +better attenuates the interference sources, allowing for a better +estimation of the signal subspace than the Euclidean mean. +We continue with examining the direction estimation to +the desired source. The estimated direction is defined as the +direction leading to the maximal value of the beam pattern, +namely +ˆθd = arg max +θ +P(θ). +(51) +As a baseline, we compare the results with a version of the +SRP-PHAT algorithm [35]. We use its position estimation +to compute the direction of the desired source. Figure 5 + +10 +-20 +0 +20 +40 +[dB] +Riemannian Euclidean Intersection +-6[dB] +-6[dB] +-6[dB] +Riemannian +Euclidean +Intersection +-10[dB] +-10[dB] +-10[dB] +Riemannian +Euclidean +Intersection +-20[dB] +-20[dB] +-20[dB] +-20 +-10 +0 +10 +20 +30 +[dB] +Riemannian Euclidean Intersection +Oracle +Oracle +Oracle +-6[dB] +-6[dB] +-6[dB] +Riemannian +Euclidean +Intersection +Oracle +Oracle +Oracle +-10[dB] +-10[dB] +-10[dB] +Riemannian +Euclidean +Intersection +Oracle +Oracle +Oracle +-20[dB] +-20[dB] +-20[dB] +Fig. 4. +The mean output SIR for the SbSp methods. Top: practical imple- +mentation. Bottom: Using an oracle. The box indicates the 25th and 75th +percentiles, and the central line marks the median. The x-axis indicates the +input SIR, and the y-axis indicates the output SIR. Several input SIR values +are presented. +20 +40 +60 +80 +100 +120 +140 +160 +Target direction [deg] +20 +40 +60 +80 +100 +120 +140 +160 +[deg] +Riem +Euc +Itersection +SRP-PHAT +Desired +Interferece +20 +40 +60 +80 +100 +120 +140 +160 +Target direction [deg] +20 +40 +60 +80 +100 +120 +140 +160 +[deg] +Riem +Euc +Itersection +SRP-PHAT +Desired +Interferece +Fig. 5. +The DoA estimation to the desired source for input SIR of −6dB +(left) and input SIR of −10dB (right). +shows the estimated direction to the desired source for the +Riemannian DS method (blue square), the Euclidean DS +method (red circle), the intersection method (orange star), and +the SRP-PHAT method (purple triangle). The solid black line +marks the true location of the target source (at 20 different +positions). The dashed line marks the fixed location of the +interference sources. The results for input SIR −6dB and +−10dB appear on the left and right, respectively. We see that +using the Riemannian mean, the direction estimation follows +the desired source. In contrast, using the Euclidean mean (the +entire signal) results in estimating the direction of one of +the interference sources. The direction estimation using SRP- +PHAT follows the other interference source. The intersection +method is also inferior to the proposed approach, resulting in +direction estimation to the desired source or an interference +source depending on the input SIR. +Next, we examine the sensitivity of the proposed approach +to the SNR and the reverberation time. We repeat the setting +of the two interferences, as described in the first experiment. +The results are presented in Figure 6. At the top, the mean +output SIR for the DS method is presented as a function +of the reverberation time for a fixed SNR of 50dB. Several +input SIR values are shown: 0dB (asterisks), −6dB (circle), +and −10dB (triangle). The results for the Riemannian and +150 +200 +250 +300 +350 +400 +[msec] +-5 +0 +5 +10 +15 +[dB] + 0[dB] (Riem) + -6[dB] (Riem) +-10[dB] (Riem) + 0[dB] (Euc) + -6[dB] (Euc) +-10[dB] (Euc) +20 +25 +30 +35 +40 +45 +50 +SNR[dB] +-5 +0 +5 +10 +15 +[dB] + 0[dB] (Riem) + -6[dB] (Riem) +-10[dB] (Riem) + 0[dB] (Euc) + -6[dB] (Euc) +-10[dB] (Euc) +Fig. 6. The mean output SIR as a function of the reverberation times (top) +and as a function of the SNR (bottom), for 2 interference sources. The +Riemannian DS appears in blue, whereas the Euclidean DS appears in red. +Several input SIR values are presented: 0dB (asterisks), −6dB (circle), and +−10dB (triangle). We recall the SNR at the array is approximately 30dB +lower. +Euclidean DS appear in blue and red, respectively. The bottom +figure is the same as the top, only for different SNR values +for a fixed reverberation time of β = 150ms (we recall the +SNR at the array is approximately 30dB lower). We see that +the smaller the reverberation time is, the higher the output SIR +is for the Riemannian DS method. In contrast, the Euclidean +DS is less affected by the reverberation time, resulting in +relatively low output SIRs. For all values of reverberation +times, the Riemannian DS results in higher output SIR than +its Euclidean counterpart. From the bottom figure, we see +that the SNR has a large impact on the performance of the +Riemannian DS; the higher the SNR is, the higher the output +SIR becomes. Conversely, the Euclidean DS is moderately +affected by the SNR, resulting in much lower output SIR +values. A possible explanation is that the main phenomenon +limiting the performance of the Euclidean approach is the +existence of interference sources. In addition, as Proposition +2 predicts, the sensitivity of the Riemannian DS to the SNR +is higher than the Euclidean DS, and the higher the SNR is, +the larger the gap in the performance between the Riemannian +and the Euclidean approaches. +We examine the performance of the MVDR beam pattern, +given by (48). The MVDR beamformer is popular when +interference sources are present, thanks to its distortionless +response in the direction of the desired source, and its typical +narrow beams. However, in our setting, since the desired +source is accompanied by interference sources, and the direc- +tions to all the sources are unknown, the DoA estimation of the +desired source using the MVDR beamformer is outperformed +by the DS and the SbSp methods. We also examine the case of +2 desired sources. The results show similar trends and appear +in Appendix B, due to space considerations. + +11 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Segment index +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +Interference index +Fig. 7. +Left: Activation map for the 14 interference sources during the +10 segments (blue indicates ‘active’). Right: Output SIR of the different +beamformers. The input SIR of each interference is −6dB. +In the second experiment, we examine a multiple interfer- +ence setting, by considering NI = 14 interference sources. +We note that the number of interference sources is larger +than the number of microphones, NI > M = 12, which +typically limits the number of interference sources that can +be accommodated (e.g. see [21]). Furthermore, assumptions +1 and 2 implicitly restrict the number of interference sources +to be bounded by M − ND = 11. This limitation is only for +the analysis, and, in practice, improved results are obtained +even for a larger number of interference sources. The signal +contains 10 segments, demonstrating the number of segments +could be smaller than the number of interference sources. The +duration of the emitted signal is 10.24s. The input SIR for each +interference is −6dB. Each interference has a 30% probability +of being active at each segment. The position of all the sources +is set at random on the arc, as described in the first experiment. +The activation map of the interference sources at the first +Monte Carlo iteration appears in Figure 7 (left), where light +blue indicates ‘active’. On average, 5.4 interference sources +are active at the same time at each segment. An interference +source that is partially active during a segment is considered +active during the entire segment (a worst case). Note that there +are interference sources active continuously during more than +one segment, so their activation is not necessarily related to +the division of the received signal into segments. Additionally, +there exists no segment in which only one source is active. The +output SIRs are presented in Fig. 7(right). The Riemannian DS +and SbSp methods appear in blue, the Euclidean DS and SbSp +methods are in red, and the intersection is in orange. It can be +seen that the Riemannian approach is superior to the Euclidean +one, resulting in higher output SIRs. +VIII. CONCLUSION +We present a Riemannian approach for the design of dif- +ferent beamformers for interference rejection in reverberant +environments. The resulting beam pattern is used for DoA +estimation of the desired source, as it rejects the beams +directed at the interference sources. 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The projection +matrix onto the signal subspace of Γi is computed as follows: +Psig(Γi) = V (Γi) +� +V H(Γi)V (Γi) +�−1 V H(Γi). +(52) +Since each desired source is active during all the segments, its ATF, hd +j, is an eigenvector of V (Γi). Consequently it is +also an eigenvector of Psig(Γi) for all i, with an eigenvalue 1, i.e. Psig(Γi)hd +j = hd +j. As a result, it is an eigenvector of +Psig = �Ls +i=1 Psig(Γi) with eigenvalue 1 as well since Psighd +j = +��Ls +i=1 Psig(Γi) +� +hd +j = hd +j. We consider the leading ND +eigenvectors of Psig and form the matrix Psig,ND ∈ CM×ND, whose columns are the eigenvectors. The intersection beam +pattern is defined as follows: +PIntersect(θ) = dH(θ)Psig,NDP H +sig,NDd(θ). +(53) +We note that in addition to the dimension of the signal space of the matrix Psig, the intersection method requires the estimation +of the dimension of the signal space of the correlation matrix for each segment (in order to compute V (Γi)). +APPENDIX B +ADDITIONAL EXPERIMENTAL RESULTS +We repeat the setting of the first experiment, only with an additional desired source, located at (2m, 3.5m, 2.5m). Figure 8 +is the same as Fig. 3, but the mean output SIR is taken over the two interferences and the two desired sources. The left figure +presents results for input SIR of −10dB, and the right presents results for input SIR of −6dB. We see that the Riemannian +approach is superior to the Euclidean one, resulting in higher SIRs. Also, the intersection method is more sensitive to the input +SIR, resulting in higher output SIR for −6dB. Similar to the setting of one desired source, the Riemannian SbSp method is +superior to the Riemannian DS method, as opposed to the Euclidean approach, for which they are on par. This is a consequence +of the higher interference attenuation of the Riemannian approach, resulting in a better estimate of the signal space dimension. +Next, we examine the performance of the Riemannian MVDR, and the Euclidean MVDR beamformers, given by (48), for +ΓR and ΓE, respectively. Figure 9 presents the results. It is the same as Fig. 3, only for the MVDR beamformer. We see that +the Riemannian approach is superior to the Euclidean one, however, with a slight advantage. +APPENDIX C +EXTENSION TO A STREAMING DATA SETTING +In a streaming data setting, we do not have access in advance to the entire signal, and the direction estimation is updated as +more samples become available. Therefore, in this setting, we cannot compute the Riemannian mean using (17). To circumvent +this, we turn to the estimator of the Riemannian mean proposed in [36], [37], which is updated after every received segment. +Each segment, i, is processed separately using Lw STFT windows, and an estimation of the correlation matrix, Γi, is +computed using (16). Next, the estimation of the Riemannian mean, denoted as ˆRi, which is the adaptive counterpart of ΓR, +is updated using the following update step: +ˆRi = ˆR +1 +2 +i−1( ˆR +− 1 +2 +i−1Γi ˆR +− 1 +2 +i−1) +1 +i ˆR +1 +2 +i−1. +(54) +-5 +0 +5 +10 +15 +20 +25 +[dB] +Riemannian +Euclidean +DS +DS +Riemannian Euclidean Intersection +SbSp +SbSp +SbSp +Fig. 8. Output SIR of the different beamformers for two desired sources and two interference sources. Left: Input SIR is −10dB. Right: Input SIR is −6dB. + +30 +25 +20 +15nannian +Euclidean +Intersection +SbSp +SbSp +SbSp10 +白 +5 +0 +-5 +-10 +Riemannian +Euclidean +Rier +DS +DS14 +-5 +0 +5 +10 +[dB] +Riemannian Euclidean +-6[dB] +-6[dB] +Riemannian +Euclidean +-10[dB] +-10[dB] +Riemannian +Euclidean +-20[dB] +-20[dB] +Fig. 9. The mean output SIR for the Riemannian and the Euclidean MVDR beamformer in the presence of two interference sources. The x-axis indicates +the input SIR, and the y-axis indicates the output SIR. The box indicates the 25th and 75th percentiles, and the central line marks the median. Several input +SIR values are presented. +The DS beam pattern is computed using (18) with ˆRi as an estimate of ΓR, and the direction to the desired source is set +using (19). The algorithm is described in Algorithm 3. +We note that the current correlation matrix estimate has a full rank if Lw > M. This implies a latency of at least M times +the duration of the STFT window. In case the STFT is performed with overlap this latency is reduced accordingly. +Algorithm 3 Streaming DoA estimation in the presence of multiple interferences +Input: The current result of the STFT window of the received signal +Output: The estimated direction of the desired source ˆθ +1: Set ˆR0 = I +2: Repeat +1) Accumulate Lw STFT windows (that form a segment) +2) Compute its correlation matrix, Γi, using (16) +3) Compute ˆRi = ˆR +1 +2 +i−1( ˆR +− 1 +2 +i−1Γi ˆR +− 1 +2 +i−1) +1 +i ˆR +1 +2 +i−1 +4) Compute PDS(θ; ˆRi) using (18) +5) Return ˆθ = arg maxθ PDS(θ; ˆRi) +As for the Euclidean alternative, its estimator is updated in finer granularity at every STFT window, l. The estimator at the +lth update is denoted by El and is computed as follows: +El = n − 1 +n +El−1 + 1 +nz(l)zH(l). +(55) +Setting Lw = 1 in step 2.1 in Algorithm 3, and substituting (55) in step 2.3 in Algorithm 3, results in the streaming version +of the Euclidean counterpart. +APPENDIX D +ON THE PARTICULAR CHOICE OF THE RIEMANNIAN METRIC +In this work, we consider the Affine Invariant metric (also called Fisher Information metric) [38]. Another commonly-used +metric in the space of HPD matrices is the Log-Euclidean metric [39], which could be viewed as a computationally efficient +local approximation of the Affine Invariant metric. The induced Log-Euclidean distance is given by: +d2 +LE(Γ1, Γ2) = ∥ log(Γ1) − log(Γ2)∥2 +F . +(56) +In the context of this work, the Affine Invariant metric is advantageous over the Log Euclidean because it better enhances +the desired source subspace relative to the interference and noise subspace, as we show next. We remark that the derivation +is similar to the derivation in Section V, where we demonstrate the advantage of the Affine Invariant metric over Euclidean +geometry. First, we note that the following holds [40]: +tr(ΓR) ≤ tr(ΓLE), +(57) + +15 +where ΓLE denotes the Riemannian mean based on the Log-Euclidean metric. Second, the ATF to the desired source, h0, +is a common eigenvector of all the correlation matrices per segment. Consequently, according to Lemma 2 below, the mean +correlation matrices based on both metrics have the same eigenvalue associated with the common eigenvector h0. More +specifically, denoting λ0(Γ) ≡ hH +0 Γh0 +∥h0∥2 , we get λ0(ΓR) = λ0(ΓLE). +Lemma 2. Let {Γj}j be the set of HPD matrices, having a common eigenvalue λ0, associated with the common eigenvector +h0. Then +hH +0 ΓRh0 = hH +0 ΓLEh0 = hH +0 ΓEh0 = ∥h0∥2λ0. +(58) +where ΓR, ΓLE, ΓE are the Riemannian means based on the Affine Invariant and the Log-Euclidean metrics, and the Euclidean +mean. +The rest of the eigenvectors of the correlation matrices span the interference and noise subspace. We recall that λi(Γ) is +the ith eigenvalue, so we get +λ0(ΓR) +�M−1 +i=1 λi(ΓR) +≥ +λ0(ΓLE) +�M−1 +i=1 λi(ΓLE) +, +(59) +which follows from Lemma 2, and (57). +We see that the Riemannian mean induced by the Affine Invariant metric captures better the desired signal subspace in +comparison to the Riemannian mean induced by the Log Euclidean metric and the Euclidean mean (according to (24)), +entailing an advantage in SbSp methods, for example. +APPENDIX E +PROOFS OF THEORETICAL RESULTS +A key observation in our analysis is that although there is no explicit expression for the Riemannian mean, in general, under +assumptions 1 and 2, the correlation matrices {Γl} commute, and the Riemannian mean has an explicit form, according to +Proposition 5. We note that in practice we use Algorithm 1 to compute the Riemannian mean because the assumptions do not +strictly hold. +For completeness, we prove Proposition 5, which is a property of Karcher’s mean [28]. +Proposition 5. The Riemannian mean of K commuting HPD matrices, {Γl}K +l=1 is given by +ΓR = +K +� +l=1 +Γ +1 +K +l . +(60) +Proof. The Karcher mean, ΓR, is the solution of the Karcher equation [26], [28], [41], [42] +K +� +l=1 +log(Γ +1 +2 +R Γ−1 +l +Γ +1 +2 +R ) = 0 +(61) +We set ΓR = �K +l=1 Γ +1 +K +l , and use the fact the matrices commute: +K +� +l=1 +log(Γ +1 +2 +R Γ−1 +l +Γ +1 +2 +R ) = +K +� +l=1 +log(ΓRΓ−1 +l +) += +K +� +l=1 +log +� +Γ−1 +l +K +� +l=1 +Γ +1 +K +l +� += log +K +� +l=1 +� +Γ−1 +l +K +� +l=1 +Γ +1 +K +l +� += log I += 0. +(62) + +16 +A. Proof of Proposition 1 +Proof. The proof relies on Lemma 1, which we prove first. +Lemma 1. The Riemannian or the Euclidean mean of the population correlation matrices of the segments (25) over the entire +interval can be written in the same parametric form as: +Γ = σ2 +0h0hH +0 + +NI +� +j=1 +µ2 +jhjhH +j + σ2 +vI. +(63) +The Riemannian mean ΓR is obtained by setting the parameters µj to +µ2 +j = (σ2 +j ∥hj∥2 + σ2 +v)τj(σ2 +v)1−τj − σ2 +v +∥hj∥2 +, +(64) +and the Euclidean mean ΓE is obtained by setting +µ2 +j = σ2 +j τj. +(65) +Proof. We start by expressing the correlation matrix of the ith segment: +Γi = σ2 +0h0hH +0 + +� +j∈Ji +σ2 +j hjhH +j + σ2 +vI, +(66) +where Ji is the set of indices of the active interference sources during the ith segment. The vectors h0, and {hj}NI +j=1 are all +orthogonal, thus they are eigenvectors of Γi, ∀i = 1, ..., Ls. +We denote by upper tilde the normalized version of a vector, i.e. ˜h = +h +∥h∥. So, +˜hH +0 Γi˜h0 = σ2 +0∥h0∥2 + σ2 +v +∀i +˜hH +j Γi˜hj = σ2 +j ∥hj∥2 + σ2 +v +i ∈ Lj +˜hH +j Γl˜hj = σ2 +v +i /∈ Lj +(67) +The correlation matrices for each segment, {Γi}Ls +i=1, commute with each other because they share their eigenvectors: +h0, {hj}LI +j=1, and {vl}M−NI−1 +l=1 +, where {vj}M−NI−1 +j=1 +are the eigenvectors spanning the common noise subspace of all the +matrices. +Following Proposition 5, it holds true that +ΓR = +Ls +� +j=1 +Γ +1 +Ls +j . +(68) +We compose ΓR using a transformation for the eigenvalues for each Γi: +˜hH +j ΓR˜hj = (σ2 +j ∥hj∥2 + σ2 +v)τj(σ2 +v)1−τj, +∀j +˜hH +0 ΓR˜h0 = σ2 +0∥h0∥2 + σ2 +v, +(69) +and the rest of the eigenvectors have an eigenvalue of σ2 +v, with multiplicity of M − NI − 1. The matrix ΓR meets all the +requirements. +The proof for the Euclidean mean follows from its definition and (66). +We compute the output SIR for a general matrix with a similar structure as in Lemma 1. The correlation matrix is: +Γ = σ2 +0hH +0 h0 + +NI +� +l=1 +µ2 +l hH +l hl + σ2 +vI, +(70) +and the SIR becomes: +SIRj(Γ) = +dH +0 +� +σ2 +0hH +0 h0 + �NI +l=1 µ2 +l hH +l hl + σ2 +vI +� +d0 +dH +j +� +σ2 +0hH +0 h0 + �NI +l=1 µ2 +l hH +l hl + σ2vI +� +dj += σ2 +0|⟨dH +0 , h0⟩|2 + �NI +l=1 µ2 +l |⟨dH +0 , hl⟩|2 + σ2 +vM +σ2 +0|⟨dH +j , h0⟩|2 + �NI +l=1 µ2 +l |⟨dH +j , hl⟩|2 + σ2vM += +σ2 +0M∥h0∥2κ + �NI +l=1 µ2 +l M∥hl∥2ρ + σ2 +vM +σ2 +0M∥h0∥2ρ + �NI +l=1,l̸=j µ2 +l M∥hl∥2ρ + µ2 +jM∥hj∥2κ + σ2vM += 1 + (σ2 +0∥h0∥2 − µ2 +j∥hj∥2)κ + (µ2 +j∥hj∥2 − σ2 +0∥h0∥2)ρ +σ2 +0∥h0∥2ρ + �NI +l=1,l̸=j µ2 +l ∥hl∥2ρ + µ2 +j∥hj∥2κ + σ2v +(71) + +17 +We note that the SIR depends on the number of microphones implicitly through the norm of the ATFs ∥hj∥, and ∥h0∥. +Since κ, ρ ≥ 0, and κ > ρ, the smaller µ2 +j, µ2 +l ∀l are, the higher the SIR is. Using Lemma 1, we identify the coefficients +for the Riemannian mean as: +µ2 +l = (σ2 +l ∥hl∥2 + σ2 +v)τl(σ2 +v)1−τl − σ2 +v +∥hl∥2 +, +(72) +and for the Euclidean mean as +µ2 +l = σ2 +l τl. +It is left to show that the coefficients for the Riemannian SIR are smaller than for the Euclidean SIR. So, we examine the +conditions under which +(σ2 +l ∥hl∥2 + σ2 +v)τl(σ2 +v)1−τl − σ2 +v +∥hl∥2 +< σ2 +l τj +(σ2 +l ∥hl∥2 + σ2 +v)τl(σ2 +v)1−τl < σ2 +l ∥hl∥2τj + σ2 +v +(73) +Equation (73) holds due to the weighted arithmetic mean and geometric mean inequality [43, pp. 111–112, Theorem 10.5] +[44]. +We see that the Riemannian mean of the correlation matrices leads to the geometric mean of the noise power and the +interference power plus the noise power. In contrast, the common practice that is the Euclidean mean of the correlation +matrices leads to the arithmetic mean of the powers. +B. Proof of Lemma 2 +Proof. For the Euclidean geometry, the proof is straightforward. +For the Riemannian geometry, since, in general, there is no close form expression for the Riemannian mean, we use its +definition as the minimizer of +ΓR = arg min +Γ +� +j +d2 +R(Γ, Γj). +(74) +For the Affine Invariant metric, the following holds +d2 +R(Γ1, Γ2) = +� +l +log2 � +λl(Γ +− 1 +2 +1 +Γ2Γ +− 1 +2 +1 +) +� +. +(75) +Since the matrix Γ +− 1 +2 +j +ΓRΓ +− 1 +2 +j +is a function of the matrices Γj, we have that h0 is also its eigenvector. We notice that +min log2 � +λ0(Γ +− 1 +2 +j +ΓRΓ +− 1 +2 +j +) +� += min log2 +�λ0(ΓR) +λ0(Γj) +� += 0, +(76) +for λ0(ΓR) = λ0(Γj). This is the minimum possible value for every j, so it must hold that λ0(ΓR) = λ0(Γj). +For the Log-Euclidean distance, we use eigenvalue decomposition and denote by {ui} and {vk} the eigenvectors of ΓLE +and Γj, respectively. We have +d2 +LE(ΓLE, Γj) = ∥ ln(ΓLE) − ln(Γj)∥2 +F += ∥ +� +i +ln λi(ΓLE)uiuH +i + ln λ0(ΓLE)h0hH +0 +− +� +k +ln λk(Γj)vkvH +k − ln λ0(Γj)h0hH +0 ∥2 +F += ∥C + (ln λ0(ΓLE) − ln λ0(Γj))h0hH +0 ∥2 +F += tr +� +CCH ++ (ln λ0(ΓLE) − ln λ0(Γj))2 · ∥h0∥2 · h0hH +0 +� +, +(77) +which is minimal for every j, for λ0(ΓLE) = λ0(Γj). The last equality is due to the orthogonality between h0 and the rest of +the eigenvectors. + +18 +C. Proof of Proposition 2 +We start by proving Proposition 2, i.e. showing that +∂ +∂σ2v +SIRj(ΓR) < +∂ +∂σ2v +SIRj(ΓE) < 0, +(78) +and continue with examining the conditions under which +∂ +∂σ2v +SIRj(ΓR) < 0, +(79) +∂ +∂σ2v +SIRj(ΓE) < 0. +(80) +Proof. The proof employs the following technical Lemma +Lemma 3. For µ2 +j given by (33), the following holds: +1) +∂ +∂σ2v µ2 +j ≥ 0 and µ2 +j ≥ 0. +2) +∂ +∂σ2v +�NI +l=1,l̸=j µ2 +l ∥hl∥2 ≥ 0 and �NI +l=1,l̸=j µ2 +l ∥hl∥2 ≥ 0 +Proof. +∂ +∂σ2v +µ2 +j = +∂ +∂σ2v +� +(σ2 +j ∥hj∥2 + σ2 +v)τj(σ2 +v)1−τj − σ2 +v +∥hj∥2 +� += +1 +∥hj∥2 +∂ +∂σ2v +� +(σ2 +j ∥hj∥2 + σ2 +v)τj(σ2 +v)1−τj − σ2 +v +� += +1 +∥hj∥2 +� +τj · +� +σ2 +j ∥hj∥2(σ2 +v) +1 +τj −1 + (σ2 +v) +1 +τj +�τj−1 +· +�� 1 +τj +− 1 +� +σ2 +j ∥hj∥2(σ2 +v) +1 +τj −2 + 1 +τj +(σ2 +v) +1 +τj −1 +� +− 1 +� +. +(81) +We focus on the expression in the bracket, rewrite it as a fraction, and show that it is larger than 1. +� +1 +τj − 1 +� +σ2 +j ∥hj∥2(σ2 +v) +1 +τj −2 + 1 +τj (σ2 +v) +1 +τj −1 +1 +τj · +� +σ2 +j ∥hj∥2(σ2v) +1 +τj −1 + (σ2v) +1 +τj +�1−τj += (1 − τj) σ2 +j ∥hj∥2(σ2 +v) +1 +τj −2 + (σ2 +v) +1 +τj −1 +� +σ2 +j ∥hj∥2(σ2v) +1 +τj −1 + (σ2v) +1 +τj +�1−τj += (1 − τj) σ2 +j ∥hj∥2(σ2 +v) +1 +τj −2 + (σ2 +v) +1 +τj −1 +(σ2v) +1 +τj −1 � +σ2 +j ∥hj∥2(σ2v)−1 + 1 +�1−τj += (1 − τj) σ2 +j ∥hj∥2(σ2 +v)−1 + 1 +� +σ2 +j ∥hj∥2(σ2v)−1 + 1 +�1−τj += 1 + (1 − τj) σ2 +j ∥hj∥2(σ2 +v)−1 +� +1 + σ2 +j ∥hj∥2(σ2v)−1�1−τj +≥ 1. +(82) +The last inequality follows from Bernoullie”s inequality (1 + x)r ≤ 1 + rx, by setting 0 < r = (1 − τj) < 1, and x = +σ2 +j ∥hj∥2(σ2 +v)−1 > −1. Since µ2 +j = 0 for σ2 +v = 0 and +∂ +∂σ2v µ2 +j ≥ 0, it holds that µ2 +j ≥ 0. The above holds for all j thus it also +holds for their sum. +We recall the expression for the SIR from (71): +SIRj(Γ) = 1 + (σ2 +0∥h0∥2 − µ2 +j∥hj∥2)κ + (µ2 +j∥hj∥2 − σ2 +0∥h0∥2)ρ +σ2 +0∥h0∥2ρ + �NI +l=1,l̸=j µ2 +l ∥hl∥2ρ + µ2 +j∥hj∥2κ + σ2v +. +(83) +We denote the following functions, which are the numerator and the denominator of the expression of the Riemannian SIR: +fR(σ2 +v) = (σ2 +0∥h0∥2 − µ2 +j∥hj∥2)κ + (µ2 +j∥hj∥2 − σ2 +0∥h0∥2)ρ +(84) +gR(σ2 +v) = σ2 +0∥h0∥2ρ + +NI +� +l=1,l̸=j +µ2 +l ∥hl∥2ρ + µ2 +j∥hj∥2κ + σ2 +v, +(85) +where µ2 +j is given by (33). Similarly, we define fE, and gE with µ2 +j given by (34). +The derivative is +∂ +∂σ2v +SIRj(σ2 +v) = f ′(σ2 +v)g(σ2 +v) − g′(σ2 +v)f(σ2 +v) +g2(σ2v) +, +(86) + +19 +where f and g represent fR or fE and gR or gE, respectively, and (·)′ denotes the derivative with respect to σ2 +v. In the +proof of Proposition 1, we show that µ2 +j for the Riemannian mean in (33) is smaller than its Euclidean counterpart in (34). In +combination with Lemma 3, we have that , 0 < gR(σ2 +v) < gE(σ2 +v), so we focus on the numerator. +We show that +∂ +∂σ2 +v SIRj(ΓR) < +∂ +∂σ2 +v SIRj(ΓE) < 0, by proving the following claims: +1) f ′ +R(σ2 +v)gR(σ2 +v) < f ′ +E(σ2 +v)gE(σ2 +v) = 0 +2) g′ +R(σ2 +v)fR(σ2 +v) > g′ +E(σ2 +v)fE(σ2 +v) > 0 +Proof of Claim 1 +Since µ2 +j for the Euclidean mean does not depend on σ2 +v, the same holds for fE(σ2 +v) and f ′ +E(σ2 +v) = 0. For the Riemannian +case, using Lemma 3 we have that f ′ +R(σ2 +v) = µ2 +j∥hj∥2(ρ−κ) < 0 and gR(σ2 +v) > 0, so f ′ +R(σ2 +v)·gR(σ2 +v) < f ′ +E(σ2 +v)·gE(σ2 +v) = 0. +Proof of Claim 2 +Since µ2 +j for the Riemannian mean in (33) is smaller than its Euclidean counterpart in (34), we have fR(σ2 +v) > fE(σ2 +v). +Additionally, if (30) holds for µ2 +j in (34) then fE(σ2 +v) > 0. +It is left to show that g′ +R(σ2 +v) ≥ g′ +E(σ2 +v) > 0. This holds due to Lemma 3: +g′ +R(σ2 +v) = ρ +NI +� +l=1,l̸=j +∂ +∂σ2v +µ2 +l ∥hl∥2 + ∥hj∥2κ ∂ +∂σ2v +µRiem +j ++ 1 +≥ 1 += g′ +E(σ2 +v). +(87) +Following the proof of Proposition 2, since f ′ +E(σ2 +v) = 0, it follows that iff g′ +E(σ2 +v)fE(σ2 +v) > 0 then +∂ +∂σ2v SIRj(ΓE) < 0. Since +g′ +E(σ2 +v) = 1, we have that +∂ +∂σ2v SIRj(ΓE) < 0 iff fE(σ2 +v) > 0. From the expression of fE(σ2 +v) it follows that fE(σ2 +v) > 0 iff +condition (30) holds. So, it is a sufficient and a necessary condition for +∂ +∂σ2 +v SIRj(ΓE) < 0 to hold. +For the Riemannian mean, the condition under which +∂ +∂σ2 +v SIRj(ΓR) < 0 is more easily met. Additionally, it is a sufficient +but not a necessary condition, as we show next. +First, we note that condition (30) can be recast as +σ2 +0∥h0∥2 ≥ µj∥hj∥2, ∀j, +(88) +where µ2 +j given by (34). +For the Riemannian mean, under condition (88) with µ2 +j given by (33) it holds that g′ +R(σ2 +v)fR(σ2 +v) > 0. Since +f ′ +R(σ2 +v)gR(σ2 +v) < 0, condition (88) with µ2 +j given by (33) is a sufficient but not a necessary condition for +∂ +∂σ2v SIRj(ΓR) < 0 +to hold, as opposed to the Euclidean case. +In the proof of Proposition 1 it is shown that the parameter µj for the Riemannian mean in (33) is smaller than its Euclidean +counterpart in (34). Therefore, condition (88) is more easily met when µj is given by (33) than when µj is given by (34). As +a consequence, it is possible for +∂ +∂σ2v SIRj(ΓE) to be positive, while +∂ +∂σ2v SIRj(ΓR) is negative. +D. Proof of Proposition 3 +Proof. Under Assumption 1 the ATF of the desired source, h0, is an eigenvector of Γj for all j with the same eigenvalue λ0. +Then, according to Lemma 2 the following holds +hH +0 ΓRh0 = hH +0 ΓEh0 = λ0. +(89) +For the Riemannian and the Euclidean mean, the following holds [28]: +ΓR ⪯ ΓE. +(90) +Thus, for all j: +hH +j ΓRhj ≤ hH +j ΓEhj +(91) +and therefore +NI +� +j=1 +hH +j ΓRhj ≤ +NI +� +j=1 +hH +j ΓEhj, +(92) +which completes the proof. + +20 +E. Proof of Proposition 4 +Proof. The Riemannian mean is ΓR = Γ +1 +2 +1 Γ +1 +2 +2 , since the matrices commute. The matrices Γ1 and Γ2 are expressed using their +eigenvectors, allowing the derivation of Γ +1 +2 +1 and Γ +1 +2 +2 . Finally, ΓR is computed via their product: +ΓR = σ2 +dhd(hd)H + µ2 +1hi +1(hi +1)H + µ2 +2hi +2(hi +2)H + σ2 +vI +(93) +with µ2 +j = +((α2σ2 +j ∥hj∥2+σ2 +v) +1 +2 ·((1−α)2σ2 +j ∥hj∥2+σ2 +v) +1 +2 )−σ2 +v +∥hj∥2 +for j = 1, 2. +For the Euclidean mean, µ2 +j = +σ2 +j +2 (α2 + (1 − α)2), which reaches its minimum for α = 1 +2 ∀j. +From the proof of Proposition 1, the smaller the µ2 +j-s are, the higher the SIR is. We show that +((α2σ2 +j ∥hj∥2 + σ2 +v) +1 +2 · ((1 − α)2σ2 +j ∥hj∥2 + σ2 +v) +1 +2 ) − σ2 +v +∥hj∥2 +≤ σ2 +j +2 (α2 + (1 − α)2). +(94) +Rearranging results in +((α2σ2 +j ∥hj∥2 + σ2 +v) +1 +2 · ((1 − α)2σ2 +j ∥hj∥2 + σ2 +v) +1 +2 ) +≤ ∥hj∥2 σ2 +j +2 (α2 + (1 − α)2) + σ2 +v, +(95) +which holds due to the inequality of arithmetic and geometric means √x · y ≤ +x+y +2 , where x = α2σ2 +j ∥hj∥2 + σ2 +v and +y = (1 − α)2σ2 +j ∥hj∥2 + σ2 +v. +APPENDIX F +THEORETICAL RESULTS FOR THE TOTAL SIR +Proposition 6. For any number of microphones in the array, the following holds +SIRtot(ΓR) > SIRtot(ΓE). +(96) +Proof. We compute the output total SIR for a general matrix with a similar structure as in Lemma 1. The correlation matrix +is: +Γ = σ2 +0hH +0 h0 + +NI +� +l=1 +µ2 +l hH +l hl + σ2 +vI, +(97) +and the SIR becomes: +SIRtot(Γ) = +dH +0 +� +σ2 +0hH +0 h0 + �NI +l=1 µ2 +l hH +l hl + σ2 +vI +� +d0 +1 +NI +�NI +j=1 dH +j +� +σ2 +0hH +0 h0 + �NI +l=1 µ2 +l hH +l hl + σ2vI +� +dj += +σ2 +0|⟨dH +0 , h0⟩|2 + �NI +l=1 µ2 +l |⟨dH +0 , hl⟩|2 + σ2 +vM +1 +NI +�NI +j=1 +� +σ2 +0|⟨dH +j , h0⟩|2 + �NI +l=1 µ2 +l |⟨dH +j , hl⟩|2 + σ2vM +� += +σ2 +0M∥h0∥2κ + �NI +l=1 µ2 +l M∥hl∥2ρ + σ2 +vM +σ2 +0M∥h0∥2ρ + +1 +NI +�NI +j=1 +�NI +l=1,l̸=j µ2 +l M∥hl∥2ρ + +1 +NI +�NI +j=1 µ2 +jM∥hj∥2κ + σ2vM += 1 + +(σ2 +0∥h0∥2 − 1 +NI +�NI +j=1 µ2 +j∥hj∥2)κ + ( 1 +NI +�NI +j=1 µ2 +j∥hj∥2 − σ2 +0∥h0∥2)ρ +σ2 +0∥h0∥2ρ + NI−1 +NI +�NI +l=1 µ2 +l ∥hl∥2ρ + +1 +NI +�NI +j=1 µ2 +j∥hj∥2κ + σ2v +. +(98) +In the proof of Proposition 1 we show that µ2 +j for the Riemannian mean is smaller than µ2 +j for the Euclidean mean for all j. +Consequently, we have that �NI +j=1 µ2 +j∥hj∥2 for the Riemannian mean is smaller than for the Euclidean mean. Since κ > ρ ≥ 0, +it holds that SIRtot(ΓR) > SIRtot(ΓE). +Proposition 7. If +σ2 +0∥h0∥2 ≥ 1 +NI +NI +� +j=1 +σ2 +j τj∥hj∥2, +(99) +then +∂ +∂σ2v +SIRtot(ΓR) < +∂ +∂σ2v +SIRtot(ΓE) < 0. +(100) +Proof. The proof follows the same steps as the proof of Proposition 2 + diff --git a/WdE1T4oBgHgl3EQfvgUI/content/tmp_files/load_file.txt b/WdE1T4oBgHgl3EQfvgUI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00aa9b571c5b0b7ac9eafa2f8954369a9f35a379 --- /dev/null +++ b/WdE1T4oBgHgl3EQfvgUI/content/tmp_files/load_file.txt @@ -0,0 +1,1144 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf,len=1143 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='03399v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='SP] 9 Jan 2023 1 On Interference-Rejection using Riemannian Geometry for Direction of Arrival Estimation Amitay Bar and Ronen Talmon Senior Member, IEEE Abstract—We consider the problem of estimating the direction of arrival of desired acoustic sources in the presence of multiple acoustic interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' All the sources are located in noisy and reverberant environments and are received by a microphone array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We propose a new approach for designing beamformers based on the Riemannian geometry of the manifold of Hermitian positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Specifically, we show theoretically that incorporating the Riemannian mean of the spatial correlation matrices into frequently-used beamformers gives rise to beam patterns that reject the directions of interference sources and result in a higher signal-to-interference ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We experimentally demonstrate the advantages of our approach in designing several beamformers in the presence of simultaneously active multiple interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Index Terms—Array Signal Processing, Direction of Arrival Estimation, Interference Rejection, Hermitian Positive Semidefi- nite Matrices, Riemannian Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' INTRODUCTION E STIMATION of the direction of arrival (DoA) of an acoustic source is prevalent in signal processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' it is an important step in many tasks, such as source localization, beamforming, source separation, spectrum sensing, and speech enhancement [1], to name but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Despite the large research attention it has drawn in the past decades, acoustic DoA estimation is still considered a challenging open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Especially in noisy and reverberant environments and in the presence of interference sources, it continues to be an active research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Acoustic source localization, and particularly DoA estima- tion, are often addressed using beamforming [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Many beam- formers have been proposed over the years for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' One class of beamformers is based on the steered response power of a beamformer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For example, considering the maximum likelihood criterion for a single source, the output power of the beamformer from all the directions is computed, and the DoA is identified as the direction with the maximal power [3]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Another example is the Minimum Variance Distortionless Response (MVDR) beamformer [7]–[9], which was first introduced by Capon [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The MVDR beamformer extracts the DoA of each of the existing sources, maintaining a unit gain at their direction while minimizing the response from other directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' An important generalization of the MVDR beamformer is the Linearly Constrained Minimum Variance (LCMV) beamformer [11], obtained by minimizing the output A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Bar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Talmon are with the Viterbi Faculty of Electrical and Com- puter Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail: amitayb@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='il;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ronen@ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='il).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This work was supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 802735-ERC-DIFFOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' power under multiple linear constraints, and can be used for DoA estimation as well [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Another line of beamformers is derived based on a subspace approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', by identifying the subspace of the desired sources, which is assumed to contain only a small portion of the noise and the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' A prominent subspace method, which is also used for DoA estimation, is MUltiple Signal Classification (MUSIC) [13]– [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this paper, we consider DoA estimation in a reverberant enclosure consisting of desired sources along with interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We assume the desired sources are constantly active, whereas the interference sources are only intermittently active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The number of sources, their locations, and their times of activity are all unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently, their identification as desired or interference is unknown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The power of the different sources is also unknown, and the interference sources could, in fact, be stronger than the desired sources with overlapping activity periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Our goal is to estimate the DoA of the desired sources in the presence of possibly simultaneously active, multiple interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This setting poses a major challenge to the common practice in existing methods that rely on maximal power because estimating the DoA of the strongest sources might result in distinct beams in the direction of the interference sources rather than the desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, these beams could mask the beams pointing at the directions of desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We address this challenge from a geometric standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Our approach relies on the observation that the frequently-used beamformers implicitly consider Euclidean geometry when processing sample correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Therefore, since the sample correlation matrices are Hermitian Positive Definite (HPD) matrices, important geometric information is not fully utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Instead, we propose a new approach for beamforming design that is based on the Riemannian geometry of the manifold of HPD matrices [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Concretely, we analyze the received signal in short time windows and consider the Riemannian mean [17] of the sample correlation matrices in these windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then, we leverage particular spectral proper- ties of the Riemannian mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In [19], it was shown that the Riemannian mean of HPD matrices preserves shared spectral components and attenuates unshared spectral components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently, the continual activity of the desired sources and the intermittent activity of the interference sources enable us to associate desired sources with shared spectral components and interference sources with unshared spectral components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By combining the above, we show that the incorporation of the Riemannian mean into the beamformer design leads to 2 interference rejection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', gives rise to beam patterns that implicitly reject the beams pointing at the interference sources and preserve the beams pointing at the desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The resulting beam patterns are, in turn, used for the estimation of the DoA of the desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Importantly, our approach is applicable to a large number of beamformers used for DoA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By incorporating our Riemannian approach, we present new implementations of several beamformers for DoA estimation of the desired sources: the Delay and Sum (DS) beamformer, subspace-based beamformers, and the MVDR beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We show that the Riemannian geometry preserves better the desired sources subspace in comparison to the Euclidean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, we analytically show that, when in- corporated into a DS beamformer, the proposed Riemannian approach results in a higher Signal to Interference Ratio (SIR) compared to its Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, we show that the lower the noise is, the advantage of the Riemannian approach over the Euclidean approach increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Experimentally, we showcase the performance of the differ- ent beamformers in reverberant environments, including in the presence of simultaneously active multiple interference sources, whose locations are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In particular, we focus on demonstrating the advantages of the Riemannian approach over the standard Euclidean approach, providing empirical support for the analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We conclude the introduction with three remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' First, a similar setting to ours, consisting of desired sources accom- panied by interference sources, was considered in [20] and [21] but in the context of signal enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In [20], a single desired source and a single interference source were considered, and in [21], multiple desired sources and multiple interference sources were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' However, in both works, it was assumed that there is at least one segment for each source, desired or interference, in which it is the only active source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, in [21], the number of the desired sources and their activity patterns were assumed to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Second, in the context of radar, the Riemannian geometry of the Toeplitz HPD matrices was used in [22] and [23] for target detection by comparing Riemannian distances to a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the radar settings, [24] estimated the correlation matrix as a linear combination of correlation matrices, with weights that are based on the Riemannian distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Third, in this paper, we demonstrate the Riemannian approach for designing beam- formers that reject interference sources for DoA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' However, other applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', signal enhancement, could also benefit from beam patterns that reject interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In section II, we present a brief background on the HPD manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section III, we formulate the problem and the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section IV we describe the proposed approach and present the algorithm for DoA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section V, we provide a theoretical analysis of the proposed approach for the DS beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section VI, extensions of the approach to other beamformers are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Section VII shows simulation results demon- strating our Riemannian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Lastly, we conclude the work in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' BACKGROUND ON THE HPD MANIFOLD An HPD matrix, Γ ∈ Cn×n, is a Hermitian matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ = ΓH, where ΓH is the conjugate transpose of Γ, whose real eigenvalues are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Associating the space of HPD matrices with the Affine Invariant metric [25] constitutes a Riemannian manifold, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The distance between two matrices Γ1 and Γ2, induced by the Affine Invariant metric, is given by d2 R(Γ1, Γ2) = ��� log � Γ − 1 2 2 Γ1Γ − 1 2 2 � ��� 2 F , (1) where ∥ · ∥F is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian mean, ΓR, of a set of point {Γi|Γi ∈ M} is defined by the Fr´echet mean as follows ΓR ≡ arg min Γ∈M � i d2 R(Γ, Γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (2) In general, there is no closed-form expression for the Rieman- nian mean of more than two matrices, and a solution can be found using an iterative procedure [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The computation of the Riemannian mean requires two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Logarithm map, which projects an HPD matrix Γi ∈ M to the tangent space of the HPD manifold at Γ, is given by LogΓ(Γi) = Γ 1 2 log(Γ− 1 2 ΓiΓ− 1 2 )Γ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (3) The Exponential map, which projects a vector T from the tangent space at Γ, is given by ExpΓ(T ) = Γ 1 2 exp(Γ− 1 2 T Γ− 1 2 )Γ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (4) A discussion about the choice of the Affine Invariant metric appears in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' PROBLEM FORMULATION We consider the problem of localizing ND desired sources in the presence of NI interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' All the sources are static and located in a reverberant environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The signals are received at a noisy microphone array of M microphones, which are positioned in known, but possibly arbitrary, posi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The acoustic environment between each source and each microphone is modeled by the Acoustic Impulse Response (AIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The signal at the mth microphone is given by: zm(n) = ND � j=1 sd j(n) ∗ hd jm(n) + NI � j=1 si j(n) ∗ hi jm(n) + vm(n), (5) where sd j(n) is the jth desired source, si j(n) is the jth interference source, and vm(n) is the mth microphone noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We denote by hd jm(n) the AIR between the mth microphone and the jth desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Similarly, we denote by hi jm(n) the AIR between the mth microphone and the jth interference source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The sources are characterized by their activation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The desired sources are active during the entire interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the interference sources are only partially active, namely active during segments of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, we assume that the desired sources, the interference sources, and the noise are all uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The noise is assumed to be spatially white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 3 The received signal is processed using the Short-Time Fourier Transform (STFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We denote by sd j(l, k) the STFT at the lth window and the kth frequency of sd j(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The notation for si j(l, k), hd jm(l, k), hi jm(l, k), and vm(l, k) follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then, the STFT at the lth window and the kth frequency of the received signal is given by zm(l, k) = ND � j=1 sd j(l, k)hd jm(l, k) + NI � j=1 si j(l, k)hi jm(l, k) + vm(l, k), (6) where we assume the length of the window is much larger than the AIR length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We stack the received signals, {zm(l, k)}m, from all the microphones to obtain a column vector z(l, k) ∈ CM×1 z(l, k) = [z1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' zM(l, k)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (7) Its explicit expression is z(l, k) = Hd(l, k)sd(l, k) + Hi(l, k)si(l, k) + v(l, k), (8) where sd(l, k) and si(l, k) denote the stacked STFT repre- sentations of the desired sources and the interference sources, respectively, and are given by sd(l, k) = [sd 1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' sd ND(l, k)]⊤ si(l, k) = [si 1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' si NI(l, k)]⊤, (9) and the noise term is v(l, k) = [v1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' vM(l, k)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (10) The Acoustic Transfer Functions (ATFs) from the jth desired source and the jth interference source to the microphone array are hd j(l, k) = [hd j1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hd jM(l, k)]⊤ j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', ND hi j(l, k) = [hi j1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hi jM(l, k)]⊤ j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', NI, (11) and in a matrix form Hd(l, k) = [hd 1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hd ND(l, k)] Hi(l, k) = [hi 1(l, k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hi NI(l, k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (12) Henceforth, we focus on a single frequency bin and omit the frequency index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Throughout the paper, we refer to z(l) as the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since all the sources are static the ATFs do not change over time, so in the following, we omit their STFT window index l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Our goal is to estimate the direction to the desired sources, given z(l), l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' , LSTFT, where LSTFT is the number of STFT windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The main challenge is the existence of interference sources, positioned at unknown locations with possibly high signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' PROPOSED APPROACH Typically, the DoA estimation of a desired source is based on the output of a beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this section, we present the proposed approach applied to the Delay-and-Sum (DS) beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section VI, we extend the proposed approach to other beamformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We consider arbitrary indexing of the microphones in the array and designate the first microphone as the reference microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Let d(θ) denote the steering vector of the array to direction θ relative to the first (reference) microphone, which is given by d(θ) = [1, ejφ2(θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', ejφM(θ)]⊤, (13) where φm(θ) is the phase of the received signal at the mth microphone with respect to the first microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For example, for a uniform linear array and the typical microphone indexing, we have φm(θ) = 2π · m δ λ sin θ, where λ is the wavelength of the received signal, and δ is the distance between the microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The weights for the DS beamformer are set according to the steering vector, and its output is given by yDS(θ, l) = dH(θ)z(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (14) We refer to the squared absolute value of the output of the beamformer as the beam pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By (14), the DS beam pattern is given by |yDS(θ, l)|2 = |dH(θ)z(l)|2 = dH(θ)z(l)zH(l)d(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (15) Note that the rank-1 matrix z(l)zH(l) could be viewed as the sample correlation matrix of the population correlation matrix E[z(l)zH(l)] based on one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since the desired source is constantly active and assumed to be at a fixed location during the entire interval, we propose to improve the correlation estimation by averaging z(l)zH(l) over multiple STFT windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We divide the STFT of the signal into Ls disjoint segments, each consisting of Lw consecutive STFT windows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', LSTFT = Ls ·Lw (for more details regarding the partitioning see Section V-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then, the sample correlation matrix over each segment is computed as follows Γi = 1 Lw i·Lw � l=(i−1)·Lw+1 z(l)zH(l), (16) where i is the segment index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To obtain a full rank correlation matrix, the number of STFT windows is set to be larger than the dimension of the matrix (the number of microphones), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', Lw ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The incorporation of the Riemannian geometry is realized by viewing each matrix Γi as a point on the HPD manifold [25] and considering their Riemannian mean, denoted as ΓR and given by: ΓR = arg min Γ∈M Ls � i=1 d2 R(Γ, Γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (17) In general, there is no closed-form solution to (17) on the HPD manifold for more than two points [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Therefore, Algorithm 1 proposed in [27] is used to compute the Riemannian mean of the Ls correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Once ΓR is at hand, the DS beam pattern, PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR), is computed by PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR) = dH(θ)ΓRd(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (18) In the case of a single desired source and assuming the direct path is dominant in the AIR, the direction to it is set as 4 the direction achieving the maximum value of the DS beam pattern, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', ˆθ = arg max θ PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (19) In the case of ND desired sources, the ND directions are set according to the ND strongest lobes in the beam pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The algorithm for a single desired source is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Algorithm 1 Riemannian mean for the HPD manifold [27] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Input: a set of K HPD matrices {Γj}K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Output: the Riemannian mean ΓR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1: Compute ΓR = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j=1 Γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='2: do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1) Compute the Euclidean mean in the tangent plane: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='P = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j=1 LogΓR(Γj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='2) Update ΓR = ExpΓR(P) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='3) Stop if ∥P∥F < ǫ (∥ · ∥F is the Frobenius norm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Algorithm 2 Direction estimation in the presence of multiple ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='interference sources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Input: The received signal in the STFT domain {z(l)}LSTFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='l=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='Output: The estimated direction of the desired source ˆθ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1: Divide {z(l)}LSTFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='l=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='into Ls consecutive segments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='2: For each segment i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' compute the correlation matrix Γi using (16) 3: Compute ΓR of the set {Γi}LS i=1 using Algorithm 1 4: Compute PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR) according to (18) 5: Return ˆθ = arg maxθ PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR) As a baseline, we consider the common practice of the DS beam pattern computation, which is typically based on the sample correlation matrix over the entire interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓE), where ΓE = 1 LSTFT LSTFT � l=1 z(l)zH(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (20) We observe that computing the Euclidean mean of the correlation matrices per segment, {Γi}Ls i=1, results in ΓE in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' So, ΓE in (20) is the Euclidean counterpart of ΓR, the correlation matrix resulting from the Riemannian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We will show that our Riemannian approach exploits the assumption that the desired source is constantly active and at a fixed location, whereas the interference sources are inter- mittent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' More specifically, we will show both theoretically in Section V and empirically in Section VII that the Riemannian mean attenuates the intermittent interferences while preserving the constantly active sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the standard Eu- clidean mean accumulates all the sources, and as a result, the main lobe could deviate from the direction of a desired source, and even focus on an interference source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We show in Section V and Section VII that our proposed approach results in a beam pattern that rejects the interference sources, allowing the beamformer to extract the DoA of the desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We remark that the proposed approach only requires that the desired sources are the only sources active during the entire interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Unlike other works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' [21]), we do not need to know the activation times of each interference, nor the number of interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, we do not assume that there exists a segment, at which a desired source is the only active source, namely, it could always be accompanied by interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In terms of complexity, the Riemannian approach requires the computation of the Riemannian mean of the correlation matrices, which is more complex than the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' However, the dimension of the correlation matrices is deter- mined by the number of microphones in the array, which is typically not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, there is an efficient estimator for the Riemannian mean, which is updated iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Appendix C, we present the estimator along with an imple- mentation for the streaming data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Performance Evaluation To evaluate the performance of the proposed approach, we define the output Signal to Interference Ratio (SIR) as follows: SIRj(Γ) = P(θd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ) P(θi j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ) , (21) where P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ) is the beam pattern computed using the corre- lation matrix Γ, θd is the direction of a desired source, and θi j is the direction of the jth interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' When using the DS beamformer, the output SIR becomes SIRj(Γ) = dH(θd)Γd(θd) dH(θi j)Γd(θi j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (22) This measure of performance is used because the main challenge in this setting is the presence of interference sources rather than the microphone noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ANALYSIS In this section, we analyze the proposed approach which is based on Riemannian geometry and compare it to its Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proofs of the statements appear in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We begin with a short derivation, demonstrating that Rie- mannian geometry preserves better the desired source sub- space in comparison to Euclidean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consider a single desired source and assume its ATF, denoted by h0, is a com- mon eigenvector of all the correlation matrices per segment associated with the same eigenvalue (this assumption is made formal in Assumption 1 in the sequel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this case, according to Lemma 2 (see Appendix D), h0 is an eigenvector of both means and it is associated with the same eigenvalue, namely λ0(ΓR) = λ0(ΓE), where λi(Γ) is the ith eigenvalue of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In addition, all other eigenvectors span the interference and noise subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By the following relation between the means of the two geometries [28] ΓR ⪯ ΓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (23) 5 we get that λ0(ΓR) �M−1 i=1 λi(ΓR) ≥ λ0(ΓE) �M−1 i=1 λi(ΓE) , (24) due to the equality of the numerators and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The inequality in (24) implies that the desired source subspace is more dominant relative to the subspace of the interference and noise in the Riemannian mean compared to the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the remainder of this section, we extend this analysis and present additional results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumptions To make the analysis tractable, we consider a single de- sired source and multiple interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Therefore, we simplify the notations by omitting the superscripts (·)i and (·)d associated with the interference sources and the desired sources and setting the index of the desired source to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the purpose of analysis, we make the following assump- tions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hH 0 hj = 0, ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hH l hj = 0, ∀l ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It follows from Assumption 1 and Assumption 2 that the ATFs, associated with the desired source and the interference sources are all uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' These are common assumptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that we do not assume there exists a segment at which only one of the sources is active (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', as in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In case an interference source is only partially active during a segment, we consider it active during the entire segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the analysis, we consider the population correlation matrix of the received signal, neglecting the estimation errors stemming from the finite sample in a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The population correlation matrix of the ith segment is given by Γi = σ2 0h0hH 0 + HΛiHH + σ2 vIM×M, (25) where the diagonal matrix Λi captures the signal power of the interference sources and is given by: Λi = diag � σ2 1(i) · Ii∈L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' , σ2 NI(i) · Ii∈LNI � , (26) where H = Hi(l, k) due to the omission of the indices and σ2 j (i) = E[|sj(n)|2|n ∈ ith segment] is the expected signal power of the jth interference source at the ith segment, Lj is the set of segments at which the jth interference source is active, and Ii∈Lj is an indicator function, attaining the value of 1 when the jth interference is active during the ith segment and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We denote by τj = |Lj| Ls the relative number of segments during which the jth interference source is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We assume the same expected power at all the segments in the interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', σ2 j (i) = σ2 j for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' , NI and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' , Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We continue with defining the Signal to Noise Ratio (SNR) as SNR = σ2 0 σ2v , (27) where σ2 0 and σ2 v are the power of the desired source and the power of the noise, respectively, and are assumed fixed over the interval as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The power of the desired source in (27) is considered without the effect of the acoustic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that the actual SNR at the microphones is typically significantly lower due to the attenuation of the acoustic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since we focus on a single frequency bin, (27) is the narrowband SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To capture the correlation between the steering vectors and the ATFs, we define ρrs = |⟨dr, hs⟩|2 ∥dr∥2 · ∥hs∥2 = |⟨dr, hs⟩|2 M∥hs∥2 , (28) where r, s = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', NI, indicating the desired source or an interference source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We conclude the preliminaries of the analysis with two additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ρrr is fixed ∀r, and ρrs is fixed ∀r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumption 3 implies that the correlation between the ATFs and the steering vectors depends only on whether they are associated with the same source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Following Assumption 3, henceforth we denote κ = ρrr and ρ = ρrs for r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' κ > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Assumption 4 is typically made in the context of source localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It implies that the correlation between a steering vector to a source and the ATF associated with that source is higher than the correlation between a steering vector to a source and the ATF associated with a different source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Main Results Our first result states that the output SIR (22) of the Riemannian-based DS beam pattern is higher than the output SIR of the Euclidean-based DS beam pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For every interference source j, the following holds SIRj(ΓR) > SIRj(ΓE), (29) for any number of microphones in the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Examining the dependency of the output SIR on the noise power σ2 v leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' If σ2 0∥h0∥2 ≥ σ2 j τj∥hj∥2, ∀j, (30) then ∂ ∂σ2v SIRj(ΓR) < ∂ ∂σ2v SIRj(ΓE) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (31) Namely, the lower the noise power is the higher the SIR is, and the improvement in SIRj(ΓR) is greater than the improve- ment in SIRj(ΓE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since we established that the Riemannian approach is better than the Euclidean one in terms of the SIR in Proposition 1, Proposition 2 implies that increasing the SNR further increases the gap between the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Nevertheless, it also indicates that the performance of the Riemannian approach in terms of the SIR is more sensitive 6 to noise compared to the Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Note that this statement holds under condition (30), which implies that the received power of the desired source is stronger than the received power of each interference source, considering the attenuation stemming from the activity duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' See more details in Appendix E-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proofs of Proposition 1 and Proposition 2 rely on the following lemma, which is important in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian or the Euclidean mean of the population correlation matrices of the segments (25) over the entire interval can be written in the same parametric form as: Γ = σ2 0h0hH 0 + NI � j=1 µ2 jhjhH j + σ2 vI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (32) The Riemannian mean ΓR is obtained by setting the parame- ters µj to µ2 j = (σ2 j ∥hj∥2 + σ2 v)τj(σ2 v)1−τj − σ2 v ∥hj∥2 , (33) and the Euclidean mean ΓE is obtained by setting µ2 j = σ2 j τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (34) We note that only assumptions 1 and 2 are necessary for this lemma to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In addition, we note that if the interference sources are always active, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', |Lj| = Ls, ∀j, it holds that ΓR = ΓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Lemma 1 shows that both ΓR and ΓE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', the population correlation matrix of the segments in (25), can be decom- posed into three terms associated with the desired source, the interference sources, and the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By (32), the desired source term and the noise term (the first and third terms) are the same in both the Riemannian and the Euclidean means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the coefficients in (33) and (34) imply that the amplitude of the interference sources term (the second term) depends on the used geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By further inspecting the expressions of {µ2 j}, we see that the interference attenuation using the Riemannian geometry in (33) is more involved than its Euclidean counterpart in (34), depending not only on the interference power and the duration of activity but also on the noise power and the corresponding ATF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, considering µj in (34), the condition (30) could be viewed as the dominance of the desired source after the attenuation of the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Discussion about the condition (30) for µj in the Riemannian case in (33) appears in Appendix E-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Next, we examine a family of correlation matrices that pertain to the same parametric form as in (32) in Lemma 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', Γa = h0hH 0 + NI � j=1 ajhjhH j + σ2 vI, (35) for some coefficients a = [a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', aNI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Without loss of generality, we set the coefficient of h0hH 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that Γa is in accordance with (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For any j, we have that Γopt ≡ arg max Γa SIRj(Γa) = h0hH 0 + σ2 vI, (36) where a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently, SIRj(Γopt) = dH 0 (h0hH 0 + σ2 vI)d0 dH j (h0hH 0 + σ2vI)dj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (37) Considering vanishing noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', when the noise power approaches zero, the following result stems from Lemma 1 by considering the limit limσ2v→0 µ2 j = 0 using (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' lim σ2v→0 ΓR = Γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (38) According to Corollary 1, the Riemannian mean approaches the optimal correlation matrix as the noise becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' By adding a condition on the presence of the interference sources, from Lemma 1 and (33) we also have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For any interference source j, if τj < 1 2, then lim σ2 j →∞,σ2v→0 ΓR = Γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (39) Additionally, if τj < 1 2 for all j, then lim σ2 j →∞∀j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=',NI,σ2 v→0 ΓR = Γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (40) Corollary 2 implies that for vanishing noise, even when all the interference sources have infinite power, the desired source is still the dominant source in the DS beam pattern using the Riemannian mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Following (37) it holds that limσ2 j →∞,σ2v→0 SIRj(Γopt) = κ ρ > 1 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Euclidean mean it holds that limσ2 j →∞,σ2v→0 SIRj(ΓE) = ρ κ < 1 < κ ρ = limσ2 j →∞,σ2v→0 SIRj(Γopt) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Note that we consider noise power approaching zero rather than strictly zero, because when σ2 v = 0, the correlation matrix is singular, and therefore, lies outside the HPD manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, in practice, noise is always present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To illustrate the obtained expressions for the Riemannian and the Euclidean SIR, we present the following simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consider an anechoic environment without at- tenuation, for which κ = 1 and ρ = 0, and two interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Each interference source is active at a different segment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', Ls = 2, L1 = {1}, and L2 = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' All the sources have the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this setting, for the Riemannian geometry, we have SIRj(ΓR) = � M σ2v + 1, (41) and for Euclidean geometry, we have: SIR(ΓE) = 2(M + σ2 v) M + 2σ2v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (42) Therefore, in the limit of σ2 v → 0, or M → ∞, we have SIR(ΓR) = ∞, whereas SIR(ΓE) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We conclude this analysis with a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' First, we note that ΓE leads to the ML estimator by taking ˆθ0 = arg maxθ PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓE) for the interference-free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this case, the Riemannian mean coincides with the Euclidean 7 mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', the proposed method coincides with the ML es- timator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The main advantage of the proposed method lies in attenuating the interference sources while preserving the de- sired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Second, under assumptions 1 and 2, the number of sources, both desired and interference, are limited by the number of microphones in the array, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', NI + ND < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section V-C we alleviate Assumption 2, which removes this re- striction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Third, following the same techniques in the proof of Proposition 1 and Proposition 2, similar results are derived for an alternative definition of the SIR: SIRtot(Γ) ≡ dH 0 Γd0 �NI j=1 dH j Γdj , which captures the ratio between the desired source and the sum of all interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' See Appendix F for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Relation to Signal Enhancement For signal enhancement in reverberant environments, the estimation of the ATF of the desired source is typically required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In our setting, there is no segment at which the desired source is the only active source, and therefore, the ATF estimation is done in the presence of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In such a case, the following quantity could be of interest: SIRj(Γ) = hH 0 Γh0 hH j Γhj , (43) which is different than (22) in the use of the ATFs instead of the steering vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Similarly to Proposition 1, the following Proposition 3 examines the performance in terms of the SIR defined in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Here, assumptions 2-4 are not required, and therefore, the ATFs of the interference sources could be correlated, and the number of sources is not limited by the number of microphones in the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Under Assumption 1, for all j we have: SIRj(ΓR) ≥ SIRj(ΓE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (44) Another interesting component in signal enhancement is the Relative Transfer Function (RTF) between different micro- phones [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We compute the RTFs with respect to the first microphone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', hj hj(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since the RTFs are proportional to the ATFs, assumption 1 holds for the RTFs, and therefore, all the derived results apply to the RTFs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Segments and the Interference Sources Activity In this section, we investigate the effect of misalignment between the segments and the activity of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We consider two interference sources and two seg- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We denote by α the offset between the segments and the activity of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For simplicity, we consider alternately active interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Suppose the first interference source is active during α ∈ [0, 1] of the first segment and during 1−α of the second segment, and suppose the second interference source is active during 1 − α and α of the first and second segments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this case, the correlation matrices of the two segments are given by: Γ1(α) = σ2 0h0hH 0 + α2σ2 1h1hH 1 + (1 − α)2σ2 2h2hH 2 + σ2 vI Γ2(α) = σ2 0h0hH 0 + (1 − α)2σ2 1h1hH 1 + α2σ2 2h2hH 2 + σ2 vI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (45) The correlation matrices in (45) depend on α, and as a result, their Riemannian mean ΓR(α) and their Euclidean mean ΓE(α) depend on α as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Examining the dependency of the SIR on α leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For any α ∈ [0, 1], we have SIR(ΓR(α)) ≥ SIR(ΓE(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (46) Proposition 4 states that for every misalignment between the segments and the activity of the interference sources, the Riemannian mean leads to higher SIR in comparison to its Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Equality in (46) is obtained for α = 1 2, which means 50% offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In this case, it holds that Γ1 = Γ2, and both means are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Empirically, we found that the advantage of the Riemannian mean over the Euclidean mean decreases as the offset between the segments and the activity of the interference sources increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We leave the question of optimal partitioning of the STFT windows into segments to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' EXTENSION TO OTHER BEAMFORMERS In this section, to broaden its applicability, we demonstrate the incorporation of the Riemannian approach in other beam- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Each beamformer generates a beam pattern from which the directions to the desired sources are estimated according to the highest peaks in the beam pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' As a subspace (SbSp) approach, we implement MUSIC [13] in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' given ND desired sources, we take the leading ND eigenvectors of the correlation matrix, Γ, and construct the signal subspace matrix, U(Γ) ∈ CM×ND, whose columns are the ND eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then, the SbSp beam pattern is defined by PSbSp(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ) = dH(θ)U(Γ)UH (Γ)d(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (47) We note that the appropriate number of eigenvectors ND (the dimension of the subspace) needs to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Similarly to the DS beamformer based on the DS beam pattern in (18) and (19), the Riemannian and the Euclidean SbSp methods are given by PSbSp(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR) and PSbSp(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓE), respectively, according to (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The SbSp method could also benefit from our Riemannian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Recalling assumptions 1 and 2 and the structure of the mean correlation matrices ΓR and ΓE in (32), we see that h0 is an eigenvector of both ΓR and ΓE, spanning the signal subspace (we assume a single desired source for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Moreover, the vectors {hj} are also eigenvectors of ΓR and ΓE, spanning the subspace of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' SbSp methods typically focus on the principal eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Following (32), the principal eigenvector is determined according to the largest coefficient among σ0 and {µj}NI j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The parameter µj in (33) for the Riemannian mean ΓR is smaller than µj in (34) for the Euclidean mean 8 ΓE, as proven in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' As a result, the signal sub- space is more dominant relative to the interference subspace when considering the Riemannian mean in comparison to the Euclidean mean, implying better results for the Riemannian SbSp approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For an interference source with sufficiently high power, the leading eigenvector of ΓE could span the interference subspace rather than the signal subspace, whereas the leading eigenvector of ΓR spans the signal subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In Section VII, we demonstrate these SbSp methods empirically and show the advantage of the Riemannian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Our approach is also applicable to the MVDR beamformer [10], whose beam pattern is given by PMVDR(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Γ) = 1 dH(θ)Γ−1d(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (48) The typical beam pattern of the MVDR beamformer is ob- tained by using ΓE in (48), namely PMVDR(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We propose to use the Riemannian mean by setting Γ to be ΓR in (48) to obtain PMVDR(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ΓR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we demonstrate the performance of the pro- posed approach based on Riemannian geometry, and compare it to Euclidean geometry, implicitly considered by the common practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, we compare our approach to a heuristic method, based on the intersection of subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The intersec- tion leads to the rejection of non-common components, such as the interference sources subspace, and preserves common components, such as the desired source subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We refer to it as the intersection beamformer (see Appendix A for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We consider a reverberant enclosure of dimensions 5m × 4m × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5m consisting of a microphone array with M = 12 microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The AIRs between the different sources and the array are generated based on the image method [32], as im- plemented by the simulator in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The sampling frequency is 16KHz, and the length of the AIRs is set to 2048 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='The emitted signals are generated as white Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The desired source is constantly active, whereas the interference sources are active only intermittently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The received signal is transformed to the time-frequency domain using STFT with a window size of 1024 samples with 50% overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We test all methods using a single frequency bin, of index 250, chosen according to the microphone spacing, which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='36cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Here, the correlation matrix estimation is based on 16 STFT windows, which results in a segment duration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='024s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' All the sources are positioned on a 140◦ arc of radius 2m from the center of the array on the XY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The heights of the interference sources vary randomly, uniformly distributed between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5m and 3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The height of the desired source is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='8m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 1 presents the room layout, where the (two) interferences are marked by green squares, the desired source by a red star, and the microphone array by blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The leftmost microphone is positioned at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='0436m, 1m, 2m), and the rest are positioned 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='36cm apart along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' While we focus on this specific configuration, we note that we tested other configurations that yielded similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 0 1 2 3 4 5 x[m] 1 2 3 y[m] Microphone Target Interference Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The reverberant room with the microphone array (blue circles), the desired source (red star), and the interference sources (green squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Left: a 3D view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Right: a 2D view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We examine the performance of both the DS and the SbSp methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Algorithm 2 is used for the proposed Riemannian DS, and the common practice is implemented by replacing step 3 in Algorithm 2 with (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The SbSp methods require knowing the dimension of the signal space of the mean corre- lation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the intersection method only, the dimension of the signal space of each segment is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since the desired source may not be the strongest source received at the array, we need to consider all the active sources, and not merely the strongest one when estimating the dimension of the signal subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To estimate the dimension, we implement a heuristic algorithm, based on the spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We consider the dimension of the signal space to be the number of eigenvalues higher than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For all methods, the threshold for the mean correlation matrix is the mean value plus the standard deviation of the eigenvalues (normalized to a unit sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the intersection method, the threshold for the correlation matrix of each segment is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5 times the mean of the eigenvalues of the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Apart from this practical implementation, we also present results for an oracle implementation, assuming the dimension of the signal space is perfectly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For quantitative evaluation, we use two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The first is the mean of the empirical output SIR with respect to all the interference sources, which is given by: SIR = 1 NI NI � j=1 P(θd) P(θi j) , (49) where P is the beam pattern computed using the evaluated method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The second metric is the directivity [34, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='2], which is given by D(Γ) = P(θd) 1 2 � π 0 P(θ) sin(θ)dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (50) In the first experiment, we consider two interference sources, each active at a single, but disjoint, segment, resulting in a signal of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='048s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The reverberation time is set to 150ms, and the SNR is 50dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To remove the dependency on the particular layout, the SNR follows the definition in (27), which is with respect to the sources’ emitted power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that the effective SNR at the microphone is significantly lower (as the power of the source is attenuated by the AIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Empirically, the effective SNR at the microphones is approximately 30dB lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, since we focus on a single frequency bin, we consider the narrowband SNR, which is typically much higher than the broadband SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Microphone Target Interference4 352 1 0 4 2 4 3 2 1 y[m] 0 0 x[m]9 0 30 60 90 120 150 180 20 15 10 5 0 R (Riem) E (Euc) Desired Interference 0 30 60 90 120 150 180 20 15 10 5 0 1 (1st segment) 2 (2nd segment Desired Interference Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The DS beam pattern using ΓR in solid blue and ΓE in dashed red (top), and Γ1 and Γ2 in different shades of orange (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The black solid line indicates the direction of the desired source, and the dashed black lines indicate the directions to the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Input SIR is −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We start with an example of the DS beam pattern (see (18)), computed using ΓR and ΓE, which is presented at the top of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian DS beam pattern is shown in solid blue and the Euclidean DS beam pattern is shown in dashed red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Both beam patterns are in a dB (log) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The directions to the desired source and the interference sources are represented by a black solid line and a dashed black line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that by using ΓR the main lobe is directed towards the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the beam pattern using ΓE is peaked at 2 different directions, none of which is the direction to the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The bottom of Figure 2 is the same as the top, only with the beam pattern computed using the correlation matrix of each of the two segments, presented in different shades of orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Even though the main lobes of the two beam patterns are not pointing toward the desired source, the Riemannian mean leads to a beam pattern with the main lobe directed at the desired source, whereas the lobes to other directions are highly attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We emphasize that viewing the Euclidean DS beam pattern in addition to the beam pattern of each segment separately does not allow correct identification of the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Next, we randomly generate 200 different pairs of positions for the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For each pair, the target source is located at 20 different equally spaced directions along the arc (with the height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='8m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Thus, in total, 4000 different scenarios are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 3 presents the mean output SIR (top) and the directivity (bottom) for the DS method using the correlation matrix estimates: ΓR (based on Riemannian geometry) in blue and ΓE (based on Euclidean geometry) in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The box indicates the 25th and 75th percentiles, and the line marks the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We test the different methods, Riemannian or Euclidean, in varying input SIR values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' the SIR with respect to the source’s power (excluding the AIRs and the beamformer processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian DS method attains high output SIR values, even for strong interference sources (high input SIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the Euclidean DS method results in relatively low output SIR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The gap in the output SIR values 15 10 5 0 5 10 15 [dB] Riemannian Euclidean 6[dB] 6[dB] Riemannian Euclidean 10[dB] 10[dB] Riemannian Euclidean 20[dB] 20[dB] 10 5 0 5 [dB] Riemannian Euclidean 6[dB] 6[dB] Riemannian Euclidean 10[dB] 10[dB] Riemannian Euclidean 20[dB] 20[dB] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The mean output SIR (top) and the directivity (bottom) for two interference sources, for the Riemannian and the Euclidean DS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The x-axis indicates the input SIR, and the y-axis indicates the output SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The box indicates the 25th and 75th percentiles, and the central line marks the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Several input SIR values are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' between the Riemannian DS, and the Euclidean DS is up to 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' These results coincide with Proposition 1, stating that SIRj(ΓR) > SIRj(ΓE), for every interference source j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We emphasize that the Euclidean mean, ΓE, is equivalent to the common practice of using the entire signal for a single correlation matrix estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since both the mean output SIR and the directivity present similar trends, and due to space considerations, in the following, we only present the mean output SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 4 is the same as Figure 3, but presenting the SbSp method with the addition of the intersection method, which appears in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The top subfigure presents the results for the practical implementation that includes estimating the dimension, whereas the bottom subfigure presents the results for the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian approach outper- forms its Euclidean counterpart, by approximately 20dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In addition, the oracle SbSp method is better than the practical SbSp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In comparison to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 3(top), it can be seen that the Riemannian SbSp method results in higher output SIRs than the Riemannian DS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, for the Euclidean approach, the SbSp method yields slightly lower SIRs than the DS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The reason is that the Riemannian mean better attenuates the interference sources, allowing for a better estimation of the signal subspace than the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We continue with examining the direction estimation to the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The estimated direction is defined as the direction leading to the maximal value of the beam pattern, namely ˆθd = arg max θ P(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (51) As a baseline, we compare the results with a version of the SRP-PHAT algorithm [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We use its position estimation to compute the direction of the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 5 10 20 0 20 40 [dB] Riemannian Euclidean Intersection 6[dB] 6[dB] 6[dB] Riemannian Euclidean Intersection 10[dB] 10[dB] 10[dB] Riemannian Euclidean Intersection 20[dB] 20[dB] 20[dB] 20 10 0 10 20 30 [dB] Riemannian Euclidean Intersection Oracle Oracle Oracle 6[dB] 6[dB] 6[dB] Riemannian Euclidean Intersection Oracle Oracle Oracle 10[dB] 10[dB] 10[dB] Riemannian Euclidean Intersection Oracle Oracle Oracle 20[dB] 20[dB] 20[dB] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The mean output SIR for the SbSp methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Top: practical imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Bottom: Using an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The box indicates the 25th and 75th percentiles, and the central line marks the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The x-axis indicates the input SIR, and the y-axis indicates the output SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Several input SIR values are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 20 40 60 80 100 120 140 160 Target direction [deg] 20 40 60 80 100 120 140 160 [deg] Riem Euc Itersection SRP-PHAT Desired Interferece 20 40 60 80 100 120 140 160 Target direction [deg] 20 40 60 80 100 120 140 160 [deg] Riem Euc Itersection SRP-PHAT Desired Interferece Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The DoA estimation to the desired source for input SIR of −6dB (left) and input SIR of −10dB (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' shows the estimated direction to the desired source for the Riemannian DS method (blue square), the Euclidean DS method (red circle), the intersection method (orange star), and the SRP-PHAT method (purple triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The solid black line marks the true location of the target source (at 20 different positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The dashed line marks the fixed location of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The results for input SIR −6dB and −10dB appear on the left and right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that using the Riemannian mean, the direction estimation follows the desired source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, using the Euclidean mean (the entire signal) results in estimating the direction of one of the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The direction estimation using SRP- PHAT follows the other interference source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The intersection method is also inferior to the proposed approach, resulting in direction estimation to the desired source or an interference source depending on the input SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Next, we examine the sensitivity of the proposed approach to the SNR and the reverberation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We repeat the setting of the two interferences, as described in the first experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The results are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' At the top, the mean output SIR for the DS method is presented as a function of the reverberation time for a fixed SNR of 50dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Several input SIR values are shown: 0dB (asterisks), −6dB (circle), and −10dB (triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The results for the Riemannian and 150 200 250 300 350 400 [msec] 5 0 5 10 15 [dB] 0[dB] (Riem) 6[dB] (Riem) 10[dB] (Riem) 0[dB] (Euc) 6[dB] (Euc) 10[dB] (Euc) 20 25 30 35 40 45 50 SNR[dB] 5 0 5 10 15 [dB] 0[dB] (Riem) 6[dB] (Riem) 10[dB] (Riem) 0[dB] (Euc) 6[dB] (Euc) 10[dB] (Euc) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The mean output SIR as a function of the reverberation times (top) and as a function of the SNR (bottom), for 2 interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian DS appears in blue, whereas the Euclidean DS appears in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Several input SIR values are presented: 0dB (asterisks), −6dB (circle), and −10dB (triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We recall the SNR at the array is approximately 30dB lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Euclidean DS appear in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The bottom figure is the same as the top, only for different SNR values for a fixed reverberation time of β = 150ms (we recall the SNR at the array is approximately 30dB lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the smaller the reverberation time is, the higher the output SIR is for the Riemannian DS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the Euclidean DS is less affected by the reverberation time, resulting in relatively low output SIRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For all values of reverberation times, the Riemannian DS results in higher output SIR than its Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' From the bottom figure, we see that the SNR has a large impact on the performance of the Riemannian DS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' the higher the SNR is, the higher the output SIR becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Conversely, the Euclidean DS is moderately affected by the SNR, resulting in much lower output SIR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' A possible explanation is that the main phenomenon limiting the performance of the Euclidean approach is the existence of interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In addition, as Proposition 2 predicts, the sensitivity of the Riemannian DS to the SNR is higher than the Euclidean DS, and the higher the SNR is, the larger the gap in the performance between the Riemannian and the Euclidean approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We examine the performance of the MVDR beam pattern, given by (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The MVDR beamformer is popular when interference sources are present, thanks to its distortionless response in the direction of the desired source, and its typical narrow beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' However, in our setting, since the desired source is accompanied by interference sources, and the direc- tions to all the sources are unknown, the DoA estimation of the desired source using the MVDR beamformer is outperformed by the DS and the SbSp methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We also examine the case of 2 desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The results show similar trends and appear in Appendix B, due to space considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 11 1 2 3 4 5 6 7 8 9 10 Segment index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Interference index Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Left: Activation map for the 14 interference sources during the 10 segments (blue indicates ‘active’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Right: Output SIR of the different beamformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The input SIR of each interference is −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the second experiment, we examine a multiple interfer- ence setting, by considering NI = 14 interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that the number of interference sources is larger than the number of microphones, NI > M = 12, which typically limits the number of interference sources that can be accommodated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Furthermore, assumptions 1 and 2 implicitly restrict the number of interference sources to be bounded by M − ND = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This limitation is only for the analysis, and, in practice, improved results are obtained even for a larger number of interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The signal contains 10 segments, demonstrating the number of segments could be smaller than the number of interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The duration of the emitted signal is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='24s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The input SIR for each interference is −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Each interference has a 30% probability of being active at each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The position of all the sources is set at random on the arc, as described in the first experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The activation map of the interference sources at the first Monte Carlo iteration appears in Figure 7 (left), where light blue indicates ‘active’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' On average, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='4 interference sources are active at the same time at each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' An interference source that is partially active during a segment is considered active during the entire segment (a worst case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Note that there are interference sources active continuously during more than one segment, so their activation is not necessarily related to the division of the received signal into segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, there exists no segment in which only one source is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The output SIRs are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 7(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian DS and SbSp methods appear in blue, the Euclidean DS and SbSp methods are in red, and the intersection is in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It can be seen that the Riemannian approach is superior to the Euclidean one, resulting in higher output SIRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' CONCLUSION We present a Riemannian approach for the design of dif- ferent beamformers for interference rejection in reverberant environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The resulting beam pattern is used for DoA estimation of the desired source, as it rejects the beams directed at the interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We analytically show that the DS beamformer, based on the Riemannian geometry of the HPD manifold, results in a higher output SIR than the typical DS beamformer, which implicitly considers the Euclidean ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We extend our approach to other beamformers, such as subspace-based beamformers and the MVDR, experimentally demonstrating superior output SIR and better DoA estimations in comparison to their 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 13 APPENDIX A THE INTERSECTION BEAMFORMER Another beamformer we examine is based on the observation that the desired signal subspace is the intersection of subspaces spanned by eigenvectors of the correlation matrix of the different segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' From each segment, i, we extract the desired signal subspace from the correlation matrix, Γi, to obtain V (Γi) whose columns are the ND leading eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The projection matrix onto the signal subspace of Γi is computed as follows: Psig(Γi) = V (Γi) � V H(Γi)V (Γi) �−1 V H(Γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (52) Since each desired source is active during all the segments, its ATF, hd j, is an eigenvector of V (Γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently it is also an eigenvector of Psig(Γi) for all i, with an eigenvalue 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Psig(Γi)hd j = hd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' As a result, it is an eigenvector of Psig = �Ls i=1 Psig(Γi) with eigenvalue 1 as well since Psighd j = ��Ls i=1 Psig(Γi) � hd j = hd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We consider the leading ND eigenvectors of Psig and form the matrix Psig,ND ∈ CM×ND, whose columns are the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The intersection beam pattern is defined as follows: PIntersect(θ) = dH(θ)Psig,NDP H sig,NDd(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (53) We note that in addition to the dimension of the signal space of the matrix Psig, the intersection method requires the estimation of the dimension of the signal space of the correlation matrix for each segment (in order to compute V (Γi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' APPENDIX B ADDITIONAL EXPERIMENTAL RESULTS We repeat the setting of the first experiment, only with an additional desired source, located at (2m, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5m, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 8 is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 3, but the mean output SIR is taken over the two interferences and the two desired sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The left figure presents results for input SIR of −10dB, and the right presents results for input SIR of −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian approach is superior to the Euclidean one, resulting in higher SIRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Also, the intersection method is more sensitive to the input SIR, resulting in higher output SIR for −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Similar to the setting of one desired source, the Riemannian SbSp method is superior to the Riemannian DS method, as opposed to the Euclidean approach, for which they are on par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This is a consequence of the higher interference attenuation of the Riemannian approach, resulting in a better estimate of the signal space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Next, we examine the performance of the Riemannian MVDR, and the Euclidean MVDR beamformers, given by (48), for ΓR and ΓE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Figure 9 presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 3, only for the MVDR beamformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian approach is superior to the Euclidean one, however, with a slight advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' APPENDIX C EXTENSION TO A STREAMING DATA SETTING In a streaming data setting, we do not have access in advance to the entire signal, and the direction estimation is updated as more samples become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Therefore, in this setting, we cannot compute the Riemannian mean using (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' To circumvent this, we turn to the estimator of the Riemannian mean proposed in [36], [37], which is updated after every received segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Each segment, i, is processed separately using Lw STFT windows, and an estimation of the correlation matrix, Γi, is computed using (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Next, the estimation of the Riemannian mean, denoted as ˆRi, which is the adaptive counterpart of ΓR, is updated using the following update step: ˆRi = ˆR 1 2 i−1( ˆR − 1 2 i−1Γi ˆR − 1 2 i−1) 1 i ˆR 1 2 i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (54) 5 0 5 10 15 20 25 [dB] Riemannian Euclidean DS DS Riemannian Euclidean Intersection SbSp SbSp SbSp Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Output SIR of the different beamformers for two desired sources and two interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Left: Input SIR is −10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Right: Input SIR is −6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 30 25 20 15nannian Euclidean Intersection SbSp SbSp SbSp10 白 5 0 5 10 Riemannian Euclidean Rier DS DS14 5 0 5 10 [dB] Riemannian Euclidean 6[dB] 6[dB] Riemannian Euclidean 10[dB] 10[dB] Riemannian Euclidean 20[dB] 20[dB] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The mean output SIR for the Riemannian and the Euclidean MVDR beamformer in the presence of two interference sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The x-axis indicates the input SIR, and the y-axis indicates the output SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The box indicates the 25th and 75th percentiles, and the central line marks the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Several input SIR values are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The DS beam pattern is computed using (18) with ˆRi as an estimate of ΓR, and the direction to the desired source is set using (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The algorithm is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that the current correlation matrix estimate has a full rank if Lw > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This implies a latency of at least M times the duration of the STFT window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In case the STFT is performed with overlap this latency is reduced accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Algorithm 3 Streaming DoA estimation in the presence of multiple interferences Input: The current result of the STFT window of the received signal Output: The estimated direction of the desired source ˆθ 1: Set ˆR0 = I 2: Repeat 1) Accumulate Lw STFT windows (that form a segment) 2) Compute its correlation matrix, Γi, using (16) 3) Compute ˆRi = ˆR 1 2 i−1( ˆR − 1 2 i−1Γi ˆR − 1 2 i−1) 1 i ˆR 1 2 i−1 4) Compute PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ˆRi) using (18) 5) Return ˆθ = arg maxθ PDS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ˆRi) As for the Euclidean alternative, its estimator is updated in finer granularity at every STFT window, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The estimator at the lth update is denoted by El and is computed as follows: El = n − 1 n El−1 + 1 nz(l)zH(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (55) Setting Lw = 1 in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 in Algorithm 3, and substituting (55) in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='3 in Algorithm 3, results in the streaming version of the Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' APPENDIX D ON THE PARTICULAR CHOICE OF THE RIEMANNIAN METRIC In this work, we consider the Affine Invariant metric (also called Fisher Information metric) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Another commonly-used metric in the space of HPD matrices is the Log-Euclidean metric [39], which could be viewed as a computationally efficient local approximation of the Affine Invariant metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The induced Log-Euclidean distance is given by: d2 LE(Γ1, Γ2) = ∥ log(Γ1) − log(Γ2)∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (56) In the context of this work, the Affine Invariant metric is advantageous over the Log Euclidean because it better enhances the desired source subspace relative to the interference and noise subspace, as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We remark that the derivation is similar to the derivation in Section V, where we demonstrate the advantage of the Affine Invariant metric over Euclidean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' First, we note that the following holds [40]: tr(ΓR) ≤ tr(ΓLE), (57) 15 where ΓLE denotes the Riemannian mean based on the Log-Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Second, the ATF to the desired source, h0, is a common eigenvector of all the correlation matrices per segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently, according to Lemma 2 below, the mean correlation matrices based on both metrics have the same eigenvalue associated with the common eigenvector h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' More specifically, denoting λ0(Γ) ≡ hH 0 Γh0 ∥h0∥2 , we get λ0(ΓR) = λ0(ΓLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Let {Γj}j be the set of HPD matrices, having a common eigenvalue λ0, associated with the common eigenvector h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then hH 0 ΓRh0 = hH 0 ΓLEh0 = hH 0 ΓEh0 = ∥h0∥2λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (58) where ΓR, ΓLE, ΓE are the Riemannian means based on the Affine Invariant and the Log-Euclidean metrics, and the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The rest of the eigenvectors of the correlation matrices span the interference and noise subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We recall that λi(Γ) is the ith eigenvalue, so we get λ0(ΓR) �M−1 i=1 λi(ΓR) ≥ λ0(ΓLE) �M−1 i=1 λi(ΓLE) , (59) which follows from Lemma 2, and (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian mean induced by the Affine Invariant metric captures better the desired signal subspace in comparison to the Riemannian mean induced by the Log Euclidean metric and the Euclidean mean (according to (24)), entailing an advantage in SbSp methods, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' APPENDIX E PROOFS OF THEORETICAL RESULTS A key observation in our analysis is that although there is no explicit expression for the Riemannian mean, in general, under assumptions 1 and 2, the correlation matrices {Γl} commute, and the Riemannian mean has an explicit form, according to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We note that in practice we use Algorithm 1 to compute the Riemannian mean because the assumptions do not strictly hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For completeness, we prove Proposition 5, which is a property of Karcher’s mean [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian mean of K commuting HPD matrices, {Γl}K l=1 is given by ΓR = K � l=1 Γ 1 K l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (60) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Karcher mean, ΓR, is the solution of the Karcher equation [26], [28], [41], [42] K � l=1 log(Γ 1 2 R Γ−1 l Γ 1 2 R ) = 0 (61) We set ΓR = �K l=1 Γ 1 K l , and use the fact the matrices commute: K � l=1 log(Γ 1 2 R Γ−1 l Γ 1 2 R ) = K � l=1 log(ΓRΓ−1 l ) = K � l=1 log � Γ−1 l K � l=1 Γ 1 K l � = log K � l=1 � Γ−1 l K � l=1 Γ 1 K l � = log I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (62) 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Proposition 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proof relies on Lemma 1, which we prove first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian or the Euclidean mean of the population correlation matrices of the segments (25) over the entire interval can be written in the same parametric form as: Γ = σ2 0h0hH 0 + NI � j=1 µ2 jhjhH j + σ2 vI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (63) The Riemannian mean ΓR is obtained by setting the parameters µj to µ2 j = (σ2 j ∥hj∥2 + σ2 v)τj(σ2 v)1−τj − σ2 v ∥hj∥2 , (64) and the Euclidean mean ΓE is obtained by setting µ2 j = σ2 j τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (65) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We start by expressing the correlation matrix of the ith segment: Γi = σ2 0h0hH 0 + � j∈Ji σ2 j hjhH j + σ2 vI, (66) where Ji is the set of indices of the active interference sources during the ith segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The vectors h0, and {hj}NI j=1 are all orthogonal, thus they are eigenvectors of Γi, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=', Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We denote by upper tilde the normalized version of a vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ˜h = h ∥h∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' So, ˜hH 0 Γi˜h0 = σ2 0∥h0∥2 + σ2 v ∀i ˜hH j Γi˜hj = σ2 j ∥hj∥2 + σ2 v i ∈ Lj ˜hH j Γl˜hj = σ2 v i /∈ Lj (67) The correlation matrices for each segment, {Γi}Ls i=1, commute with each other because they share their eigenvectors: h0, {hj}LI j=1, and {vl}M−NI−1 l=1 , where {vj}M−NI−1 j=1 are the eigenvectors spanning the common noise subspace of all the matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Following Proposition 5, it holds true that ΓR = Ls � j=1 Γ 1 Ls j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (68) We compose ΓR using a transformation for the eigenvalues for each Γi: ˜hH j ΓR˜hj = (σ2 j ∥hj∥2 + σ2 v)τj(σ2 v)1−τj, ∀j ˜hH 0 ΓR˜h0 = σ2 0∥h0∥2 + σ2 v, (69) and the rest of the eigenvectors have an eigenvalue of σ2 v, with multiplicity of M − NI − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The matrix ΓR meets all the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proof for the Euclidean mean follows from its definition and (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We compute the output SIR for a general matrix with a similar structure as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The correlation matrix is: Γ = σ2 0hH 0 h0 + NI � l=1 µ2 l hH l hl + σ2 vI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (70) and the SIR becomes: SIRj(Γ) = dH 0 � σ2 0hH 0 h0 + �NI l=1 µ2 l hH l hl + σ2 vI � d0 dH j � σ2 0hH 0 h0 + �NI l=1 µ2 l hH l hl + σ2vI � dj = σ2 0|⟨dH 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' h0⟩|2 + �NI l=1 µ2 l |⟨dH 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hl⟩|2 + σ2 vM σ2 0|⟨dH j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' h0⟩|2 + �NI l=1 µ2 l |⟨dH j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hl⟩|2 + σ2vM = σ2 0M∥h0∥2κ + �NI l=1 µ2 l M∥hl∥2ρ + σ2 vM σ2 0M∥h0∥2ρ + �NI l=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='l̸=j µ2 l M∥hl∥2ρ + µ2 jM∥hj∥2κ + σ2vM = 1 + (σ2 0∥h0∥2 − µ2 j∥hj∥2)κ + (µ2 j∥hj∥2 − σ2 0∥h0∥2)ρ σ2 0∥h0∥2ρ + �NI l=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='l̸=j µ2 l ∥hl∥2ρ + µ2 j∥hj∥2κ + σ2v (71) 17 We note that the SIR depends on the number of microphones implicitly through the norm of the ATFs ∥hj∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' and ∥h0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since κ, ρ ≥ 0, and κ > ρ, the smaller µ2 j, µ2 l ∀l are, the higher the SIR is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Using Lemma 1, we identify the coefficients for the Riemannian mean as: µ2 l = (σ2 l ∥hl∥2 + σ2 v)τl(σ2 v)1−τl − σ2 v ∥hl∥2 , (72) and for the Euclidean mean as µ2 l = σ2 l τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It is left to show that the coefficients for the Riemannian SIR are smaller than for the Euclidean SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' So, we examine the conditions under which (σ2 l ∥hl∥2 + σ2 v)τl(σ2 v)1−τl − σ2 v ∥hl∥2 < σ2 l τj (σ2 l ∥hl∥2 + σ2 v)τl(σ2 v)1−τl < σ2 l ∥hl∥2τj + σ2 v (73) Equation (73) holds due to the weighted arithmetic mean and geometric mean inequality [43, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 111–112, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='5] [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We see that the Riemannian mean of the correlation matrices leads to the geometric mean of the noise power and the interference power plus the noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In contrast, the common practice that is the Euclidean mean of the correlation matrices leads to the arithmetic mean of the powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Lemma 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Euclidean geometry, the proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Riemannian geometry, since, in general, there is no close form expression for the Riemannian mean, we use its definition as the minimizer of ΓR = arg min Γ � j d2 R(Γ, Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (74) For the Affine Invariant metric, the following holds d2 R(Γ1, Γ2) = � l log2 � λl(Γ − 1 2 1 Γ2Γ − 1 2 1 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (75) Since the matrix Γ − 1 2 j ΓRΓ − 1 2 j is a function of the matrices Γj, we have that h0 is also its eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We notice that min log2 � λ0(Γ − 1 2 j ΓRΓ − 1 2 j ) � = min log2 �λ0(ΓR) λ0(Γj) � = 0, (76) for λ0(ΓR) = λ0(Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This is the minimum possible value for every j, so it must hold that λ0(ΓR) = λ0(Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Log-Euclidean distance, we use eigenvalue decomposition and denote by {ui} and {vk} the eigenvectors of ΓLE and Γj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We have d2 LE(ΓLE, Γj) = ∥ ln(ΓLE) − ln(Γj)∥2 F = ∥ � i ln λi(ΓLE)uiuH i + ln λ0(ΓLE)h0hH 0 − � k ln λk(Γj)vkvH k − ln λ0(Γj)h0hH 0 ∥2 F = ∥C + (ln λ0(ΓLE) − ln λ0(Γj))h0hH 0 ∥2 F = tr � CCH + (ln λ0(ΓLE) − ln λ0(Γj))2 · ∥h0∥2 · h0hH 0 � , (77) which is minimal for every j, for λ0(ΓLE) = λ0(Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The last equality is due to the orthogonality between h0 and the rest of the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Proposition 2 We start by proving Proposition 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' showing that ∂ ∂σ2v SIRj(ΓR) < ∂ ∂σ2v SIRj(ΓE) < 0, (78) and continue with examining the conditions under which ∂ ∂σ2v SIRj(ΓR) < 0, (79) ∂ ∂σ2v SIRj(ΓE) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (80) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proof employs the following technical Lemma Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For µ2 j given by (33), the following holds: 1) ∂ ∂σ2v µ2 j ≥ 0 and µ2 j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 2) ∂ ∂σ2v �NI l=1,l̸=j µ2 l ∥hl∥2 ≥ 0 and �NI l=1,l̸=j µ2 l ∥hl∥2 ≥ 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ∂ ∂σ2v µ2 j = ∂ ∂σ2v � (σ2 j ∥hj∥2 + σ2 v)τj(σ2 v)1−τj − σ2 v ∥hj∥2 � = 1 ∥hj∥2 ∂ ∂σ2v � (σ2 j ∥hj∥2 + σ2 v)τj(σ2 v)1−τj − σ2 v � = 1 ∥hj∥2 � τj · � σ2 j ∥hj∥2(σ2 v) 1 τj −1 + (σ2 v) 1 τj �τj−1 �� 1 τj − 1 � σ2 j ∥hj∥2(σ2 v) 1 τj −2 + 1 τj (σ2 v) 1 τj −1 � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (81) We focus on the expression in the bracket, rewrite it as a fraction, and show that it is larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj (σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 + (σ2v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�1−τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='= (1 − τj) σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −2 + (σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 + (σ2v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�1−τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='= (1 − τj) σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −2 + (σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='(σ2v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='τj −1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2v)−1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�1−τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='= (1 − τj) σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v)−1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2v)−1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='�1−τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='= 1 + (1 − τj) σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='v)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='1 + σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='j ∥hj∥2(σ2v)−1�1−τj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (82) The last inequality follows from Bernoullie”s inequality (1 + x)r ≤ 1 + rx, by setting 0 < r = (1 − τj) < 1, and x = σ2 j ∥hj∥2(σ2 v)−1 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since µ2 j = 0 for σ2 v = 0 and ∂ ∂σ2v µ2 j ≥ 0, it holds that µ2 j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The above holds for all j thus it also holds for their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We recall the expression for the SIR from (71): SIRj(Γ) = 1 + (σ2 0∥h0∥2 − µ2 j∥hj∥2)κ + (µ2 j∥hj∥2 − σ2 0∥h0∥2)ρ σ2 0∥h0∥2ρ + �NI l=1,l̸=j µ2 l ∥hl∥2ρ + µ2 j∥hj∥2κ + σ2v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (83) We denote the following functions, which are the numerator and the denominator of the expression of the Riemannian SIR: fR(σ2 v) = (σ2 0∥h0∥2 − µ2 j∥hj∥2)κ + (µ2 j∥hj∥2 − σ2 0∥h0∥2)ρ (84) gR(σ2 v) = σ2 0∥h0∥2ρ + NI � l=1,l̸=j µ2 l ∥hl∥2ρ + µ2 j∥hj∥2κ + σ2 v, (85) where µ2 j is given by (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Similarly, we define fE, and gE with µ2 j given by (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The derivative is ∂ ∂σ2v SIRj(σ2 v) = f ′(σ2 v)g(σ2 v) − g′(σ2 v)f(σ2 v) g2(σ2v) , (86) 19 where f and g represent fR or fE and gR or gE, respectively, and (·)′ denotes the derivative with respect to σ2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the proof of Proposition 1, we show that µ2 j for the Riemannian mean in (33) is smaller than its Euclidean counterpart in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In combination with Lemma 3, we have that , 0 < gR(σ2 v) < gE(σ2 v), so we focus on the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We show that ∂ ∂σ2 v SIRj(ΓR) < ∂ ∂σ2 v SIRj(ΓE) < 0, by proving the following claims: 1) f ′ R(σ2 v)gR(σ2 v) < f ′ E(σ2 v)gE(σ2 v) = 0 2) g′ R(σ2 v)fR(σ2 v) > g′ E(σ2 v)fE(σ2 v) > 0 Proof of Claim 1 Since µ2 j for the Euclidean mean does not depend on σ2 v, the same holds for fE(σ2 v) and f ′ E(σ2 v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Riemannian case, using Lemma 3 we have that f ′ R(σ2 v) = µ2 j∥hj∥2(ρ−κ) < 0 and gR(σ2 v) > 0, so f ′ R(σ2 v)·gR(σ2 v) < f ′ E(σ2 v)·gE(σ2 v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Claim 2 Since µ2 j for the Riemannian mean in (33) is smaller than its Euclidean counterpart in (34), we have fR(σ2 v) > fE(σ2 v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, if (30) holds for µ2 j in (34) then fE(σ2 v) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' It is left to show that g′ R(σ2 v) ≥ g′ E(σ2 v) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' This holds due to Lemma 3: g′ R(σ2 v) = ρ NI � l=1,l̸=j ∂ ∂σ2v µ2 l ∥hl∥2 + ∥hj∥2κ ∂ ∂σ2v µRiem j + 1 ≥ 1 = g′ E(σ2 v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (87) Following the proof of Proposition 2, since f ′ E(σ2 v) = 0, it follows that iff g′ E(σ2 v)fE(σ2 v) > 0 then ∂ ∂σ2v SIRj(ΓE) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since g′ E(σ2 v) = 1, we have that ∂ ∂σ2v SIRj(ΓE) < 0 iff fE(σ2 v) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' From the expression of fE(σ2 v) it follows that fE(σ2 v) > 0 iff condition (30) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' So, it is a sufficient and a necessary condition for ∂ ∂σ2 v SIRj(ΓE) < 0 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Riemannian mean, the condition under which ∂ ∂σ2 v SIRj(ΓR) < 0 is more easily met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Additionally, it is a sufficient but not a necessary condition, as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' First, we note that condition (30) can be recast as σ2 0∥h0∥2 ≥ µj∥hj∥2, ∀j, (88) where µ2 j given by (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Riemannian mean, under condition (88) with µ2 j given by (33) it holds that g′ R(σ2 v)fR(σ2 v) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since f ′ R(σ2 v)gR(σ2 v) < 0, condition (88) with µ2 j given by (33) is a sufficient but not a necessary condition for ∂ ∂σ2v SIRj(ΓR) < 0 to hold, as opposed to the Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' In the proof of Proposition 1 it is shown that the parameter µj for the Riemannian mean in (33) is smaller than its Euclidean counterpart in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Therefore, condition (88) is more easily met when µj is given by (33) than when µj is given by (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' As a consequence, it is possible for ∂ ∂σ2v SIRj(ΓE) to be positive, while ∂ ∂σ2v SIRj(ΓR) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Proposition 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Under Assumption 1 the ATF of the desired source, h0, is an eigenvector of Γj for all j with the same eigenvalue λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Then, according to Lemma 2 the following holds hH 0 ΓRh0 = hH 0 ΓEh0 = λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (89) For the Riemannian and the Euclidean mean, the following holds [28]: ΓR ⪯ ΓE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (90) Thus, for all j: hH j ΓRhj ≤ hH j ΓEhj (91) and therefore NI � j=1 hH j ΓRhj ≤ NI � j=1 hH j ΓEhj, (92) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proof of Proposition 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The Riemannian mean is ΓR = Γ 1 2 1 Γ 1 2 2 , since the matrices commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The matrices Γ1 and Γ2 are expressed using their eigenvectors, allowing the derivation of Γ 1 2 1 and Γ 1 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Finally, ΓR is computed via their product: ΓR = σ2 dhd(hd)H + µ2 1hi 1(hi 1)H + µ2 2hi 2(hi 2)H + σ2 vI (93) with µ2 j = ((α2σ2 j ∥hj∥2+σ2 v) 1 2 ·((1−α)2σ2 j ∥hj∥2+σ2 v) 1 2 )−σ2 v ∥hj∥2 for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For the Euclidean mean, µ2 j = σ2 j 2 (α2 + (1 − α)2), which reaches its minimum for α = 1 2 ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' From the proof of Proposition 1, the smaller the µ2 j-s are, the higher the SIR is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We show that ((α2σ2 j ∥hj∥2 + σ2 v) 1 2 · ((1 − α)2σ2 j ∥hj∥2 + σ2 v) 1 2 ) − σ2 v ∥hj∥2 ≤ σ2 j 2 (α2 + (1 − α)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (94) Rearranging results in ((α2σ2 j ∥hj∥2 + σ2 v) 1 2 · ((1 − α)2σ2 j ∥hj∥2 + σ2 v) 1 2 ) ≤ ∥hj∥2 σ2 j 2 (α2 + (1 − α)2) + σ2 v, (95) which holds due to the inequality of arithmetic and geometric means √x · y ≤ x+y 2 , where x = α2σ2 j ∥hj∥2 + σ2 v and y = (1 − α)2σ2 j ∥hj∥2 + σ2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' APPENDIX F THEORETICAL RESULTS FOR THE TOTAL SIR Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' For any number of microphones in the array, the following holds SIRtot(ΓR) > SIRtot(ΓE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (96) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' We compute the output total SIR for a general matrix with a similar structure as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The correlation matrix is: Γ = σ2 0hH 0 h0 + NI � l=1 µ2 l hH l hl + σ2 vI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (97) and the SIR becomes: SIRtot(Γ) = dH 0 � σ2 0hH 0 h0 + �NI l=1 µ2 l hH l hl + σ2 vI � d0 1 NI �NI j=1 dH j � σ2 0hH 0 h0 + �NI l=1 µ2 l hH l hl + σ2vI � dj = σ2 0|⟨dH 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' h0⟩|2 + �NI l=1 µ2 l |⟨dH 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hl⟩|2 + σ2 vM 1 NI �NI j=1 � σ2 0|⟨dH j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' h0⟩|2 + �NI l=1 µ2 l |⟨dH j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' hl⟩|2 + σ2vM � = σ2 0M∥h0∥2κ + �NI l=1 µ2 l M∥hl∥2ρ + σ2 vM σ2 0M∥h0∥2ρ + 1 NI �NI j=1 �NI l=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content='l̸=j µ2 l M∥hl∥2ρ + 1 NI �NI j=1 µ2 jM∥hj∥2κ + σ2vM = 1 + (σ2 0∥h0∥2 − 1 NI �NI j=1 µ2 j∥hj∥2)κ + ( 1 NI �NI j=1 µ2 j∥hj∥2 − σ2 0∥h0∥2)ρ σ2 0∥h0∥2ρ + NI−1 NI �NI l=1 µ2 l ∥hl∥2ρ + 1 NI �NI j=1 µ2 j∥hj∥2κ + σ2v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (98) In the proof of Proposition 1 we show that µ2 j for the Riemannian mean is smaller than µ2 j for the Euclidean mean for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Consequently, we have that �NI j=1 µ2 j∥hj∥2 for the Riemannian mean is smaller than for the Euclidean mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Since κ > ρ ≥ 0, it holds that SIRtot(ΓR) > SIRtot(ΓE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' If σ2 0∥h0∥2 ≥ 1 NI NI � j=1 σ2 j τj∥hj∥2, (99) then ∂ ∂σ2v SIRtot(ΓR) < ∂ ∂σ2v SIRtot(ΓE) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' (100) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} +page_content=' The proof follows the same steps as the proof of Proposition 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf'} diff --git a/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf b/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6104f3fd8c8c76b0541dc0554ea64cf6c4febfe4 --- /dev/null +++ b/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f1e995020fb2b4ea6a5a256b22d081c5a5d731a4ebdd779e7694f3c0edf58523 +size 1374416 diff --git a/YdFOT4oBgHgl3EQf9zRP/vector_store/index.faiss b/YdFOT4oBgHgl3EQf9zRP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..87894b35925cf0b041926493f33b7a347c2b3d42 --- /dev/null +++ b/YdFOT4oBgHgl3EQf9zRP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:faf4bbacb766b4961c0279b34e23d570679c0692c7ff48970b7d37d4b0323e80 +size 13762605 diff --git a/YdFOT4oBgHgl3EQf9zRP/vector_store/index.pkl b/YdFOT4oBgHgl3EQf9zRP/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5311e33ee1d777284363b8dd05d686ed4acfbbc1 --- /dev/null +++ b/YdFOT4oBgHgl3EQf9zRP/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f44ec862f929af7e3d8747109159e285934452c960acf685b8512b1b8db5b202 +size 403924 diff --git a/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/2301.03776v1.pdf.txt b/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/2301.03776v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..80e02bbb3c895d0a4a40aa9e0a66588018780ec5 --- /dev/null +++ b/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/2301.03776v1.pdf.txt @@ -0,0 +1,1701 @@ +SUPERSOLVABLE SATURATED MATROIDS AND +CHORDAL GRAPHS +DILLON MAYHEW AND ANDREW PROBERT +Abstract. A matroid is supersolvable if it has a maximal chain of +flats each of which is modular. A matroid is saturated if every round +flat is modular. In this article we present supersolvable saturated ma- +troids as analogues to chordal graphs, and we show that several results +for chordal graphs hold in this matroid context. In particular, we con- +sider matroid analogues of the reduced clique graph and clique trees for +chordal graphs. The latter is a maximum-weight spanning tree of the +former. We also show that the matroid analogue of a clique tree is an +optimal decomposition for the matroid parameter of tree-width. +1. Introduction +The study of chordal graphs is well established, and dates to work by +Dirac [3] and Berge [1]. Our contribution here is to consider a new analogue +of chordality for matroids. A graph is chordal if every cycle with at least +four vertices has a chord. This leads fairly directly to the definition of a +chordal matroid used by Cordovil, Forge, and Klein [2]. If C is a circuit in a +matroid, then a chord of C is an element z /∈ C such that there is a partition +of C into parts A and B where A ∪ z and B ∪ z are both circuits. We will +say that a matroid is C-chordal if every circuit with size at least four has a +chord. (Cordovil et al. call such a matroid chordal, but we will try to avoid +confusion by reserving that term solely for graphs.) +In this article we concentrate on a different matroid analogue for chordal- +ity. An alternative characterisation of chordal graphs is due to Dirac [3]: a +vertex is simplicial if its neighbours are pairwise adjacent. Now G is chordal +if and only if it has a simplicial vertex v such that G − v is chordal. This +definition is well suited for matroid purposes, because the edges not incident +with a simplicial vertex comprise a modular hyperplane in the corresponding +graphic matroid. (A flat F is modular if r(F)+r(F ′) = r(F ∩F ′)+r(F ∪F ′) +for every flat F ′. A hyperplane is modular if and only if it has a non-empty +intersection with every rank-two flat of the matroid.) Now we can recur- +sively consider the class of matroids M such that M is in M if and only +if M has a modular hyperplane H where restricting M to H produces a +matroid in M. The class M is exactly the family of supersolvable matroids, +introduced by Stanley [11]. +Figure 1 shows a geometric representation of a rank-four matroid, M. +We see that the hyperplane F3 = {1, 2, 3, 4, 5, 6, 7} is modular, since every +1 +arXiv:2301.03776v1 [math.CO] 10 Jan 2023 + +2 +MAYHEW AND PROBERT +1 +3 +2 +4 +5 +6 +7 +8 +9 +10 +Figure 1. A supersolvable matroid +rank-two flat has a non-empty intersection with F3. In the same way, F2 = +{1, 2, 3, 4} is a modular hyperplane of the restriction to F3, and F1 = {1} +is a modular hyperplane of the restriction to F2. Finally, ∅ is a modular +hyperplane of the restriction to F1. It follows that M is supersolvable. +It turns out that the condition of supersolvability is not strong enough +for our purposes because supersolvable matroids may fail to have properties +shared by all graphic matroids. To expand on this point, we consider matroid +analogues of cliques in a graph. Let F be a flat of a matroid. Then F is +round if there is no pair of flats (F1, F2) such that F = F1 ∪ F2 and F1 +and F2 are properly contained in F. Let G be a graph and let F be a flat +of the graphic matroid M(G). Then F is round if and only if G[F] is a +clique (Proposition 3.7). Therefore we think of round flats as the matroid +analogues of cliques. In graphic matroids every round flat is modular but this +is not true for matroids in general, nor is it true for supersolvable matroids. +For example, if M is the matroid in Figure 1, then {4, 6, 7, 8, 9, 10} is a +round hyperplane, since it cannot be expressed as the union of two flats +that it properly contains. However, it is not modular, since it has an empty +intersection with the rank-two flat {3, 5}. +We define a matroid to be saturated if every round flat is modular. Thus +saturated matroids can be thought of as analogues to graphs. To this condi- +tion, we add the condition of supersolvability to obtain our matroid analogue +of chordal graphs. So our fundamental objects of study are supersolvable +and saturated matroids. The graphic matroid M(G) is supersolvable and +saturated if and only G is chordal (Corollary 3.8 and Proposition 3.9). Many +other examples arise: for example, the matroids that are constructed using +generalised parallel connections, starting with the projective geometries of +a given order. Any such matroid is supersolvable and saturated. +The class of supersolvable saturated matroids is properly contained in the +class of C-chordal matroids (Proposition 3.6). So our focus is on a proper +subclass of C-chordal matroids. The relationships between the conditions + +SUPERSOLVABLE AND SATURATED MATROIDS +3 +of supersovability, saturation, and C-chordality are illustrated in Figure 2. +We will justify this Venn diagram in Section 3.1. +C-chordal +Supersolvable +Saturated +F7 +Fig. 1 +U3,6 +W4 +M∗(K3,3) +Figure 2. Three matroid definitions +Our main focus is showing that many facts about chordal graphs have +analogues in the class of supersolvable saturated matroids. In particular, +Section 4 introduces one of our main ideas: the rotunda graph of such a +matroid. A rotunda is a maximal round flat. The vertices of the rotunda +graph are the rotunda of the matroid. Assume that R1 and R2 are distinct +rotunda with a non-empty intersection and that (F1, F2) is a pair of modular +flats of M such that E(M) = F1∪F2 and neither F1 nor F2 is equal to E(M). +If Ri ⊆ Fi for i = 1, 2 and F1 ∩ F2 = R1 ∩ R2, then we make R1 and R2 +adjacent in the rotunda graph. The idea of a rotunda graph is analogous +to the reduced clique graph introduced by Galinier, Habib, and Paul in [4] +(where it is called a clique graph). If G is a chordal graph, then the vertices +of the reduced clique graph of G are the maximal cliques of G. If C and C′ +are maximal cliques then they are adjacent if C ∩ C′ ̸= ∅ and any path from +a vertex of C − C′ to a vertex of C′ − C uses a vertex of C ∩ C′. +If G is a chordal graph then the reduced clique graph of G and the rotunda +graph of M(G) need not be the same, but this is only because G may have +low connectivity. In Proposition 4.4 we show that when G is 2-connected +the reduced clique graph of G and the rotunda graph of M(G) are identical. +We can go further than this: the class of reduced clique graphs and the class +of rotunda graphs are identical. +Theorem 1.1. Let H be a graph. Then H is isomorphic to the rotunda +graph of a supersolvable saturated matroid if and only if H is isomorphic to +the reduced clique graph of a chordal graph. +We prove this theorem in Section 4.1. It tells us that although a super- +solvable saturated matroid may be far from graphic, the structure of its + +4 +MAYHEW AND PROBERT +rotunda will be mirrored by the structure of maximal cliques in a chordal +graph. +Knowing that these two classes of graphs are identical allows us to deduce +facts about the structure of rotunda graphs from the facts about reduced +clique graphs that we list in [9]. +For example, in [9] we show that the +reduced clique graph of a chordal graph may have induced cycles of length +three, four, or six, but not five. Therefore the same statement applies to +rotunda graphs. We conjecture that a reduced clique graph cannot have an +induced cycle of length greater than six, so we therefore conjecture that the +same statement holds for rotunda graphs. In [9] we show that no rotunda +graph can be isomorphic to a cycle of length at least four. Thus the class of +rotunda graphs is properly contained in the class of graphs with no induced +cycle of length five. We also believe that every chordal graph is isomorphic +to the rotunda graph of some supersolvable saturated matroid, and that +there is a polynomial-time algorithm for recognising when a given graph is +isomorphic to some rotunda graph. +A clique tree of the graph G is a tree whose nodes are the maximal +cliques of G, where the set of maximal cliques containing an arbitrary vertex +v ∈ V (G) induces a subtree of T. Clique trees were introduced by Gavril [5], +who showed that G has a clique tree if and only if G is chordal. The analogue +for a supersolvable saturated matroid M is a rotunda tree. In this case the +nodes of the rotunda tree are the rotunda of M, and the rotunda containing +an arbitrary element x ∈ E(M) induces a subtree. A matroid may have a +rotunda tree without being supersolvable and saturated. For example, the +matroid in Figure 1 is not saturated, but it does have a rotunda tree (having +two nodes, corresponding to {1, 2, 3, 4, 5, 6, 7} and {4, 6, 7, 8, 9, 10}). +Galinier et al. [4] weight the edges of reduced clique graphs. The edge +that joins maximal cliques C and C′ is weighted with |C ∩ C′|. They then +prove that a spanning tree of the reduced clique graph is a clique tree if +and only if it has maximum total weight amongst all spanning trees. (Their +proof contains a flaw, which we explain and correct in [9].) In our analogous +result we weight the edges of rotunda graphs. The edge that joins rotunda +R and R′ is weighted with the rank of R ∩ R′. (Our techniques are general +enough that we could also weight it with |R ∩ R′|). In Section 5 we prove +the following. +Theorem 1.2. Let M be a connected supersolvable and saturated matroid. +Every rotunda tree of M is a spanning tree of the rotunda graph of M. +Every edge of the rotunda graph is contained in a rotunda tree. Moreover, a +spanning tree is a rotunda tree if and only if it has maximum weight amongst +all spanning trees. +In Section 6 we concentrate on tree-decompositions of optimal width. In +unpublished work, Heggernes [7] observed that a clique tree of a chordal +graph is an optimal decomposition of the graph with respect to the pa- +rameter of tree-width. A matroid analogue of tree-width was developed by + +SUPERSOLVABLE AND SATURATED MATROIDS +5 +Hlinˇen´y and Whittle [8], and in Theorem 6.5 we prove the matroid ana- +logue of Heggernes’s observation: any rotunda tree of a supersolvable and +saturated matroid is an optimal decomposition with respect to the matroid +parameter of tree-width. +We refer to [10] for the foundations of matroid theory. +2. Preliminaries +2.1. Chordal graphs. Let G be a graph. If X is a set of vertices in G, then +G[X] is the subgraph induced by X. We say that a path P is X-avoiding +if any vertex of X in P is a terminal vertex of P. A clique of G is a set of +pairwise adjacent vertices. We blur the distinction between a subgraph, its +vertex set, and its edge set. So for example we may refer to a clique of the +graph G as being a flat in the cyclic matroid M(G). +If C is a cycle of a graph, then a chord is an edge that joins two distinct +vertices of the cycle without being an edge of the cycle or parallel to any +such edge. A graph is chordal if every cycle with at least four vertices has a +chord. Thus a graph is chordal if and only if has no induced cycle with more +than three vertices. Clearly every induced subgraph of a chordal graph is +chordal. +Let G be a graph, and let v be a vertex of G. If deleting v from G produces +a graph with more connected components than G, then v is a cut-vertex of +G. A connected graph with no cut-vertex is 2-connected. +An ordering v1, . . . , vn of the vertices in a graph is a perfect elimination +order if the neighbours of vi amongst vi+1, . . . , vn form a clique, for each i. +A proof of the following can be found in [6, Theorem 4.1]. +Proposition 2.1. A graph is chordal if and only if it has a perfect elimi- +nation order. +2.2. Modularity. Let M be a matroid. The flat F is modular if r(F) + +r(F ′) = r(F ∪ F ′) + r(F ∩ F ′) whenever F ′ is a flat. Note that the entire +ground set is trivially a modular flat. We also see that the unique rank-zero +flat is modular. The following is proved in [10, Proposition 6.9.2]. +Proposition 2.2. Let F be a flat of the matroid M. Then F is modular +if and only if r(F) + r(F ′) = r(F ∪ F ′) whenever F ′ is a flat such that +F ∩ F ′ = ∅. +It follows easily that if F is a hyperplane, then F is modular if and only +if r(F ∩ L) = 1 whenever L is a rank-2 flat not contained in F. We often +use an equivalent definition. +Proposition 2.3. Let F be a flat of the matroid M. Then F is modular if +and only if there is no circuit C ⊆ F ∪ F ′ containing elements from both F +and F ′, whenever F ′ is a flat that is disjoint from F. +Proof. Let F ′ be an arbitrary flat that is disjoint from F. +There is no +circuit of M|(F ∪ F ′) that contains elements of both F and F ′ if and only + +6 +MAYHEW AND PROBERT +if r(F) + r(F ′) = r(F ∪ F ′) [10, Proposition 4.2.1]. Now the result follows +by Proposition 2.2. +□ +The next result combines Proposition 6.9.5 and Corollary 6.9.8 from [10]. +Proposition 2.4. Let F and F ′ be modular flats of the matroid M. Then +F ∩ F ′ is a modular flat of M. If F ⊆ X ⊆ E(M) then F is a modular flat +of M|X. +Proposition 2.5. Let F be a modular flat of the matroid M and let C be a +circuit of M such that C∩F is non-empty. Then cl(C−F)∩F is non-empty. +Proof. If cl(C − F) ∩ F = ∅ then Proposition 2.3 is violated, since cl(F − C) +is a flat that is disjoint from F, but C is a circuit that contains elements +from both F and cl(C − X). +□ +Let H be a modular hyperplane of the matroid M, and let C∗ be the +complementary cocircuit. Let x and y be distinct rank-one flats contained +in C∗. Then r(H ∩ cl(x ∪ y)) = 1, because H is modular. We say that +the rank-one flat H ∩ cl(x ∪ y) is the projection of x and y onto H, and +we denote this flat with PH(x, y). If x and y are elements of C∗ such that +r({x, y}) = 2, then we also use PH(x, y) to stand for PH(cl({x}), cl({y})). +Proposition 2.6. Let H be a modular hyperplane of the matroid M. Let +X be a subset of E(M) − H and let P be the union ∪PH(x, y), where {x, y} +ranges over all pairs of distinct rank-one flats in X. Let U be a subset of H +such that U contains P. Then cl(U) = cl(U ∪ X) ∩ H. +Proof. Note that cl(U) is contained in H. Thus it is obvious that cl(U) is +a subset of cl(U ∪ X) ∩ H. Let us assume that the containment is proper, +and let z be an element that is in cl(U ∪ X) ∩ H but not cl(U). Thus z +is not in U. There is some circuit C ⊆ U ∪ X ∪ z that contains z. Let us +assume that we have chosen C so that C − H is as small as possible. If +C − H is empty, then C certifies that z is in cl(U), contrary to hypothesis, +so C − H ̸= ∅. If C − H contains a single element x, then C certifies that x +is in cl(H) = H, which is a contradiction. Therefore we can choose x and y +to be distinct elements of C −H. Let p be an element in PH(x, y). Thus p is +in P and {x, y, p} is a circuit. Note that z ̸= p, since z is not in P ⊆ U. We +perform strong circuit elimination on C and {x, y, p} to obtain the circuit +C′ ⊆ (C − x) ∪ {p, z} such that z is in C′. Thus C′ is a subset of U ∪ X ∪ z, +but C′ −H is smaller than C. Now our choice of C is contradicted, and this +completes the proof. +□ +Proposition 2.7. Let H be a modular hyperplane of the connected matroid +M. Then M|H is connected. +Proof. Assume that M|H is not connected, and let (U, V ) be a separation of +M|H. Because M is connected, there are circuits of M that contain elements +from both U and V . Amongst such circuits choose C so that C − H is as +small as possible. Let u be an element in C ∩ U and let v be an element + +SUPERSOLVABLE AND SATURATED MATROIDS +7 +from C ∩ V . Note that C − H is not empty since (U, V ) is a separation +of M|H. Furthermore, C − H does not contain a single element, or else +that element would be in cl(H) = H. Therefore we choose distinct elements +x, y ∈ C − H. Let p be an element in PH(x, y), so that {x, y, p} is a circuit +of M. Because p is in H we can assume without loss of generality that p is +in U. We perform strong circuit elimination on C and {x, y, p} to obtain a +circuit C′ ⊆ (C − x) ∪ {y, p} that contains v. Note that C′ contains p, or +else it is a proper subset of C. Thus C′ contains elements from both U and +V , but |C′ − H| < |C − H|, and we have a contradiction. Therefore M|H is +connected. +□ +2.3. Roundness. A proper flat of a matroid is one that is not equal to the +entire ground set. +Definition 2.8. Let M be a matroid. +A vertical cover of M is a pair +(F, F ′) of proper flats such that F ∪ F ′ = E(M). If, in addition, F and F ′ +are modular flats, then (F, F ′) is a modular cover. A matroid is round if it +has no vertical cover. +Thus a matroid is round if and only if there is no partition (U, U′) of +E(M) such that neither U nor U ′ is spanning. Such a partition is said to +be a vertical separation. If X is a subset of E(M), then we say that X is +round if M|X is round. If F is a round flat of the matroid M and F is +contained in the subset X ⊆ E(M), then clearly F is a round flat of M|X. +A round flat is maximal if it is not properly contained in a round flat. For +brevity, we refer to a maximal round flat as a rotunda. The set of rotunda +of a matroid M is denoted by R(M). +Proposition 2.9. Let R and R′ be distinct rotunda. Let (F, F ′) be a vertical +cover such that R ⊆ F and R′ ⊆ F ′ and F ∩ F ′ = R ∩ R′. Then R ⊈ F ′ and +R′ ⊈ F. +Proof. It suffices to prove that R is not contained in F ′. Assume this fails. +Then R is contained in F ∩ F ′ = R ∩ R′, implying that R is a subset of R′. +This is impossible since R and R′ are distinct rotunda. +□ +The next result follows from work in [12], but we include a proof for +completeness. +Proposition 2.10. Let H be a modular hyperplane of the matroid M. Let +X be a subset of the cocircuit E(M) − H. Then +{PH(x, y): x, y ∈ X, r({x, y}) = 2} +is round. +Proof. Let P be the union of all projections onto H of pairs of distinct, +non-parallel, elements in X. Thus our aim is to show that P is round. We +assume for a contradiction that (F, F ′) is a vertical cover of M|P, so that +F and F ′ are proper flats of M|P and F ∪ F ′ = P. Note that if X contains + +8 +MAYHEW AND PROBERT +fewer than three rank-one flats, then P is either empty or consists of a single +rank-one flat. In this case P is trivially round, so we must assume that X +contains at least three rank-one flats. +Let x, y, and z be distinct rank-one flats in X. Assume that PH(x, y) +and PH(x, z) are both in F. +We claim that PH(y, z) is also in F. +If z +is in cl(x ∪ y), then cl(x ∪ y) = cl(x ∪ z) = cl(y ∪ z), and it follows that +PH(x, y) = PH(x, z) = PH(y, z), so the claim is true. Therefore we will +assume that r(x ∪ y ∪ z) = 3. Let Z be cl(x ∪ y ∪ z). Since H is a modular +hyperplane and Z is not contained in H, it follows that r(H ∩ Z) = 2. Now +PH(x, y) and PH(x, z) are rank-one flats contained in H ∩ Z. If they are +not distinct, then y and z are both in the closure of x ∪ PH(x, y). This +implies that z is in cl(x ∪ y), contrary to earlier hypothesis. It follows that +PH(x, y) ∪ PH(x, z) spans H ∩ Z, and in particular spans PH(y, z). Thus +PH(y, z) is in F, as claimed. Symmetrically, if PH(x, y) and PH(x, z) are +both in F ′, then so is PH(y, z). +We think of the rank-one flats that have a non-empty intersection with +X as the vertices of a complete graph. If x and y are two such flats, then we +colour the edge between x and y red if PH(x, y) is in F, and blue if it is in +F ′. Notice that an edge may be both red and blue. The previous paragraph +shows that if the edges xy and xz are both red (blue), then the edge yz is +also red (blue). +Let x be a vertex in this complete graph and assume that every edge +incident with x is red. +Then every edge is red, and it follows that P is +contained in F. This is impossible since F is a proper flat of M|P. Similarly, +it is not possible for every edge incident with x to be blue. +Therefore we can assume that the edge between x and y is red but not +blue, and the edge between x and z is blue but not red. However, if the +edge yz is red, then xz is red, and if yz is blue then xy is blue. In either +case we have a contradiction, so the proof is complete. +□ +Proposition 2.11. Let H be a modular hyperplane of the matroid M and +let C∗ be the complementary cocircuit. Let (F1, F2) be a vertical cover of +M|H. Let P be the union ∪PH(x, y), where x and y range over all distinct +rank-one flats contained in C∗. Then P is contained in Fi for some i, and +(Fi ∪ C∗, F3−i) is a vertical cover of M. Moreover, if (F1, F2) is a modular +cover, then so is (Fi ∪ C∗, F3−i). +Proof. Proposition 2.10 says that P is a round subset of H. Thus (F1 ∩ +P, F2 ∩ P) is not a vertical cover of M|P, so either F1 ∩ P or F2 ∩ P is equal +to P. We assume the former without any loss of generality, so P ⊆ F1. +Proposition 2.6 implies that F1 is equal to cl(F1 ∪ C∗) ∩ H. It follows that +cl(F1∪C∗) = F1∪C∗. Now F1∪C∗ is a proper flat of M because F1 is a proper +flat of M|H. Similarly, F2 is a proper flat of M. As (F1 ∪C∗)∪F2 = E(M), +it follows that (F1 ∪ C∗, F2) is a vertical cover of M. +Now we assume that (F1, F2) is a modular cover of M|H. Then F2 is a +modular flat of M|H so it immediately follows from [10, Proposition 6.9.7] + +SUPERSOLVABLE AND SATURATED MATROIDS +9 +that F2 is also a modular flat in M. It remains only to prove that F1 ∪ C∗ +is a modular flat of M. To this end, assume that F is a flat of M that is +disjoint from F1 ∪ C∗. Thus F is a flat of M|(F2 − F1). If we can show +that there is no circuit of M|((F1 ∪ C∗) ∪ F) containing elements from both +F and F1 ∪ C∗, then the result will follow from Proposition 2.3. Assume +that C is such a circuit, chosen so that C ∩ C∗ is as small as possible. Let +f be an element of C ∩ F. If C ∩ C∗ = ∅, then Proposition 2.3 implies +that F1 is not a modular flat of M|H, which is a contradiction. Therefore +C ∩C∗ ̸= ∅. If C ∩C∗ contains a single element, x, then C certifies that x is +in cl(H) = H, a contradiction. Therefore we let x and y be distinct elements +in C ∩ C∗. Let p be in PH(x, y). Thus {x, y, p} is a circuit and p is in P, +and hence in F1. We perform strong circuit elimination on C and {x, y, p} +to obtain C′ ⊆ (C − x) ∪ {y, p}, a circuit that contains f. It must contain +p, since otherwise it is properly contained in C. But now C′ is contained +in F1 ∪ C∗ ∪ F, and it contains elements from both F and F1 ∪ C∗. Since +C′ ∩ C∗ is strictly smaller than C ∩ C∗, we have contradicted our choice of +C, so the proof is complete. +□ +The following result provides a partial converse to Proposition 2.11. +Proposition 2.12. Let H be a modular hyperplane of the matroid M and +let C∗ be the complementary cocircuit. Let (F, F ′) be a modular cover of +M such that F ′ is contained in H. If F ′ ̸= H, then (F ∩ H, F ′ ∩ H) is a +modular cover of M|H. +Proof. Note that C∗ is contained in F because F ′ contains no element of C∗. +Let P be the union of PH(x, y) where x and y range over distinct rank-one +flats in C∗. Since F contains C∗, and P is spanned by C∗ it follows that +P is a subset of F. Now F ∩ H is the intersection of two modular flats, so +Proposition 2.4 implies that it is a modular flat of M, and hence of M|H. +Because F ′ is contained in H it is also true that F ′∩H = F ′ is a modular flat +of M|H. By hypothesis F ′ ∩H is a proper flat of M|H. Furthermore, F ∩H +is a proper flat of M|H, or else F contains H ∪ C∗ = E(M), contradicting +the fact that F is a proper flat of M. Therefore (F ∩H, F ′ ∩H) is a modular +cover of M|H. +□ +Proposition 2.13. Let H be a modular hyperplane of the matroid M and +let C∗ be the complementary cocircuit. If F is a round flat not contained in +H, then F ⊆ cl(C∗). +Proof. Assume this fails. Then F ∩ C∗ does not span F. It is also true that +F ∩H does not span F, as cl(F ∩H) ⊆ cl(H) = H and F is not contained in +H. Therefore (F ∩H, F ∩C∗) is a vertical cover of M|F, and this contradicts +the fact that M|F is round. +□ +Proposition 2.14. Let H be a modular hyperplane of the matroid M and let +C∗ be the complementary cocircuit. Then cl(C∗) is a rotunda. Furthermore, +every other rotunda of M is contained in H. + +10 +MAYHEW AND PROBERT +Proof. Let R be cl(C∗). Assume that R is not round, and let (F, F ′) be a +vertical cover of M|R. Let P be the union ∪PH(x, y), where x and y range +over all distinct rank-one flats contained in C∗. Note that P is contained in +R∩H. Proposition 2.10 says that P is round. It follows that one of F ∩P or +F ′ ∩ P is equal to P. Without loss of generality we will assume the former. +If F ′ contains C∗, then it contains R, which is impossible as (F, F ′) is a +vertical cover of R. Therefore we choose x ∈ C∗ − F ′. The same argument +shows we can choose y ∈ C∗ − F. Note that x and y are not parallel, since +x is in F − F ′ and y is in F ′ − F. Let p be in PH(x, y), so that p is in P, +and hence in F. As {x, y, p} is a circuit and both x and p belong to the flat +F it follows that y is in F, contrary to assumption. Therefore R is round. +Let Z be any flat that properly contains R. Note that Z ∩H is a flat that +does not contain any element of C∗. Therefore (Z ∩H, R) is a vertical cover +of Z, a contradiction. This shows that R is a maximal round flat, which is +to say, a rotunda. +Finally, let Z be a rotunda that is not contained in H. By Proposition +2.13, we see that Z is contained in R. As Z and R are both rotunda it now +follows that Z = R. +□ +Proposition 2.15. Let H be a modular hyperplane of the matroid M. Let +C∗ be the complementary cocircuit. Then cl(C∗) ∩ H is round. +Proof. Let R be cl(C∗). Assume for a contradiction that (F, F ′) is a vertical +cover of R ∩ H. Let P be the union ∪PH(x, y) where x and y range over all +distinct rank-one flats contained in C∗. Note that P is contained in R ∩ H. +Proposition 2.10 says that P is round. Therefore (F ∩ P, F ′ ∩ P) is not a +vertical cover of P, so we can assume without loss of generality that P is +contained in F. Applying Proposition 2.6, we see that cl(F ∪ C∗) ∩ H is +equal to F. Thus cl(C∗) ∩ H = R ∩ H is contained in F. This contradicts +the fact that (F, F ′) is a vertical cover of R ∩ H. +□ +3. Supersolvability and saturation +The following definition was introduced by Stanley [11]. +Definition 3.1. The rank-r matroid M is supersolvable if it has a chain of +modular flats F0 ⊆ F1 ⊆ · · · ⊆ Fr, where r(Fi) = i for each i. +We can give an equivalent, recursive, definition: if r(M) > 0 then M +is supersolvable if it contains a modular hyperplane H such that M|H is +supersolvable. Note that every rank-zero matroid is trivially supersolvable. +Definition 3.2. A matroid is saturated if every round flat is modular. +Proposition 3.3. Let F be a flat of the saturated matroid M. Then M|F +is saturated. +Proof. Let R be a round flat of M|F. Then R is a round flat of M so it is +modular in M. Now [10, Proposition 6.9.5] implies that R is a modular flat +of M|F. +□ + +SUPERSOLVABLE AND SATURATED MATROIDS +11 +If M is supersolvable and saturated and H is a modular hyperplane such +that M|H is supersolvable, then it follows from Proposition 3.3 that M|H +is supersolvable and saturated. +Proposition 3.4. Let M be a saturated matroid. Let H be a modular hy- +perplane of M and let C∗ be the complementary cocircuit. If C∗ is non- +spanning, then (H, cl(C∗)) is a modular cover of M. +Proof. Certainly H is a proper flat of M, and C∗ is non-spanning by hy- +pothesis. Therefore (H, cl(C∗)) is a vertical cover. We have assumed that +H is a modular flat. Proposition 2.14 says that cl(C∗) is round. Since M is +saturated, it follows that cl(C∗) is modular, so the proof is complete. +□ +Proposition 3.5. Let M be a matroid. Then M is supersolvable if and +only if each of its connected components is supersolvable. Similarly M is +saturated if and only if each of its connected components is saturated. +Proof. This result will follow by an easy inductive argument if we can prove +it in the case when M has exactly two connected components. Therefore we +will assume that M = M1⊕M2, where M1 and M2 are non-empty connected +matroids. For i = 1, 2, let ri be r(Mi). +Assume that M1 and M2 are supersolvable. For i = 1, 2, let F i +0 ⊆ F i +1 ⊆ +· · · ⊆ F i +ri be a chain of modular flats in Mi such that each F i +j has rank j. +Using [10, Corollary 6.9.10] we see that each F 1 +j ∪ F 2 +k is a modular flat of +M. Now it is easy to confirm that the chain +F 1 +0 ⊆ F 1 +1 ⊆ · · · ⊆ F 1 +r1 ⊆ F 1 +r1 ∪ F 2 +1 ⊆ F 1 +r1 ∪ F 2 +2 · · · ⊆ F 1 +r1 ∪ F 2 +r2 +certifies that M is supersolvable. +For the other direction, assume that M is supersolvable. Assume for a +contradiction that either M1 or M2 is not supersolvable. We will assume +that amongst such counterexamples, M is as small as possible. Now M has a +modular hyperplane H such that M|H is supersolvable. The complement of +H is a cocircuit, and is therefore contained in either M1 or M2. Without loss +of generality we assume that H contains E(M2). Now M|H = (M1|H)⊕M2. +The minimality of M means that M1|H and M2 are both supersolvable. But +[10, Corollary 6.9.10] implies that H ∩ E(M1) is a modular flat of M1. It is +the complement of a cocircuit of M1, so H ∩ M1 is a modular hyperplane +of M1, and restricting to this hyperplane produces a supersolvable matroid. +This shows that M1 too is supersolvable, so the proof of this direction is +complete. +From [10, Corollary 6.9.10] we see that E(M1) and E(M2) are modular +flats of M. It follows from [10, Proposition 6.9.5] that a flat of Mi is modular +in M if and only if it is modular in Mi. If F is a round flat of M then +F ⊆ E(M1) or F ⊆ E(M2) because otherwise (F ∩ E(M1), F ∩ E(M2)) is a +vertical cover of M|F. In fact, the round flats of M are exactly the round +flats of M1 along with the round flats of M2. From these considerations we +can easily see that M is saturated if and only if M1 and M2 are saturated. +□ + +12 +MAYHEW AND PROBERT +3.1. Chordality for matroids. We shall start this section by justifying +the Venn diagram in Figure 2. Recall that if C is a matroid circuit, then a +chord of C is an element x /∈ C such that A ∪ z and B ∪ z are both circuits +for some partition of C into sets A and B. A matroid is C-chordal if every +circuit with at least four elements has a chord. +As we discussed in the introduction, the matroid in Figure 1 is supersolv- +able but not saturated. To see that it is not C-chordal, note that {3, 5, 6, 7} +has no chord. Because the only round flats of U3,6 are the empty set, the +singleton sets, and the entire ground set, we can easily confirm that every +round flat is modular, so U3,6 is saturated. It has no modular hyperplane, +so it is not supersolvable, and no circuit has a chord so it is not C-chordal. +Recall that W4 is the rank-three matroid with ground set {a, b, c, d, e, f} and +non-spanning circuits {a, b, d}, {b, c, e}, and {a, c, f}. It is easy to confirm +that every circuit of size four has a chord. However no line is modular, so +W4 is not supersolvable, and it also follows that it is not saturated. +We will leave as an exercise the fact that the Fano matroid F7 is supersolv- +able, saturated, and C-chordal. Cordovil et al. note that M∗(K3,3) is not su- +persolvable [2]. It is an easy exercise to see that it is saturated and C-chordal. +Finally, let M be the rank-three matroid with ground set {p, a, b, c, d, e, f, x} +where the non-spanning circuits are {p, a, b, c}, {p, d, e, f}, {a, d, x}, {b, e, x}, +and {c, f, x}. Now {p, a, b, c} and {p, d, e, f} are both modular hyperplanes, +and we can easily confirm that M is supersolvable. +On the other hand, +{a, d, x} is a round hyperplane that has empty intersection with the rank- +two flat {b, f}. Hence {a, d, x} is not modular and therefore M is not satu- +rated. On the other hand, a simple case-analysis shows that M is C-chordal. +We can finish the justification of Figure 2 by proving that every supersolv- +able saturated matroid is C-chordal. In fact, we prove something slightly +stronger. +Proposition 3.6. Let C be a circuit in the supersolvable saturated matroid +M and assume that |C| ≥ 4. There exist distinct elements x, y ∈ C and an +element z /∈ C such that {x, y, z} and (C − {x, y}) ∪ z are circuits of M. +Proof. Let M be a smallest possible counterexample to the result. If r(M) ≤ +2 then the result holds vacuously, so r(M) ≥ 3. +Let H be a modular +hyperplane of M such that M|H is supersolvable and saturated. Let C∗ be +the complement of H. +Choose C to be an arbitrary circuit of M such that |C| ≥ 4. If C is a +circuit of M|H, then the result holds by induction. Therefore C ∩C∗ is non- +empty. Because H is a flat it follows that C ∩ C∗ contains distinct elements +x and y. Let L be cl({x, y}). Note that L contains an element in PH(x, y), +so that L is a rank-two flat containing at least three rank-one flats. Now it +is easy to confirm that L is a round flat. Since M is saturated, it follows +that L is modular. Note that C contains exactly two elements of L because +|C| ≥ 4. Now +r(cl(C − L) ∩ L) = r(C − L) + r(L) − r(C) = (|C| − 2) + 2 − (|C| − 1) = 1. + +SUPERSOLVABLE AND SATURATED MATROIDS +13 +Therefore we choose an element z which is in cl(C−L)∩L. Note that neither +x nor y is in cl(C − L), or else C properly contains a circuit. Therefore z +is in L − {x, y} and {x, y, z} is a circuit. +Let C′ ⊆ (C − L) ∪ z be a +circuit that contains z. Now (C′ ∪ {x, y}) − z contains a circuit, by circuit +elimination with C′ and {x, y, z}. But (C′ ∪ {x, y}) − z is a subset of C, so +(C′ ∪ {x, y}) − z = C. It follows that C′ = (C − L) ∪ z. Thus (C − L) ∪ z +and {x, y, z} are both circuits and M is not a counterexample after all. +□ +In the next results we justify using supersolvable saturated matroids as +analogues for chordal graphs. +Proposition 3.7. Let G be a graph, and let F be a flat of M(G). Then F +is round if and only if G[F] is a clique. +Proof. Let M be M(G). Assume that G[F] is not a clique. Let u and v be +distinct vertices in G[F] that are not adjacent. Let U be the set of edges +in F that are incident with u, and let U ′ be F − U. If f ∈ F is an edge +incident with u, there is no cycle contained in U ′ ∪ f that contains f. This +shows that cl(U ′) is a proper flat of M|F. The same argument shows that +cl(U) is a proper flat of M|F, so (cl(U), cl(U ′)) is a vertical cover of M|F. +Thus F is not round. +For the other direction, assume that G[F] is a clique, but that (U, U′) is +a vertical cover of G[F]. We colour the edges of F red if they are in U, and +blue if they are in V . Note that an edge may be both red and blue. Let v +be an arbitrary vertex of G[F]. The set of edges incident with v spans F, +since G[F] is a clique. If all the edges of F incident with v are red, then U +contains F, a contradiction. By symmetry, we can now let e, f ∈ F be edges +incident with v so that e is red but not blue, and f is blue but not red. Let +g be the edge of F so that {e, f, g} is the edge-set of a triangle. If g is red, +then f is also red, and if g is blue, then e is blue, and in either case we have +a contradiction. +□ +Corollary 3.8. Let G be a graph. Then M(G) is a saturated matroid. +Proof. From Proposition 3.7 we see that every round flat of M(G) is a clique +of G, and any such flat is modular by [10, Proposition 6.9.11]. The result +follows. +□ +The next result is a consequence of [11, Proposition 2.8]. +Proposition 3.9. Let G be a graph. Then G is chordal if and only if M(G) +is supersolvable. +The next result implies the known fact [6, Proposition 4.16] that in a +chordal graph the number of maximal cliques does not exceed the number +of vertices. +Proposition 3.10. Let M be a supersolvable matroid. Then M has at most +r(M) rotunda. + +14 +MAYHEW AND PROBERT +Proof. Let H be a modular hyperplane of M such that M|H is supersolvable. +Any rotunda of M that is contained in H is a rotunda of M|H. But M|H +has at most r(M)−1 rotunda by induction, and Proposition 2.14 says there +is exactly one rotunda of M that is not a rotunda of M|H. +The result +follows. +□ +4. Reduced clique graphs and rotunda graphs +Let G be a chordal graph. The clique graph of G, denoted C(G), has +the maximal cliques of G as its vertices. Two distinct maximal cliques are +adjacent in C(G) if and only if they have at least one vertex in common. +Our focus will be the reduced clique graph, CR(G), which was introduced in +[4]. The vertices of CR(G) are again the maximal cliques of G. Let C1 and +C2 be distinct maximal cliques of G. We say that C1 and C2 are a separating +pair if there is at least one vertex in C1 ∩ C2 and any path from a vertex +of C1 − C2 to a vertex of C2 − C1 uses a vertex in C1 ∩ C2. Now CR(G) is +the subgraph of C(G) where two maximal cliques are adjacent if and only +if they form a separating pair. We now define a matroid analogue of this +graph. +Definition 4.1. Let M be a supersolvable saturated matroid. Recall that +R(M) is the family of rotunda of M. The rotunda graph R(M) is the graph +with R(M) as its vertex set. The rotunda R1 and R2 are adjacent in R(M) +if R1 ∩ R2 ̸= ∅ and there is a modular cover (F1, F2) such that Ri ⊆ Fi for +i = 1, 2, and F1 ∩ F2 = R1 ∩ R2. In this case we say that the modular cover +(F1, F2) certifies the adjacency of R1 and R2. +The next result allows us to prove statements about rotunda graphs in- +ductively. +Proposition 4.2. Let M be a supersolvable saturated matroid and let H +be a modular hyperplane of M such that M|H is supersolvable. Let C∗ be +the complement of H and let R be cl(C∗). Then R is a rotunda of M and +either: +(a) R ∩ H is a rotunda of M|H and +R(M) = (R(M|H) − {R ∩ H}) ∪ {R}, +or +(b) R ∩ H is properly contained in a rotunda of M|H and +R(M) = R(M|H) ∪ {R}. +If case (a) holds then R(M|H) is obtained from R(M) by relabelling R as +R ∩ H. If case (b) holds then R(M|H) is obtained from R(M) by deleting +R. +Proof. Note that M|H is saturated as well as supersolvable (Proposition +3.3). Proposition 2.14 says that R is a rotunda of M, and that moreover it +is the unique rotunda of M that is not contained in H. Now it is an easy + +SUPERSOLVABLE AND SATURATED MATROIDS +15 +exercise to prove that every other rotunda of M is a rotunda of M|H. This +shows R(M) ⊆ R(M|H) ∪ {R}. +Proposition 2.15 says that R ∩ H is a round flat of M|H. First assume +that R ∩ H is a maximal round flat of M|H. Then R ∩ H is a rotunda of +M|H but not of M, since R ∩ H is properly contained in R. So in this case +R(M) is contained in (R(M|H) − {R ∩ H}) ∪ {R}. Now let Z be a rotunda +of M|H that is not equal to R ∩ H. We will prove that Z is a rotunda of +M. Assume otherwise. Because Z is a round flat of M|H, and hence of M, +it is properly contained in a rotunda of M. Let this rotunda be Z′. Now Z′ +is not contained in H, because in this case Z and Z′ would both be rotunda +of M|H, and then Z cannot be properly contained in Z′. So Z′ is a rotunda +of M that is not contained in H, and hence Z′ = R. Thus Z is contained +in R ∩ H. Because Z is not properly contained in a round flat of M|H we +deduce that Z = R ∩ H, contrary to hypothesis. Thus Z is a rotunda of M +and we have shown that when R ∩ H is a rotunda of M|H, the set R(M) is +equal to (R(M|H) − {R ∩ H}) ∪ {R} and case (a) holds. +Next we assume that R ∩ H is not a rotunda of M|H. We have already +shown that R(M) is contained in R(M|H) ∪ {R}. +Let Z be a rotunda +of M|H and assume that Z is not a rotunda of M. Then Z is properly +contained in Z′, a rotunda of M. As in the previous paragraph, Z′ = R, so +Z is contained in R ∩ H. Again we deduce that Z = R ∩ H, and we have a +contradiction to Z being a rotunda of M|H. So in the case R(M) is equal +to R(M|H) ∪ {R}. Furthermore, R ∩ H is a round flat of M|H but not a +rotunda, so it must be properly contained in a rotunda of M|H. Thus case +(b) holds. +Assume case (a) holds. +We let Z1 and Z2 be distinct rotunda of M, +where Z1 is not equal to R. Thus Z1 is a rotunda of M|H. Either Z2 is +equal to R or it is not. In the former case Z2 ∩ H = R ∩ H and in the latter +Z2 ∩ H = Z2. In either case Z2 ∩ H is a rotunda of M|H. We will prove +that Z1 and Z2 ∩ H are adjacent in R(M|H) if and only if Z1 and Z2 are +adjacent in R(M), and this will show that R(M|H) is obtained from R(M) +by relabelling R as R ∩ H. +Assume that (F1, F2) is a modular cover of M that certifies the adjacency +of Z1 and Z2 in R(M). Thus F1 and F2 are proper modular flats of M and +F1 ∪ F2 = E(M). Moreover F1 ∩ F2 = Z1 ∩ Z2. Assume that either F1 or F2 +contains H. Since Proposition 2.9 implies that neither F1 nor F2 contains +Z1 ∪ Z2, we deduce that Z2 = R and F1 = H. Now +R ∩ H ⊆ F1 ∩ F2 = Z1 ∩ Z2 +so Z1 contains R ∩ H. Since Z1 and R ∩ H are both rotunda of M|H, we +see that Z1 = R ∩ H, and in this case Z1 is properly contained in Z2. This +is impossible, so F1 ∩ H or F2 ∩ H are proper flats of M|H. Moreover, their +union is equal to H. +Since F1 and F2 are modular flats of M it follows that F1 ∩H and F2 ∩H +are modular flats of M (Proposition 2.4), and hence modular flats of M|H. + +16 +MAYHEW AND PROBERT +Furthermore, +(F1 ∩ H) ∩ (F2 ∩ H) = (F1 ∩ F2) ∩ H = (Z1 ∩ Z2) ∩ H = Z1 ∩ (Z2 ∩ H). +Now we see that (F1 ∩ H, F2 ∩ H) is a modular cover of M|H, and that it +certifies the adjacency of Z1 and Z2 ∩ H in R(M|H). +For the other direction, assume Z1 and Z2 ∩ H are adjacent in R(M|H), +and let (F1, F2) be a modular cover of M|H that certifies their adjacency. +Let P be the union ∪PH(x, y), where x and y range over distinct rank-one +flats in C∗. We apply Proposition 2.11 and see that P is contained in either +F1 or F2. +Assume that Z2 = R. Then P is contained in Z2 ∩ H ⊆ F2. In this +case Proposition 2.11 implies that (F1, F2 ∪ C∗) is a modular cover of M. +Moreover, +F1 ∩ (F2 ∪ C∗) = F1 ∩ F2 = Z1 ∩ (Z2 ∩ H) = Z1 ∩ Z2. +Thus (F1, F2 ∪ C∗) certifies the adjacency of Z1 and Z2 in R(M). Next we +assume that Z2 ̸= R, so that Z1 and Z2 are both rotunda of M|H. We again +apply Proposition 2.11 and see that (Fi ∪ C∗, F3−i) is a modular cover of M +for some i ∈ {1, 2}, and as before we can see that (Fi ∪ C∗, F3−i) certifies +the adjacency of Z1 and Z2 in R(M). Thus we are now finished with case +(a). +Assume case (b) holds. Let Z1 and Z2 be two rotunda of M|H. We can +use exactly the same arguments as in the previous paragraphs to show that +Z1 and Z2 are adjacent in R(M|H) if and only if they are adjacent in R(M). +Thus R(M|H) is obtained from R(M) by deleting the rotunda R and the +proof is complete. +□ +4.1. Rotunda graphs vs. reduced clique graphs. In this section we +compare rotunda graphs and reduced clique graphs. Ultimately we will show +that they are identical classes of graphs. We also consider the connection +between the reduced clique graph of G and the rotunda graph of M(G) when +G is a chordal graph. +Proposition 4.3. Let G be a chordal graph. Then the maximal cliques of G +are the rotunda of M(G), and every edge in R(M(G)) is an edge in CR(G). +Proof. The first statement follows from Proposition 3.7. Let M stand for +M(G), so that we identify the vertices of CR(G) and the vertices of R(M). +Let R1 and R2 be rotunda that are adjacent in R(M), and let C1 and C2 +be the corresponding maximal cliques of G. We will show that C1 and C2 +are adjacent in CR(G). Let (F1, F2) be a modular cover of M certifying the +adjacency of R1 and R2, so that Ri ⊆ Fi for i = 1, 2, and F1 ∩F2 = R1 ∩R2. +Because R1 and R2 are adjacent in R(M), they have a non-empty inter- +section, which means that C1 and C2 share at least two vertices. Let S be +the set of vertices in both C1 and C2. Thus |S| ≥ 2. If C1 and C2 form a +separating pair, then there is nothing left for us to prove. Therefore we will +let P be an S-avoiding path from a1 ∈ C1 − C2 to a2 ∈ C2 − C1. + +SUPERSOLVABLE AND SATURATED MATROIDS +17 +Let u be an arbitrary vertex in S. Assume that every edge of C1 incident +with u is in F2. Then R1 is contained in F2, which contradicts Proposition +2.9. Therefore we let e1 be an edge of C1 that is incident with u and not in +F2. By the same reasoning, we can let e2 be an edge of C2 that is incident +with u and not in F1. Assume that ei joins u to bi for i = 1, 2. Note that b1 +is in C1 − C2, or else e1 would be in R2 ⊆ F2. Similarly b2 is in C2 − C1. +We obtain the cycle D from P by appending the edges e1 and e2 as well +as a1b1 and a2b2. (This assumes that a1 ̸= b1; if a1 = b1 then we do not +append a1b1. The same comment applies if a2 = b2.) +Note that D is not contained in F2, as e1 is not in F2. Because F2 is +a modular flat we can apply Proposition 2.5 and deduce that there is an +element x ∈ F2 ∩ cl(D − F2). Thus D′ ∪ x is a cycle of G for some subset +D′ ⊆ D − F2. Because (D − F2) ∪ x is a circuit of M it follows that x is in +F1 as well as F2. Therefore x is in R1 ∩ R2, so x joins two vertices of S. Let +v be a vertex incident with x such that v is not u. Thus v is in the cycle D, +so v is either an internal vertex of P, or is equal to one of a1, b1, a2, or b2. +But none of the internal vertices of P is in S, and a1, b1 are in C1 −C2 while +a2, b2 are in C2 − C1. Therefore we have a contradiction that completes the +proof. +□ +From the previous result we know that R(M(G)) is a subgraph of CR(G). +To see that R(M(G)) and CR(G) need not be equal, we let G be the path +with two edges. Thus G is a tree and is therefore chordal. There are two +maximal cliques in G, and M(G) has two rotunda. However CR(G) consists +of two vertices joined by an edge, whereas R(M(G)) consists of two isolated +vertices, since the two rotunda of M(G) are disjoint. The next result shows +that sufficient connectivity prevents this situation from happening. +Proposition 4.4. Let G be a chordal graph that is 2-connected. +Then +CR(G) = R(M(G)). +Proof. We identify the vertices of CR(G) and R(M). By virtue of Propo- +sition 4.3, it suffices to show that every edge of CR(G) is also an edge of +R(M). To this end let C1 and C2 be maximal cliques of G that are adjacent +in CR(G). Let Ri be the edge set of Ci for i = 1, 2. Then R1 and R2 are +rotunda of M. We will show they are adjacent in R(M). +Set S to be the set of vertices in both C1 and C2. Since C1 and C2 are +adjacent in CR(G) it follows that S ̸= ∅. For each i = 1, 2, let ai be a vertex +in Ci − C3−i. +4.4.1. R1 ∩ R2 ̸= ∅ +Proof. This claim holds if |S| ≥ 2, because then any edge joining two vertices +of S is in R1 ∩ R2. So assume that |S| = 1 and let v be the unique vertex +of S. Now C1 and C2 form a separating pair, so a1 and a2 are in different +connected components of G − S = G − v, but this contradicts the fact that +G is 2-connected. +□ + +18 +MAYHEW AND PROBERT +Now we know that R1 and R2 are not disjoint we can complete the proof +by constructing a modular cover to certify their adjacency in R(M). Let U1 +be the set of edges that are contained in S-avoiding paths having a1 as a +terminal vertex. Observe that every edge incident with a1 is in U1. Let U2 +be the set of edges of G not in U1. Thus (U1, U2) is a partition of the edge +set. +4.4.2. (U1, U2) is a vertical separation of M. +Proof. We must prove that neither U1 nor U2 is spanning in M. Let e be any +edge incident with a2. We claim that e is not in U1. Assume otherwise, and +let P be an S-avoiding path with a1 as a terminal vertex, where P contains +e. Since e is incident with a2, we can let P ′ be a subpath of P from a1 to a2. +As C1 and C2 form a separating pair, it follows that P ′ contains a vertex of +S. But the end vertices of P ′ are a1 and a2, and neither is in S, so P ′ has an +internal vertex in C1 ∩ C2. Thus P does as well, a contradiction. Therefore +e is not in U1. +Assume that U1 is spanning. Let e be an edge incident with a2. Then e +is in U2 by the previous paragraph. Since it is in the closure of U1, we can +let D be a cycle containing e such that every other edge of D is in U1. In +particular, this means that a2 is incident with an edge of U1, contrary to +the previous paragraph. So U1 is not spanning. +Similarly, if U2 is spanning, then we let e be an edge incident with a1. +Then e is not in U2, so we can let D be a cycle that contains e, where all the +other edges of D are in U2. This implies that an edge incident with a1 is in +U2, which contradicts an earlier conclusion. Therefore (U1, U2) is a vertical +separation. +□ +For i = 1, 2, we let Fi be cl(Ui). Recall that Ri is the edge-set of Ci. +4.4.3. F1 ∩ F2 = R1 ∩ R2. +Proof. Let e be an edge that joins vertices u and v. First assume that e +is in R1 ∩ R2. Then u and v are in S. This means there is no S-avoiding +path containing e with a1 as a terminal vertex. Hence e is not in U1 so it +is in U2. However, a1 is adjacent to u and v, and the edges a1u and a1v +are in U1, so e is in cl(U1). Thus e is in F1 ∩ F2 and we have shown that +R1 ∩ R2 ⊆ F1 ∩ F2. +For the other direction, assume that e is in F1 ∩ F2. First assume that +e is in U1. Let P be an S-avoiding path with a1 as a terminal vertex such +that e is in P. We can assume that either u or v is a terminal vertex of P. +Since e is in U1 ∩ cl(U2) we can let D be a cycle such that e is in D, +and every other edge of D is in U2. Thus both u and v are incident with +edges in U2. Let e′ be an edge incident with u that is in U2. Assume for +a contradiction that u is not in S. If u is a terminal vertex of P then we +obtain a new path by adding e′ to the end of P. No internal vertex of this +new path is in S, so it implies that e′ is in U1, a contradiction to e′ being in +U2. Therefore u is not a terminal vertex of P. Since P contains e, it follows + +SUPERSOLVABLE AND SATURATED MATROIDS +19 +that u is an internal vertex of P, so v is a terminal vertex of P. In this case +we can obtain a new path from P by replacing the edge e with e′. Again we +see that e′ is in U1 and we have a contradiction. Therefore u is in S, and by +symmetry, so is v. Hence e joins two vertices of S, and is thus in R1 ∩ R2. +We must also consider the case that e is in U2 ∩ cl(U1). Let D be a cycle +that contains e, where every other edge of D is in U1. Let x be an edge +of D − e that is incident with u. Thus x is in U1. Let P be an S-avoiding +path containing x and a1 as a terminal vertex. If u is not in S, then we +can either extend P by adding the edge e, or replacing x in P with e. In +either case, the new path shows that e is in U1, a contradiction. Therefore +u, and by symmetry v, is in S, so we again see that e is in R1 ∩ R2. Hence +F1 ∩ F2 ⊆ R1 ∩ R2 and the claim is proved. +□ +Recall that a1 is in the clique C1. Every edge incident with a1 is in U1. +As every edge of C1 − a1 is spanned by two such edges, it follows that R1 is +contained in F1. We must also show that R2 is spanned by U2. Let e be an +edge in R2 and assume that it is not in F2. In particular, this means that e +is not in U2, so it is in U1. Let P be an S-avoiding path containing e that +has a1 as a terminal vertex. Let u be the first internal vertex of P that is +incident with e. Then u is not in S, as P is S-avoiding. But u is in C2, since +e is in R2. Thus u is in C2 − C1, and the subpath of P from a1 to u is an +S-avoiding path from a vertex of C1 − C2 to a vertex of C2 − C1. This is a +contradiction, as C1 and C2 form a separating pair. This shows that R2 is +contained in F2, as claimed. +Now we can complete the proof that R1 and R2 are adjacent in R(M) by +showing that (F1, F2) is a modular cover. Assume that F1 is not a modular +flat, so that by utilising Proposition 2.3 we can let F be an arbitrary flat +of M that is disjoint from F1 such that some circuit C ⊆ F ∪ F1 contains +elements of both F and F1. If each connected component of G[F] shares at +most one vertex with G[F1], then no such cycle can exist. Therefore we let +u and v be distinct vertices from the same connected component of G[F] so +that both of u and v are incident with edges in F1. Since u is incident with +an edge in F1, it is incident with an edge in U1. Let e be such an edge, and +let P be a shortest-possible S-avoiding path that contains e and has a1 as a +terminal vertex. Let f be an edge of F that is incident with u. If u is not +in S, then extending P by adding f shows that f is in U1, a contradiction. +Therefore u, and by symmetry v, is in S. This means that there is an edge g +of R1 that joins u and v. Thus g is in R1 ⊆ F1. But there is a path of G[F] +that joins u to v, so g is in cl(F) = F, and we have contradicted F ∩F1 = ∅. +Therefore F1 is a modular flat. Almost exactly the same argument shows +that F2 is a modular flat. +□ +The previous result shows that when G is a 2-connected chordal graph, +CR(G) is isomorphic to the rotunda graph of a supersolvable saturated ma- +troid. In fact, this is true even when G is not 2-connected, as we now show. + +20 +MAYHEW AND PROBERT +First we make a simple observation. +Recall from Proposition 3.5 that a +matroid is supersolvable and saturated if and only if all its components are. +Proposition 4.5. Let G be a chordal graph with connected components +H1, . . . , Hk. Then CR(G) is the disjoint union of CR(H1), . . . , CR(Hk). Sim- +ilarly, if M is a supersolvable saturated matroid with connected components +N1, . . . , Nk, then R(M) is the disjoint union of R(N1), . . . , R(Nk). +Proof. Maximal cliques in different components of G cannot be adjacent in +CR(G) because they have no vertices in common. Similarly, rotunda from +different components of M are not adjacent in R(M) because they have +empty intersection. The result follows. +□ +Lemma 4.6. Let G be a chordal graph. There is a supersolvable saturated +matroid M such that CR(G) is isomorphic to R(M). +Proof. Let H1, . . . , Hk be the connected components of G. If each CR(Hi) is +isomorphic to R(Ni) for some supersolvable saturated Ni, then Proposition +4.5 implies that CR(G) is isomorphic to R(N1 ⊕ · · · ⊕ Nk). In other words, +it suffices to prove the lemma when G is connected. In this case, we will +prove that CR(G) is isomorphic to CR(G′), where G′ is a 2-connected chordal +graph. Then Proposition 4.4 shows that CR(G′) = R(M(G′)), and M(G′) +is supersolvable and saturated by Corollary 3.8 and Proposition 3.9, so the +result will follow. +If G is 2-connected, then there is nothing left to prove, so let v1, v2, . . . , vm +be the cut-vertices of G. +We produce G′ by introducing new vertices +v′ +1, v′ +2, . . . , v′ +m and for each i making v′ +i adjacent to vi and all of the neigh- +bours of vi in G. +4.6.1. G′ is 2-connected. +Proof. Certainly G′ is connected. Assume that v is a cut-vertex of G′. Note +that for each i, the graph produced from G′ by deleting v′ +i is obtained from +G by adding m−1 new vertices and making each of them adjacent to at least +one vertex in G. Since G is connected it follows that G′ − v′ +i is connected. +Thus no vertex v′ +i is a cut-vertex of G′ so v is not equal to v′ +i for any i. Now +v is a vertex of G. If v /∈ {v1, v2, . . . , vm} then G−v is connected and G′ −v +is obtained from the connected graph G − v by adding m new vertices and +making each of them adjacent to at least one vertex in G−v. Thus G′ −v is +connected, which is a contradiction. Therefore v = vi for some i. But in G′ +the vertices vi and v′ +i are adjacent to exactly the same vertices. Therefore +G′ − vi is obtained from G′ − v′ +i by relabelling v′ +i as vi. This means that +G′ − vi is connected, and we have a contradiction. +□ +4.6.2. G′ is chordal. +Proof. We rely on Proposition 2.1. Let u1, u2, . . . , un be a perfect elimination +order of G. We produce an ordering of the vertices of G′ by inserting each +v′ +i into the order u1, u2, . . . , un immediately after vi. It is easy to verify that +this produces a perfect elimination order for G′ and the result follows. +□ + +SUPERSOLVABLE AND SATURATED MATROIDS +21 +We can complete the proof by showing that CR(G) is isomorphic to +CR(G′). It is clear that any maximal clique of G′ contains one of the ver- +tices {vi, v′ +i} if and only if it contains both. Now we can easily verify that +there is a bijective correspondence between the maximal cliques of G and +the maximal cliques of G′. If C is a maximal clique of G, then we obtain +the corresponding maximal clique of G′ by adding each vertex v′ +i such that +vi is in C. +Let C1 and C2 be distinct maximal cliques of G, and let C′ +1 and C′ +2 be +the corresponding maximal cliques of G′. We will prove that C1 and C2 are +adjacent in CR(G) if and only if C′ +1 and C′ +2 are adjacent in CR(G′). First +note that C1 ∩ C2 is non-empty if and only if C′ +1 ∩ C′ +2 is non-empty. +If C1 and C2 are not adjacent in CR(G), then either C1 ∩ C2 = ∅, or P +is a (C1 ∩ C2)-avoiding path of G from a vertex of C1 − C2 to a vertex in +C2 − C1. In the first case C′ +1 ∩ C′ +2 = ∅. In the second case, it is obvious +that P is a (C′ +1 ∩ C′ +2)-avoiding path of G′. In either case C′ +1 and C′ +2 are not +adjacent in CR(G′). +Next assume that C′ +1 and C′ +2 are not adjacent in CR(G′). If C′ +1 ∩ C′ +2 = ∅ +then C1 ∩C2 = ∅ so we have nothing left to prove. Therefore we will assume +that P is a (C′ +1∩C′ +2)-avoiding path in G′, and that P joins a vertex in C′ +1−C′ +2 +to a vertex in C′ +2 − C′ +1. If a vertex v′ +i appears anywhere in P, then we may +replace it with vi, since these two vertices have the same neighbourhoods. +Note that the resulting path is still (C′ +1∩C′ +2)-avoiding, and still joins a vertex +of C′ +1 − C′ +2 to a vertex of C′ +2 − C′ +1. Thus we can assume that P is a path +of G, and is consequently a (C1 ∩ C2)-avoiding path of G from a vertex of +C1 −C2 to a vertex of C2 −C1. This shows that C1 and C2 are not adjacent +in CR(G) so the proof is complete. +□ +We have established that every reduced clique graph is isomorphic to a +rotunda graph. Next we start moving towards proving the converse. +Definition 4.7. Let M be a connected supersolvable and saturated matroid, +and let G be a 2-connected chordal graph. Assume that θ is a function from +E(M) to the powerset of V (G). +If U is a subset of vertices in G, then +let θ−1(U) be {x ∈ E(M): θ(x) ⊆ U}. +For any subset R ⊆ E(M), let +θ(R) stand for ∪x∈Rθ(x). Thus we can think of θ as being a function from +P(E(M)) to P(V (G)) such that R ⊆ R′ if and only if θ(R) ⊆ θ(R′). Assume +that the following properties hold: +(i) |θ(x)| = 2 for every x ∈ E(M), and +(ii) for any vertex v ∈ V (G) there exists exactly one element x ∈ E(M) +such that v is in θ(x). +(iii) if R is a non-empty round flat of M, then θ(R) is a clique, +(iv) if F is a modular flat of M and U is a union of connected components +of G − θ(F), then F ∪ θ−1(U) is a modular flat of M, and +(v) the restriction of θ to R(M) is a bijection from R(M) to the maximal +cliques of CR(G) and this bijection is an isomorphism between R(M) +and CR(G). + +22 +MAYHEW AND PROBERT +If all these conditions hold, then we will say that (G, θ) is compliant with +M. +Lemma 4.8. Let M be a connected supersolvable and saturated matroid. +There exists a 2-connected chordal graph G and a function θ: E(M) → +P(V (G)) such that (G, θ) is compliant with M. +Proof. The proof is a straightforward induction, although the technical de- +tails are require some work. If M has rank at most one, then we can simply +make G a clique of the appropriate size. Now we are going to choose C∗ +to be the complement of a modular hyperplane, H. Then inductively M|H +has a compliant graph. The intersection of H with cl(C∗) is a round flat, +and therefore corresponds to a clique. We create a new maximal clique by +adding new vertices and making them adjacent to each other and to the +clique corresponding to H ∩ cl(C∗). The rest of the proof involves nothing +more than checking that this construction does indeed satisfy the conditions +for compliance. +To implement this strategy, we let M be a supersolvable saturated ma- +troid. Assume r(M) ≤ 1. We can easily see that the only rotunda of M +is E(M) itself. We let G be isomorphic to K2|E(M)|, and we consider an +arbitrary partition of V (G) into blocks of size two. We then set θ to be +an arbitrary bijection from E(M) to the blocks of the partition. It is not +hard to verify that (G, θ) is compliant with M. Therefore we assume that +r(M) > 1. +Let H be a modular hyperplane of M such that M|H is supersolvable. +Then M|H is also saturated. Proposition 2.7 says that M|H is connected. +Therefore we can apply the obvious inductive hypothesis and let G′ be a +2-connected chordal graph with a function θ′ : H → P(V (G′)) such that +(G′, θ′) is compliant with M|H. +Let C∗ be the complementary cocircuit of H, and let R be the closure +of C∗. Proposition 2.14 says that R is a rotunda, and furthermore it is the +only rotunda of M that is not contained in H. Certainly C∗ is non-empty, +and r(H) = r(M) − 1 > 0, so H is non-empty also. But (H, C∗) is not a +separation of M, since M is connected. As H is modular, we deduce that +r(R ∩ H) = r(cl(C∗) ∩ H) = r(C∗) + r(H) − r(M) > 0. +Therefore R ∩ H is non-empty and Proposition 2.15 tells us that R ∩ H is +round. +Let W be θ′(R ∩ H). Since (G′, θ′) is compliant with M|H, we see that +W is the set of vertices of a clique in G′. Note also that |W| = 2|R∩H| ≥ 2. +We produce G from G′ by adding Y , a set of 2|C∗| new vertices, and making +each of them adjacent to all the vertices of W. Note that W ∪Y is a maximal +clique of G and G′ = G − Y . Because W has at least two vertices it is easy +to see that G is 2-connected. The neighbours of any vertex in Y form a +clique in G. Therefore we can construct a perfect elimination order for G by + +SUPERSOLVABLE AND SATURATED MATROIDS +23 +prepending the vertices of Y to a perfect elimination order for G′. It follows +that G is chordal. +Consider an arbitrary partition of Y into pairs of vertices, and let φ be an +arbitrary bijection from C∗ to the blocks of this partition. Then we define +θ to be the union of θ′ and φ. Note that |θ(x)| = 2 for any x ∈ E(M), and +for any vertex v of G, there is exactly one element x ∈ E(M) such that v +is in θ(x). Therefore the remainder of the proof consists in showing that θ +satisfies conditions (iii), (iv), and (v) in Definition 4.7. +Proposition 2.13 tells us that if Z is a round flat of M then either Z ⊆ H +or Z ⊆ R. In the former case, Z is a round flat of M|H, and θ(Z) = θ′(Z) +is a clique of G, since θ′ satisfies (iii). In the latter case θ(Z) is a subset of +W ∪ Y , and again θ(Z) is a clique of G. So condition (iii) holds for (G, θ). +4.8.1. Condition (iv) in Definition 4.7 holds for (G, θ). +Proof. Let F be a modular flat of M, and let U be a union of connected +components of G − θ(F). Let D be F ∪ θ−1(U). Thus our aim is to show +that D is a flat of M. Assume that U is the empty union. In this case +D = F ∪ θ−1(U) = F and since F is a modular flat there is nothing left +to prove. +Therefore we assume that U contains at least one connected +component of G − θ(F). +Assume that D is disjoint with C∗. This means that θ(F) ∪ U is disjoint +with Y . Thus F is a modular flat of M|H. If U is not a union of connected +components in G′ −θ′(F) then there is a connected component of this graph +that contains vertices u ∈ U and v /∈ U. There is a path of G′ − θ′(F) = +G − (θ(F) ∪ Y ) from u to v. Hence u and v are in the same component of +G − θ(F). This contradicts the fact that U is a union of components in this +graph. Hence U is a union of components of G′ − θ′(F), so we can apply +the inductive assumption and see that D = F ∪(θ′)−1(U) = F ∪θ−1(U) is a +modular flat of M|H. Therefore D is a modular flat of M and we are done. +Hence we assume that D contains at least one element of C∗. +Now θ(F) ∪ U contains at least two vertices from Y . +Since any such +vertex is adjacent to every vertex in W ∪ Y , and U is a non-empty union of +connected components, it now follows that θ(F) ∪ U contains W ∪ Y . Note +that D contains θ−1(W ∪ Y ) = R = cl(C∗). +Assume that U − Y is not a union of connected components in G′ − +θ(F ∩ H). Then there is a connected component of G′ − θ(F ∩ H) that +contains vertices u ∈ U − Y and v /∈ U − Y . There is a path from u to v in +G′ − θ(F ∩ H) = G − (θ(F) ∪ Y ). Thus u and v are in the same connected +component of G − θ(F). This means that u and v are both in U, since U +is a union of connected components in this graph. Since v is not in U − Y +this means that v is in Y . But this is impossible, since v is a vertex of G′, +which is equal to G − Y . This shows that U − Y is a union of connected +components in G′ − θ(F ∩ H). + +24 +MAYHEW AND PROBERT +We note that F ∩ H is a modular flat of M|H since both F and H are +modular in M. The inductive hypothesis now tells us that +(F ∩ H) ∪ θ−1(U − Y ) +is a modular flat of M|H. Let this flat be D′. Note that because C∗ ⊆ D +we have +D = F ∪ θ−1(U) = (F ∩ H) ∪ θ−1(U − Y ) ∪ C∗ = D′ ∪ C∗. +Let P be the union ∪PH(x, y), where x and y range over all distinct rank- +one flats contained in C∗. Thus P is a subset of R ∩H ⊆ D ∩H = D′. Note +that +cl(D) = (cl(D) ∩ H) ∪ C∗ = (cl(D′ ∪ C∗) ∩ H) ∪ C∗. +Now we apply Proposition 2.6. Since P ⊆ D′ ⊆ H we see that +(cl(D′ ∪ C∗) ∩ H) ∪ C∗ = cl(D′) ∪ C∗ = D′ ∪ C∗ = D. +Thus D is a flat of M. +Assume that D is not a modular flat of M, and let F ′ be a flat of M +that is disjoint with D, chosen so that C ⊆ D ∪ F ′ is a circuit that contains +elements of both D and F ′. Choose C so that |C∩C∗| is as small as possible. +Exactly as in the proof of Proposition 2.11 we can prove that C∩C∗ contains +distinct elements x and y. We choose p to be an element in PH(x, y), and +we perform strong circuit elimination on C and {x, y, p}. In this way we +find a circuit contained in D ∪ F ′ that contains elements of both sets, and +contains fewer elements of C∗ than C. This contradiction shows that D is +a modular flat of M so condition (iv) holds. +□ +4.8.2. The restriction of θ to R(M) is a bijection between R(M) and the +maximal cliques of G. +Proof. The inductive hypothesis means that θ′ induces a bijection between +the rotunda of M|H and the maximal cliques of G′. +First assume that +R ∩ H is a rotunda of M|H, so that W = θ′(R ∩ H) is a maximal clique of +G′. Now Proposition 4.2 shows that the rotunda of M are the rotunda of +M|H, except that R ∩ H has been replaced by R. It is easy to see that he +maximal cliques of G are the maximal cliques of G′, except that W has been +replaced by W ∪ Y . We observe that θ(R) = W ∪ Y and now it follows that +θ|R(M) is a bijection between the rotunda of M and the maximal cliques of +G. +Next we assume that R ∩ H is not a rotunda of M|H. Proposition 4.2 +implies that every rotunda of M|H is also a rotunda of M. Furthermore R +is the only rotunda of M that is not a rotunda of M|H. Because R ∩ H is +not a rotunda of M|H, we can let Z be a rotunda of M|H that properly +contains R ∩ H. Now W = θ′(R ∩ H) is properly contained in θ′(Z). Since +Z is round, we see that θ(Z) = θ′(Z) is a clique that properly contains +W. Therefore W is not a maximal clique of G′. Now it is easy to see that +every maximal clique of G′ is a maximal clique of G, and that W ∪ Y is + +SUPERSOLVABLE AND SATURATED MATROIDS +25 +the only maximal clique of G that is not a maximal clique of G′. The claim +follows. +□ +We can complete the proof of Lemma 4.8 by proving that the restriction +of θ to R(M) is an isomorphism from R(M) to CR(G). Let Z and Z′ be +distinct rotunda of M. We will show that they are adjacent in R(M) if and +only if θ(Z) and θ(Z′) are adjacent in CR(G). +Case 1. Neither Z nor Z′ is equal to R. In this case both Z and Z′ are +rotunda of M|H, and θ(Z) and θ(Z′) are maximal cliques of G′. Assume +that θ(Z) and θ(Z′) are adjacent in CR(G). Then these maximal cliques +have at least one vertex in common, and there is no (θ(Z) ∩ θ(Z′))-avoiding +path in G from a vertex of θ(Z)−θ(Z′) to a vertex of θ(Z′)−θ(Z). Exactly +the same statements apply to θ′(Z) and θ′(Z′) in G′, so θ′(Z) and θ′(Z′) +are adjacent in CR(G′). The inductive assumption implies that Z and Z′ +are adjacent in R(M|H). Proposition 4.2 now implies that they are also +adjacent in R(M). +For the converse, assume that Z and Z′ are adjacent in R(M), and let +(F, F ′) be a modular cover of M that certifies the adjacency. We assume +that Z ⊆ F and Z′ ⊆ F ′. Let us assume that both F ∩ C∗ and F ′ ∩ C∗ are +non-empty. No element of F ∩ C∗ is in F ′, because any such element would +be in F ∩ F ′ = Z ∩ Z′, and this is not possible since Z and Z′ are subsets of +H = E(M) − C∗. Symmetrically, no element of F ′ ∩ C∗ is in F. So F ∩ R +does not contain any element of F ′ ∩ C∗ and F ′ ∩ R does not contain any +element of F ∩ C∗. This shows that (F ∩ R, F ′ ∩ R) is a vertical cover of +R, which is impossible as R is a round flat. Therefore either F ∩ C∗ = ∅ or +F ′ ∩C∗ = ∅. We assume the latter, so C∗ is a subset of F and F ′ is a subset +of H. +Proposition 2.9 says that F ′ does not contain Z. It therefore does not +contain H, so we can apply Proposition 2.12 and deduce that +(F ∩ H, F ′ ∩ H) = (F ∩ H, F ′) +is a modular cover of M|H. Note that +(F ∩ H) ∩ F ′ = F ∩ F ′ = Z ∩ Z′ +so Z and Z′ are adjacent in R(M|H). By the inductive hypothesis, θ′(Z) = +θ(Z) and θ′(Z′) = θ(Z′) are adjacent in CR(G′). Since Z and Z′ are rotunda +of M, neither is equal to R∩H, which is properly contained in R. Therefore +neither θ(Z) nor θ(Z′) is equal to W. Because θ(Z) and θ(Z′) are adjacent +in CR(G′) they have a non-empty intersection. +Assume that θ(Z) and θ(Z′) are not adjacent in CR(G). +Let P be a +(θ(Z)∩θ(Z′))-avoiding path of G from a vertex a ∈ θ(Z)−θ(Z′) to a vertex +b ∈ θ(Z′) − θ(Z). Because no such path can exist in G′ = G − Y , it follows +that P contains a vertex in Y . Let y and y′, respectively, be the first and +last vertices of P that are in Y . Note that y and y′ are not equal to a or b, +which are vertices of G′. Let w be the neighbour of y in the subpath of P +from y to a. Similarly let w′ be the neighbour of y′ in the subpath from y′ + +26 +MAYHEW AND PROBERT +to b. Because w and w′ are adjacent to vertices in Y , but are not in Y , they +must be in W. Thus w and w′ are adjacent, so there is a path of G′ from a +to b that avoids any vertex in θ(Z) ∩ θ(Z′). This is a contradiction, so we +conclude that θ(Z) and θ(Z′) are adjacent in R(M). +We have now completed the case that neither Z nor Z′ is equal to R. +Case 2. One of Z and Z′ is equal to R. We let Z be a rotunda of M that +is distinct from R, and we will prove that Z and R are adjacent in R(M) if +and only if θ(Z) and θ(R) = W ∪ Y are adjacent in CR(G). Observe that +Z is contained in H by Proposition 2.14. +First assume that θ(Z) and θ(R) are adjacent in CR(G). Because θ sends +distinct elements of E(M) to distinct pairs of vertices, it cannot be the case +that Z ∩ R = ∅, or else θ(Z) and θ(R) would have no vertices in common, +contradicting their adjacency in CR(G). Thus Z and R are non-disjoint. +Assume Z contains R ∩ H. If C∗ is spanning in M, then R ∩ H = H, +so θ(H) = W is a clique. In this case G = W ∪ Y is a clique, but we have +assumed that M has at least two distinct rotunda, so G has at least two +distinct maximal cliques by 4.8.2. Thus C∗ is not spanning. Proposition 3.4 +says that (H, R) is a modular cover of M. Now R ∩ H = R ∩ Z so (H, R) +certifies that Z and R are adjacent in R(M) and we have nothing left to +prove. Therefore we will assume that Z does not contain R ∩ H. Hence +Z ∩ R is a proper and non-empty subset of R ∩ H. It follows that θ(Z) +contains some, but not all, of the vertices of W. +By Proposition 2.15 we know that R ∩ H is a round flat of M|H. Let +Z0 be a rotunda of M|H that contains R ∩ H. Thus Z0 is not equal to Z, +but it may be equal to R ∩ H. Now θ′(Z0) = θ(Z0) is a maximal clique +of G′ that contains W. Assume that θ(Z) and θ(Z0) are not adjacent in +CR(G′). Because these cliques have at least one vertex of W in common, +we can let P be a (θ(Z) ∩ θ(Z0))-avoiding path of G′ from a vertex a ∈ +θ(Z) − θ(Z0) to a vertex b ∈ θ(Z0) − θ(Z). Note that P contains no vertex +of θ(Z) ∩ θ(R). But P is also a path of G, and b is adjacent to any vertex of +W −θ(Z). Thus, if necessary, we can adjoin an edge to P from b to a vertex +of W − θ(Z), and certify that θ(Z) and θ(R) are not adjacent in CR(G), +contrary to hypothesis. Therefore θ(Z) and θ(Z0) are adjacent in CR(G′), +so by induction Z and Z0 are adjacent in R(M|H). +Because Z0 contains R ∩ H, the intersection of Z and Z0 contains Z ∩ R. +Assume this containment is proper, and let e be an element of Z ∩ Z0 that +is not in Z ∩ R. Let v be a vertex in θ(e). Thus v is in θ(Z) − θ(R). Choose +w, an arbitrary vertex in W − θ(Z). Because v is in θ(Z0), which contains +W, it follows that v and w are adjacent. Since w is in θ(R) − θ(Z), we now +see that θ(Z) and θ(R) are not adjacent in CR(G), contrary to hypothesis. +We conclude that Z ∩ Z0 = Z ∩ R. +Since Z and Z0 are adjacent in R(M|H), we can let (F, F ′) be a modular +cover of M|H that certifies this adjacency, where Z ⊆ F and Z0 ⊆ F ′. +Because Z0 contains R ∩ H, it follows that F ′ contains ∪PH(x, y), where +x and y range over distinct rank-one flats contained in C∗. +Proposition + +SUPERSOLVABLE AND SATURATED MATROIDS +27 +2.11 says that (F, F ′ ∪ C∗) is a modular cover of M. Certainly Z ⊆ F and +Z0 ⊆ F ′ ∪ C∗. Furthermore, +F ∩ (F ′ ∪ C∗) = F ∩ F ′ = Z ∩ Z0 = Z ∩ R. +Thus (F, F ′ ∪ C∗) certifies that Z and R are adjacent in R(M), exactly as +desired. +For the converse, we assume that Z and R are adjacent in R(M). Thus +Z ∩ R is non-empty. Assume that Z contains R ∩ H. Then θ(Z) contains +θ(R ∩ H) = W, so θ(Z) ∩ θ(R) = W. In G − W there is no path from a +vertex of θ(R)−θ(Z) = Y to a vertex not in Y , and in particular there is no +path to a vertex in θ(Z) − θ(R). So in this case θ(Z) and θ(R) are adjacent +in CR(G) and we have nothing left to prove. Therefore we will assume that +Z does not contain R ∩ H. Hence Z ∩ R is a non-empty proper subset of +R ∩ H. Since R ∩ H is a round flat of M|H by Proposition 2.15, we can let +Z0 be a rotunda of M|H that contains R ∩ H. Thus Z0 may be equal to +R ∩ H, but it is not equal to Z. +Let (F, F ′) be a modular cover of M that certifies the adjacency of R +and Z in R(M), where R ⊆ F and Z ⊆ F ′. Because F ∩ F ′ = R ∩ Z and +Z is contained in H it follows that F ′ is contained in H. If F ′ = H, then +F ∩ F ′ contains R ∩ H, which properly contains Z ∩ R. This contradicts +F ∩ F ′ = R ∩ Z, so F ′ does not contain H. By applying Proposition 2.12, +we see that (F ∩ H, F ′ ∩ H) = (F ∩ H, F ′) is a modular cover of M|H. +Because Z0 is round, one of F ∩ Z0 and F ′ ∩ Z0 is not a proper flat of +M|Z0. That is, Z0 is contained in either F or F ′. Assume Z0 is contained +in F ′. Then R ∩ H ⊆ Z0 ⊆ F ′ and R ⊆ F so F ∩ F ′ contains R ∩ H. This +is a contradiction as F ∩ F ′ = R ∩ Z, which is a non-empty proper subset +of R ∩ H. Therefore Z0 is contained in F. We observe that +(F ∩ H) ∩ F ′ = (F ∩ F ′) ∩ H = (R ∩ Z) ∩ H = R ∩ Z = F ∩ F ′ ⊇ Z0 ∩ Z. +Assume that F ∩ F ′ properly contains Z0 ∩ Z and let e be an element of +(F ∩ F ′) − (Z0 ∩ Z). Since F ∩ F ′ = R ∩ Z it follows that e is in Z. But we +also have +e ∈ F ∩ F ′ = R ∩ Z ⊂ R ∩ H ⊆ Z0. +Thus e is in Z0∩Z after all and we have a contradiction. Thus (F ∩H)∩F ′ = +Z0 ∩ Z = R ∩ Z and the modular cover (F ∩ H, F ′) of M|H certifies that +Z0 and Z are adjacent in R(M|H). Induction now tells us that θ(Z0) and +θ(Z) are adjacent in CR(G′). +Assume that θ(Z) and θ(R) = W ∪ Y are not adjacent in CR(G). These +cliques certainly have common vertices, so we can let P be a path from +a ∈ θ(Z)−θ(R) to b ∈ θ(R)−θ(Z) such that P contains no vertex of θ(Z)∩ +θ(R) = θ(Z)∩θ(Z0). If P is a path of G′ then it certifies that θ(Z) and θ(Z0) +are not adjacent in R(M|H), contrary to our earlier conclusion. Therefore +P contains at least one vertex in Y . Consider the maximal subpath of P +from a to vertex not in Y , and let this vertex be w. +Note that w is in +W − θ(Z) ⊆ θ(Z0) − θ(Z). So this subpath certifies that θ(Z) and θ(Z0) are + +28 +MAYHEW AND PROBERT +not adjacent in R(M|H), and we have another contradiction that completes +the proof. +□ +Proof of Theorem 1.1. Lemma 4.6 shows that every reduced clique graph +is isomorphic to a rotunda graph. On the other hand, if M is a supersolv- +able saturated matroid with connected components M1, . . . , Mn, then R(M) +is the disjoint union of R(M1), . . . , R(Mn), as we observed in Proposition +4.5. Lemma 4.8 shows that each R(Mi) is isomorphic to CR(Gi) for some +2-connected chordal graph Gi. If G is the disjoint union of G1, . . . , Gn, then +CR(G) is the disjoint union of CR(G1), . . . , CR(Gn), and is thus isomorphic +to R(M). So any rotunda graph is isomorphic to a reduced clique graph. +□ +Lemma 4.9. Let M be a supersolvable saturated matroid. Then R(M) is +connected if and only if M is connected. +Proof. In Proposition 4.5 we noted that if N1, . . . , Nk are the connected +components of M, then R(M) is the disjoint union of R(N1), . . . , R(Nk). So +if M is not connected then neither is R(M). For the converse, we let M be +a connected supersolvable saturated matroid. Lemma 4.8 shows that R(M) +is isomorphic to CR(G) where G is a 2-connected chordal graph G. From +Corollary 3.1 in [9] we see that CR(G), and hence R(M), is connected. +□ +5. Clique trees and rotunda trees +Definition 5.1. Let M be a matroid and let T be a tree. Let τ be a function +from V (T) to P(E(M)). Assume that for every element x ∈ E(M) there is +at least one vertex v ∈ V (T) such that x ∈ τ(v). In this case we say that +(T, τ) is a tree-decomposition of M. If for every element x ∈ E(M) there is +exactly one vertex v ∈ V (T) such that x ∈ τ(v) then the tree-decomposition +is strict. +In other words, the tree-decomposition is strict if {τ(t)}t∈V (T) is a parti- +tion of E(M). +Let G be a graph. A clique tree of G is a pair (T, ρ) where T is a tree +and ρ is a bijection from V (T) to the set of maximal cliques of G. We insist +that for any v ∈ V (G), the set {t ∈ V (T): v ∈ ρ(t)} induces a subtree of T. +Clique trees were introduced by Gavril [5], who showed that a graph has a +clique tree if and only if it is chordal. Our next step is to define a matroid +analogue of a clique tree. +Definition 5.2. Let M be a matroid, and let (T, τ) be a tree-decomposition +of M such that τ is a bijection from V (T) to R(M). If, for every x ∈ E(M), +the set {t ∈ V (T): x ∈ τ(t)} induces a subtree of T, then (T, τ) is a rotunda +tree of M. +In the following material we must apply weights to the edges of reduced +clique graphs and rotunda graphs. Let G be a chordal graph. Let σ be a +function which takes the set +{∅} ∪ {C ∩ C′ : C and C′ are distinct maximal cliques of G} + +SUPERSOLVABLE AND SATURATED MATROIDS +29 +to non-negative integers, and where the following conditions hold: +(i) σ(∅) = 0, +(ii) if X and X′ are in the domain of σ and X is a proper subset of X′, +then σ(X) < σ(X′). +In this case σ is a legitimate weighting of G. The function σ applies a weight +to each edge of CR(G), where the weight of the edge between C and C′ is +σ(C ∩ C′). The following result is the main theorem of [9]. +Theorem 5.3. Let G be a connected chordal graph and let σ be a legitimate +weighting. Every clique tree is a spanning tree of CR(G) and every edge of +CR(G) is contained in a clique tree. Moreover, a spanning tree of CR(G) +is a clique tree if and only if it has maximum weight amongst all spanning +trees. +Galinier, Habib, and Paul [4] prove the special case of Theorem 5.3 where +σ(C ∩ C′) = |C ∩ C′|, but their proof contains a flaw which is explained in +[9]. Next we consider the matroid analogue of legitimate weightings. +Definition 5.4. Let M be a supersolvable saturated matroid. Let σ be a +function taking +{∅} ∪ {R ∩ R′ : R, R′ ∈ R(M), R ̸= R′} +to non-negative integers, where: +(i) σ(∅) = 0, +(ii) if X and X′ are in the domain of σ and X is a proper subset of X′, +then σ(X) < σ(X′). +Then σ is a legitimate weighting of M. +For examples of legitimate weightings, we may set σ(R ∩ R′) to be either +the rank or the size of R ∩ R′, for each pair of rotunda R and R′. In the +case where we use rank, the legitimacy of the weighting relies on the fact +that the intersection of two rotunda is a flat. +Now we are able to prove Theorem 1.2, which we restate in a more general +form here. +Theorem 5.5. Let M be a connected supersolvable and saturated matroid +and let σ be a legitimate weighting of M. Every rotunda tree of M is a +spanning tree of R(M) and every edge of R(M) is contained in a rotunda +tree. Moreover, a spanning tree of R(M) is a rotunda tree if and only if it +has maximum weight amongst all spanning trees. +Proof. We apply Lemma 4.8 and let G be a 2-connected chordal graph and +let θ: E(M) → P(V (G)) be a function such that (G, θ) is compliant with +M. Let H be a graph that is isomorphic to both CR(G) and R(M). Let +πG be a bijection from V (H) to the family of maximal cliques of G, and +let πM be a bijection from V (H) to R(M), such that πG and πM are both +isomorphisms. + +30 +MAYHEW AND PROBERT +Let (T, τ) be a rotunda tree of M. Define ρ to be the composition θ|R(M)◦ +τ. This means that ρ is a bijection from V (T) to the set of maximal cliques +of G. Let v be an arbitrary vertex of G, and let x be the unique element of +E(M) such that v is in θ(x). Now +(1) +{t ∈ V (T): v ∈ ρ(t)} = {t ∈ V (T): x ∈ τ(M)}. +Because the latter set induces a connected subgraph of T, so does the former. +This shows that (T, ρ) is a clique tree of G. Therefore T is (isomorphic to) a +spanning tree of H by Theorem 5.3. We have now shown that any rotunda +tree of M is a spanning tree of R(M). Moreover, if e is an arbitrary edge +of H, then there is some spanning tree T of H such that T contains e and +(T, ρ) is a clique tree of G for some bijection ρ. Let τ be the composition +(θ|R(M))−1 ◦ ρ, so that τ is a bijection from V (T) to R(M). +If x is an +arbitrary element of E(M) and v is a vertex in θ(x), then Equation (1) still +holds and we see that (T, τ) is a rotunda tree of M that contains the edge +e. Thus any edge of R(M) is contained in a rotunda tree of M. +We apply weights to the edges of H. If u and u′ are adjacent in H, then +we weight the edge between them with σ(πM(u)∩πM(u′)). It is not difficult +to see that this weighting of H is also a legitimate weighting of CR(G); that +is, if C and C′ are maximal cliques of G that are adjacent in CR(G), and +σG applies the weight σ(θ−1(C) ∩ θ−1(C′)) to the edge between C and C′, +then σG is a legitimate weighting of G. +Let T be a maximum-weight spanning tree of H. Then (T, πG) is a clique +tree of G, by Theorem 5.3. Exactly as before, we see that (T, (θ|R(M))−1◦πG) +is a rotunda tree of M. On the other hand, if T is a spanning tree of H +and (T, τ) is a rotunda tree of M, then (T, θ|R(M) ◦ τ) is a clique tree of G. +Hence T is a maximum-weight spanning tree of H. We have now proved +that the rotunda trees of M are exactly the maximum-weight spanning trees +of R(M), as claimed. +□ +It follows from Theorem 1.2 that R(M) is the exactly the union of all +rotunda trees of M. +6. Tree-decompositions +We recall the definition of graph tree-width. Let G be a graph. Let T be +a tree and let ρ be a function from V (T) to P(V (G)) such that for every +v ∈ V (G) the set {t ∈ V (T): v ∈ ρ(t)} is non-empty and induces a subtree +of T. We further insist that if u and v are adjacent vertices of G, then +u, v ∈ ρ(t) for some t ∈ V (T). Then (T, ρ) is a tree-decomposition of G, +and the sets ρ(t) are the bags of the decomposition. The width of (T, ρ) is +the maximum size of a bag, and the tree-width of G is the minimum width +taken over all tree-decompositions. +Any clique tree of a chordal graph is a tree-decomposition of optimal +width, where the bags of the tree-decomposition are exactly the maximal +cliques [7, p. 14]. We now move towards a matroid analogue of this result. + +SUPERSOLVABLE AND SATURATED MATROIDS +31 +We first introduce the notion of matroid tree-width, as developed by Hlinˇen´y +and Whittle [8]. +Recall that a tree-decomposition of a matroid M is a +tree T along with a function τ : V (T) → P(E(M)) such that every element +x ∈ E(M) is in at least one set τ(t). +Definition 6.1. Let M be a matroid and let (T, τ) be a tree-decomposition +of M. Let t be a node of T and let T1, . . . , Td be the connected components +of T − t. For each i let Fi be ∪s∈V (Ti)τ(s). We define the node-width of t to +be +� +� +� +� +d +� +i=1 +r +� +� +� +�τ(t) ∪ +d� +k=1 +k̸=i +Fk +� +� +� +� +� +� +� +� − (d − 1)r(M). +The width of (T, τ) is the maximum node-width of any node in T. The tree- +width of M (denoted tw(M)) is the smallest width of any tree-decomposition +of M. +Note that this definition is not exactly that used by Hlinˇen´y and +Whittle because in their definition the minimum ranges over strict tree- +decompositions, rather than all tree-decompositions. To see that this makes +no difference to the definition, assume that the element x ∈ E(M) is con- +tained in both τ(u) and τ(v), where u and v are distinct vertices of the +tree T. We redefine τ by removing x from τ(u). It is easy to confirm that +the width of no node is increased by this change. By repeating this process +we can produce a strict tree-decomposition with width no greater than the +width of our original decomposition. This argument shows that there ex- +ists a strict tree-decomposition whose width is as small as possible amongst +all tree-decompositions. Thus extending Hlinˇen´y and Whittle’s definition to +include non-strict tree-decompositions makes no difference to the parameter. +We can always let T be a tree with a single node, and let τ take every +element of E(M) to this node. It follows from the definition that the width +of (T, τ) is r(M). +This shows that the tree-width of any matroid M is +bounded above by r(M). +Proposition 6.2. Let M be a round matroid. Then tw(M) = r(M). +Proof. Let E be the ground set of M. +Let (T, τ) be any strict tree- +decomposition of M. We direct each edge of T in the following way. Let e +be an arbitrary edge of T and assume that e joins u1 to u2. For each i let Ti +be the connected component of T\e that contains ui. Let Ui = ∪s∈V (Ti)τ(s). +Thus (U1, U2) is a partition of E (since the tree-decomposition is strict), and +because M is round, either U1 or U2 is spanning. If Ui is spanning then we +direct e from u3−i to ui. Note that it is possible for an edge to have two +directions applied to it. +Let P be a maximum length directed path in T, and assume that t is the +final node in P. Let T1, . . . , Td be the connected components of T − t and +let Fi = ∪s∈V (Ti)τ(s). Because the edges incident with t are all directed + +32 +MAYHEW AND PROBERT +towards t, it follows that E − Fi is spanning for each i. Since F1, . . . , Fd are +pairwise disjoint, the width of t is +r(M) − +d +� +i=1 +(r(M) − r(E − Fi)) = r(M) − +d +� +i=1 +(r(M) − r(M)) = r(M). +Hence the node-width of t is equal to r(M). Thus tw(M) ≥ r(M). We have +already observed that tw(M) ≤ r(M) so the proof is complete. +□ +Hlinˇen´y and Whittle show that if N is a minor of the matroid M, then +tw(N) ≤ tw(M) [8, Proposition 3.1]. +The next result follows from this +observation and Proposition 6.2. +Corollary 6.3. Let M be a matroid and let R be a round flat of M. Then +tw(M) ≥ r(R). +Proposition 6.4. Let (T, τ) be a rotunda tree of M, a supersolvable sat- +urated matroid. +Let e be an edge of T that joins vertices u1 and u2. +For i = 1, 2, let Ti be the connected component of T\e that contains ui +and let Fi be ∪t∈V (Ti)τ(t). +Then (F1, F2) is a modular cover of M and +F1 ∩ F2 = τ(u1) ∩ τ(u2). +Proof. Note that every element of E(M) is contained in a round flat, and +hence in a rotunda. From this it follows that E(M) = F1 ∪ F2. +We apply Lemma 4.8 and we let G be a 2-connected chordal graph with +a function θ: E(M) → P(V (G)) such that (G, θ) is compliant with M. Let +ρ be the composition θ|R(M) ◦ τ so that ρ is a bijection between V (T) and +the maximal cliques of G. Exactly as in the proof of Theorem 1.2 we can +show that (T, ρ) is a clique tree of G. +Define Ri to be the rotunda τ(ui). Let F be the flat R1 ∩ R2. Note that +because R1 and R2 are adjacent in a rotunda tree of M, they are adjacent +in R(M) by Theorem 1.2. This implies that F is non-empty. Let Ci = θ(Ri) +for i = 1, 2, so that C1 and C2 are the corresponding maximal cliques of G. +Define S to be θ(F) = C1 ∩ C2. +Note that if D is a maximal clique of G, then D − S is contained in a +connected component of G − S. For i = 1, 2, let vi be an arbitrary vertex +of Ti. Then the path of T from v1 to v2 contains u1 and u2. It follows from +[9, Proposition 2.8] that ρ(v1) − S and ρ(v2) − S are contained in different +connected components of G−S. Now we let U be the union of all connected +components of G − S that contains ρ(v) − S for some v in V (T1). From +the observations in this paragraph we see that F ∪ θ−1(U) is equal to F1. +Because (G, θ) is compliant with M this means that F1 is a modular flat of +M. Symmetrically, F2 is a modular flat. +Let x be an arbitrary element in F1 ∩ F2. Let v ∈ V (T1) and v′ ∈ V (T2) +be chosen so that x is in τ(v) ∩ τ(v′). Because (T, τ) is a rotunda tree it +follows that x is in τ(w) whenever w is in the path of T from v to v′. In +particular, x is in τ(u) ∩ τ(u′) = R ∩ R′ = F. Thus F1 ∩ F2 ⊆ F. Because + +SUPERSOLVABLE AND SATURATED MATROIDS +33 +ui is in Ti for each i it follows that Ri ⊆ Fi. Therefore F = R1 ∩ R2 is a +subset of F1 ∩ F2, and now +τ(u1) ∩ τ(u2) = R1 ∩ R2 = F = F1 ∩ F2. +From this it follows that F1 ∩ F2 does not contain R1 or R2, so neither F1 +nor F2 is equal to E(M). Since F1 and F2 are proper modular flats of M +and E(M) = F1 ∪ F2 we see that (F1, F2) is a modular cover and the result +is proved. +□ +Let M be a connected supersolvable and saturated matroid. +We will +now show that a rotunda tree of M has the properties of an optimal tree- +decomposition as per Hlinˇen´y and Whittle. +Theorem 6.5. Let M be a supersolvable saturated matroid and let (T, τ) be +a rotunda tree of M. Then the width of (T, τ) is equal to tw(M). +Proof. We will show that the node-width of any t ∈ V (T) is r(τ(t)), so +that the width of (T, τ) is the maximum rank of a rotunda of M. From +Corollary 6.3 we see that tw(M) is bounded below by this rank, so having +completed this task, we will have shown that (T, τ) is a tree-decomposition +of lowest-possible rank. It will then follow that tw(M) is equal to the width +of (T, τ). +So let t be an arbitrary vertex in T and let T1, . . . , Td be the connected +components of T − t. For each i let ti be the vertex of Ti that is adjacent to +t. Define F to be τ(t), and let Fi be ∪s∈V (Ti)τ(s) for each i. We define F i +to be +F ∪ +d� +k=1 +k̸=i +Fk. +Therefore the node-width of t is +(2) +r(F 1) + · · · + r(F d) − (d − 1)r(M). +In addition, we define F >i to be +F ∪ +d� +k=i+1 +Fk. +Notice that F >1 = F 1 and that F >d = F. +6.5.1. For any i ∈ {1, . . . , d−1}, the intersection of F >i and F i+1 is F >i+1. +Proof. We note that +F >i ∩ F i+1 = (F ∪ Fi+1 ∪ · · · ∪ Fd) ∩ (F ∪ F1 ∪ · · · ∪ Fi ∪ Fi+2 ∪ · · · Fd) += (Fi+1 ∩ (F1 ∪ · · · ∪ Fi)) ∪ (F ∪ Fi+2 ∪ · · · ∪ Fd). +Now Fi+1 ∩ (F1 ∪ · · · ∪ Fi) is contained in Fi+1 ∩ F i+1. But Proposition +6.4 tells us that Fi+1 ∩ F i+1 is equal to τ(ti+1) ∩ τ(t), which is therefore +contained in τ(t) = F. Hence we can remove Fi+1 ∩ (F1 ∪ · · · ∪ Fi) from the + +34 +MAYHEW AND PROBERT +equation above and conclude that F >i ∩F i+1 is F ∪Fi+2 ∪· · ·∪Fd = F >i+1, +as claimed. +□ +Proposition 6.4 implies that (Fi, F i) is a modular cover for each i, so that +in particular F 2 is a modular flat. Now Equation (2) reduces to +r(F 1 ∩ F 2) + r(F 1 ∪ F 2) + r(F 3) + · · · + r(F d) − (d − 1)r(M) += r(F >1 ∩ F 2) + r(E(M)) + r(F 3) + · · · + r(F d) − (d − 1)r(M) += r(F >2) + r(F 3) + · · · + r(F d) − (d − 2)r(M) +where we have applied 6.5.1 in the final step. Because F 3 is a modular flat, +we can again apply 6.5.1 and reduce to +r(F >3) + r(F 4) + · · · + r(F d) − (d − 3)r(M) +By continuing this process, we find that Equation (2) is equal to +r(F >d) − (d − d)r(M) = r(F). +So the node-width of t is r(τ(t)) = r(F), exactly as we claimed, and the +theorem is proved. +□ +Now we present the central theorem for this section. Because a rotunda +tree is a tree-decomposition of optimal width for a supersolvable saturated +matroid M, we can treat it as a canonical tree decomposition of M. +Corollary 6.6. Let M be a supersolvable saturated matroid. Then tw(M) = +max{r(R): R ∈ R(M)}. +Further observe the following. Let bw(M) denote the branch-width of M. +By [8, Theorem 4.2] we see that +bw(M) − 1 ≤ tw(M) ≤ max{2 bw(M) − 2, 1}. +We see therefore that given a supersolvable saturated matroid M of branch- +width k there must be a rotunda tree of M where the rank of the largest +maximal rotunda is bounded by a function of k. As a result, we can conclude +that supersolvable saturated matroids have canonical tree decompositions of +optimal tree-width in much the same way as chordal graphs have canonical +tree decompositions where each bag is a clique of the graph. +This theorem has algorithmic implications for how we can efficiently find +the tree-width of a supersolvable saturated matroid. However, for this to +work we would need an efficient method for constructing the rotunda graph. +7. Acknowledgements +We thank Geoff Whittle, who supervised the thesis of the second author +(which includes much of the material in this article). We also thank a referee +of an earlier draft for numerous helpful comments. + +SUPERSOLVABLE AND SATURATED MATROIDS +35 +References +[1] C. 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With +a foreword by Claude Berge. +[7] Pinar Heggernes, Treewidth, partial k-trees, and chordal graphs (2006), https://www. +ii.uib.no/~pinar/chordal.pdf. +[8] Petr Hlinˇen´y and Geoff Whittle, Matroid tree-width, European J. Combin. 27 (2006), +no. 7, 1117–1128. +[9] Dillon Mayhew and Andrew Probert, Reduced clique graphs: a correction to “Chordal +graphs and their clique graphs”. In preparation. +[10] James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, +Oxford University Press, Oxford, 2011. +[11] R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. +[12] Geoff Whittle, Some Aspects of the Critical Problem for Matroids, University of Tas- +mania, 1985. PhD Thesis. + diff --git a/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/load_file.txt b/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1326cf45345d272c12c73ecc17b2c66dc740c22 --- /dev/null +++ b/_9E2T4oBgHgl3EQfRAZd/content/tmp_files/load_file.txt @@ -0,0 +1,1543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf,len=1542 +page_content='SUPERSOLVABLE SATURATED MATROIDS AND CHORDAL GRAPHS DILLON MAYHEW AND ANDREW PROBERT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid is supersolvable if it has a maximal chain of flats each of which is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid is saturated if every round flat is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this article we present supersolvable saturated ma- troids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroid context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In particular, we con- sider matroid analogues of the reduced clique graph and clique trees for chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The latter is a maximum-weight spanning tree of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Introduction The study of chordal graphs is well established, and dates to work by Dirac [3] and Berge [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Our contribution here is to consider a new analogue of chordality for matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A graph is chordal if every cycle with at least four vertices has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This leads fairly directly to the definition of a chordal matroid used by Cordovil, Forge, and Klein [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C is a circuit in a matroid, then a chord of C is an element z /∈ C such that there is a partition of C into parts A and B where A ∪ z and B ∪ z are both circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will say that a matroid is C-chordal if every circuit with size at least four has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (Cordovil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' call such a matroid chordal, but we will try to avoid confusion by reserving that term solely for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=') In this article we concentrate on a different matroid analogue for chordal- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' An alternative characterisation of chordal graphs is due to Dirac [3]: a vertex is simplicial if its neighbours are pairwise adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now G is chordal if and only if it has a simplicial vertex v such that G − v is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This definition is well suited for matroid purposes, because the edges not incident with a simplicial vertex comprise a modular hyperplane in the corresponding graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (A flat F is modular if r(F)+r(F ′) = r(F ∩F ′)+r(F ∪F ′) for every flat F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A hyperplane is modular if and only if it has a non-empty intersection with every rank-two flat of the matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=') Now we can recur- sively consider the class of matroids M such that M is in M if and only if M has a modular hyperplane H where restricting M to H produces a matroid in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The class M is exactly the family of supersolvable matroids, introduced by Stanley [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Figure 1 shows a geometric representation of a rank-four matroid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We see that the hyperplane F3 = {1, 2, 3, 4, 5, 6, 7} is modular, since every 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='03776v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='CO] 10 Jan 2023 2 MAYHEW AND PROBERT 1 3 2 4 5 6 7 8 9 10 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A supersolvable matroid rank-two flat has a non-empty intersection with F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the same way, F2 = {1, 2, 3, 4} is a modular hyperplane of the restriction to F3, and F1 = {1} is a modular hyperplane of the restriction to F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Finally, ∅ is a modular hyperplane of the restriction to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that M is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It turns out that the condition of supersolvability is not strong enough for our purposes because supersolvable matroids may fail to have properties shared by all graphic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To expand on this point, we consider matroid analogues of cliques in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a flat of a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F is round if there is no pair of flats (F1, F2) such that F = F1 ∪ F2 and F1 and F2 are properly contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph and let F be a flat of the graphic matroid M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F is round if and only if G[F] is a clique (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we think of round flats as the matroid analogues of cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In graphic matroids every round flat is modular but this is not true for matroids in general, nor is it true for supersolvable matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For example, if M is the matroid in Figure 1, then {4, 6, 7, 8, 9, 10} is a round hyperplane, since it cannot be expressed as the union of two flats that it properly contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However, it is not modular, since it has an empty intersection with the rank-two flat {3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We define a matroid to be saturated if every round flat is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus saturated matroids can be thought of as analogues to graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To this condi- tion, we add the condition of supersolvability to obtain our matroid analogue of chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So our fundamental objects of study are supersolvable and saturated matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The graphic matroid M(G) is supersolvable and saturated if and only G is chordal (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Many other examples arise: for example, the matroids that are constructed using generalised parallel connections, starting with the projective geometries of a given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Any such matroid is supersolvable and saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The class of supersolvable saturated matroids is properly contained in the class of C-chordal matroids (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So our focus is on a proper subclass of C-chordal matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The relationships between the conditions SUPERSOLVABLE AND SATURATED MATROIDS 3 of supersovability, saturation, and C-chordality are illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will justify this Venn diagram in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' C-chordal Supersolvable Saturated F7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 1 U3,6 W4 M∗(K3,3) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Three matroid definitions Our main focus is showing that many facts about chordal graphs have analogues in the class of supersolvable saturated matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In particular, Section 4 introduces one of our main ideas: the rotunda graph of such a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A rotunda is a maximal round flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The vertices of the rotunda graph are the rotunda of the matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that R1 and R2 are distinct rotunda with a non-empty intersection and that (F1, F2) is a pair of modular flats of M such that E(M) = F1∪F2 and neither F1 nor F2 is equal to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If Ri ⊆ Fi for i = 1, 2 and F1 ∩ F2 = R1 ∩ R2, then we make R1 and R2 adjacent in the rotunda graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The idea of a rotunda graph is analogous to the reduced clique graph introduced by Galinier, Habib, and Paul in [4] (where it is called a clique graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If G is a chordal graph, then the vertices of the reduced clique graph of G are the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C and C′ are maximal cliques then they are adjacent if C ∩ C′ ̸= ∅ and any path from a vertex of C − C′ to a vertex of C′ − C uses a vertex of C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If G is a chordal graph then the reduced clique graph of G and the rotunda graph of M(G) need not be the same, but this is only because G may have low connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 we show that when G is 2-connected the reduced clique graph of G and the rotunda graph of M(G) are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can go further than this: the class of reduced clique graphs and the class of rotunda graphs are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then H is isomorphic to the rotunda graph of a supersolvable saturated matroid if and only if H is isomorphic to the reduced clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We prove this theorem in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It tells us that although a super- solvable saturated matroid may be far from graphic, the structure of its 4 MAYHEW AND PROBERT rotunda will be mirrored by the structure of maximal cliques in a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Knowing that these two classes of graphs are identical allows us to deduce facts about the structure of rotunda graphs from the facts about reduced clique graphs that we list in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For example, in [9] we show that the reduced clique graph of a chordal graph may have induced cycles of length three, four, or six, but not five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore the same statement applies to rotunda graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We conjecture that a reduced clique graph cannot have an induced cycle of length greater than six, so we therefore conjecture that the same statement holds for rotunda graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In [9] we show that no rotunda graph can be isomorphic to a cycle of length at least four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus the class of rotunda graphs is properly contained in the class of graphs with no induced cycle of length five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We also believe that every chordal graph is isomorphic to the rotunda graph of some supersolvable saturated matroid, and that there is a polynomial-time algorithm for recognising when a given graph is isomorphic to some rotunda graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A clique tree of the graph G is a tree whose nodes are the maximal cliques of G, where the set of maximal cliques containing an arbitrary vertex v ∈ V (G) induces a subtree of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Clique trees were introduced by Gavril [5], who showed that G has a clique tree if and only if G is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The analogue for a supersolvable saturated matroid M is a rotunda tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case the nodes of the rotunda tree are the rotunda of M, and the rotunda containing an arbitrary element x ∈ E(M) induces a subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid may have a rotunda tree without being supersolvable and saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For example, the matroid in Figure 1 is not saturated, but it does have a rotunda tree (having two nodes, corresponding to {1, 2, 3, 4, 5, 6, 7} and {4, 6, 7, 8, 9, 10}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Galinier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' [4] weight the edges of reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The edge that joins maximal cliques C and C′ is weighted with |C ∩ C′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' They then prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum total weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (Their proof contains a flaw, which we explain and correct in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=') In our analogous result we weight the edges of rotunda graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The edge that joins rotunda R and R′ is weighted with the rank of R ∩ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (Our techniques are general enough that we could also weight it with |R ∩ R′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In Section 5 we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a connected supersolvable and saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Every rotunda tree of M is a spanning tree of the rotunda graph of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Every edge of the rotunda graph is contained in a rotunda tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, a spanning tree is a rotunda tree if and only if it has maximum weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In Section 6 we concentrate on tree-decompositions of optimal width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In unpublished work, Heggernes [7] observed that a clique tree of a chordal graph is an optimal decomposition of the graph with respect to the pa- rameter of tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid analogue of tree-width was developed by SUPERSOLVABLE AND SATURATED MATROIDS 5 Hlinˇen´y and Whittle [8], and in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 we prove the matroid ana- logue of Heggernes’s observation: any rotunda tree of a supersolvable and saturated matroid is an optimal decomposition with respect to the matroid parameter of tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We refer to [10] for the foundations of matroid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If X is a set of vertices in G, then G[X] is the subgraph induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We say that a path P is X-avoiding if any vertex of X in P is a terminal vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A clique of G is a set of pairwise adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We blur the distinction between a subgraph, its vertex set, and its edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So for example we may refer to a clique of the graph G as being a flat in the cyclic matroid M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C is a cycle of a graph, then a chord is an edge that joins two distinct vertices of the cycle without being an edge of the cycle or parallel to any such edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A graph is chordal if every cycle with at least four vertices has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus a graph is chordal if and only if has no induced cycle with more than three vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Clearly every induced subgraph of a chordal graph is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph, and let v be a vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If deleting v from G produces a graph with more connected components than G, then v is a cut-vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A connected graph with no cut-vertex is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' An ordering v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , vn of the vertices in a graph is a perfect elimination order if the neighbours of vi amongst vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , vn form a clique, for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A proof of the following can be found in [6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A graph is chordal if and only if it has a perfect elimi- nation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The flat F is modular if r(F) + r(F ′) = r(F ∪ F ′) + r(F ∩ F ′) whenever F ′ is a flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that the entire ground set is trivially a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We also see that the unique rank-zero flat is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The following is proved in [10, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a flat of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F is modular if and only if r(F) + r(F ′) = r(F ∪ F ′) whenever F ′ is a flat such that F ∩ F ′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows easily that if F is a hyperplane, then F is modular if and only if r(F ∩ L) = 1 whenever L is a rank-2 flat not contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We often use an equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a flat of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F is modular if and only if there is no circuit C ⊆ F ∪ F ′ containing elements from both F and F ′, whenever F ′ is a flat that is disjoint from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F ′ be an arbitrary flat that is disjoint from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There is no circuit of M|(F ∪ F ′) that contains elements of both F and F ′ if and only 6 MAYHEW AND PROBERT if r(F) + r(F ′) = r(F ∪ F ′) [10, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now the result follows by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ The next result combines Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F and F ′ be modular flats of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F ∩ F ′ is a modular flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F ⊆ X ⊆ E(M) then F is a modular flat of M|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a modular flat of the matroid M and let C be a circuit of M such that C∩F is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then cl(C−F)∩F is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If cl(C − F) ∩ F = ∅ then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 is violated, since cl(F − C) is a flat that is disjoint from F, but C is a circuit that contains elements from both F and cl(C − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Let H be a modular hyperplane of the matroid M, and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let x and y be distinct rank-one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then r(H ∩ cl(x ∪ y)) = 1, because H is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We say that the rank-one flat H ∩ cl(x ∪ y) is the projection of x and y onto H, and we denote this flat with PH(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If x and y are elements of C∗ such that r({x, y}) = 2, then we also use PH(x, y) to stand for PH(cl({x}), cl({y})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let X be a subset of E(M) − H and let P be the union ∪PH(x, y), where {x, y} ranges over all pairs of distinct rank-one flats in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let U be a subset of H such that U contains P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then cl(U) = cl(U ∪ X) ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that cl(U) is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus it is obvious that cl(U) is a subset of cl(U ∪ X) ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let us assume that the containment is proper, and let z be an element that is in cl(U ∪ X) ∩ H but not cl(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus z is not in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There is some circuit C ⊆ U ∪ X ∪ z that contains z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let us assume that we have chosen C so that C − H is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C − H is empty, then C certifies that z is in cl(U), contrary to hypothesis, so C − H ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C − H contains a single element x, then C certifies that x is in cl(H) = H, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we can choose x and y to be distinct elements of C −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let p be an element in PH(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus p is in P and {x, y, p} is a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that z ̸= p, since z is not in P ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We perform strong circuit elimination on C and {x, y, p} to obtain the circuit C′ ⊆ (C − x) ∪ {p, z} such that z is in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus C′ is a subset of U ∪ X ∪ z, but C′ −H is smaller than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now our choice of C is contradicted, and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the connected matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M|H is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that M|H is not connected, and let (U, V ) be a separation of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because M is connected, there are circuits of M that contain elements from both U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Amongst such circuits choose C so that C − H is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let u be an element in C ∩ U and let v be an element SUPERSOLVABLE AND SATURATED MATROIDS 7 from C ∩ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that C − H is not empty since (U, V ) is a separation of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore, C − H does not contain a single element, or else that element would be in cl(H) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we choose distinct elements x, y ∈ C − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let p be an element in PH(x, y), so that {x, y, p} is a circuit of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because p is in H we can assume without loss of generality that p is in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We perform strong circuit elimination on C and {x, y, p} to obtain a circuit C′ ⊆ (C − x) ∪ {y, p} that contains v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that C′ contains p, or else it is a proper subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus C′ contains elements from both U and V , but |C′ − H| < |C − H|, and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore M|H is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Roundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A proper flat of a matroid is one that is not equal to the entire ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A vertical cover of M is a pair (F, F ′) of proper flats such that F ∪ F ′ = E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If, in addition, F and F ′ are modular flats, then (F, F ′) is a modular cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid is round if it has no vertical cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus a matroid is round if and only if there is no partition (U, U′) of E(M) such that neither U nor U ′ is spanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Such a partition is said to be a vertical separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If X is a subset of E(M), then we say that X is round if M|X is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F is a round flat of the matroid M and F is contained in the subset X ⊆ E(M), then clearly F is a round flat of M|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A round flat is maximal if it is not properly contained in a round flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For brevity, we refer to a maximal round flat as a rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The set of rotunda of a matroid M is denoted by R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let R and R′ be distinct rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (F, F ′) be a vertical cover such that R ⊆ F and R′ ⊆ F ′ and F ∩ F ′ = R ∩ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R ⊈ F ′ and R′ ⊈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It suffices to prove that R is not contained in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume this fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R is contained in F ∩ F ′ = R ∩ R′, implying that R is a subset of R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is impossible since R and R′ are distinct rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ The next result follows from work in [12], but we include a proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let X be a subset of the cocircuit E(M) − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then {PH(x, y): x, y ∈ X, r({x, y}) = 2} is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union of all projections onto H of pairs of distinct, non-parallel, elements in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus our aim is to show that P is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We assume for a contradiction that (F, F ′) is a vertical cover of M|P, so that F and F ′ are proper flats of M|P and F ∪ F ′ = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that if X contains 8 MAYHEW AND PROBERT fewer than three rank-one flats, then P is either empty or consists of a single rank-one flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case P is trivially round, so we must assume that X contains at least three rank-one flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let x, y, and z be distinct rank-one flats in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that PH(x, y) and PH(x, z) are both in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We claim that PH(y, z) is also in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If z is in cl(x ∪ y), then cl(x ∪ y) = cl(x ∪ z) = cl(y ∪ z), and it follows that PH(x, y) = PH(x, z) = PH(y, z), so the claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will assume that r(x ∪ y ∪ z) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z be cl(x ∪ y ∪ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since H is a modular hyperplane and Z is not contained in H, it follows that r(H ∩ Z) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now PH(x, y) and PH(x, z) are rank-one flats contained in H ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If they are not distinct, then y and z are both in the closure of x ∪ PH(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This implies that z is in cl(x ∪ y), contrary to earlier hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that PH(x, y) ∪ PH(x, z) spans H ∩ Z, and in particular spans PH(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus PH(y, z) is in F, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Symmetrically, if PH(x, y) and PH(x, z) are both in F ′, then so is PH(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We think of the rank-one flats that have a non-empty intersection with X as the vertices of a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If x and y are two such flats, then we colour the edge between x and y red if PH(x, y) is in F, and blue if it is in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Notice that an edge may be both red and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The previous paragraph shows that if the edges xy and xz are both red (blue), then the edge yz is also red (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let x be a vertex in this complete graph and assume that every edge incident with x is red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then every edge is red, and it follows that P is contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is impossible since F is a proper flat of M|P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly, it is not possible for every edge incident with x to be blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we can assume that the edge between x and y is red but not blue, and the edge between x and z is blue but not red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However, if the edge yz is red, then xz is red, and if yz is blue then xy is blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In either case we have a contradiction, so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (F1, F2) be a vertical cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union ∪PH(x, y), where x and y range over all distinct rank-one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then P is contained in Fi for some i, and (Fi ∪ C∗, F3−i) is a vertical cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, if (F1, F2) is a modular cover, then so is (Fi ∪ C∗, F3−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10 says that P is a round subset of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (F1 ∩ P, F2 ∩ P) is not a vertical cover of M|P, so either F1 ∩ P or F2 ∩ P is equal to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We assume the former without any loss of generality, so P ⊆ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6 implies that F1 is equal to cl(F1 ∪ C∗) ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that cl(F1∪C∗) = F1∪C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now F1∪C∗ is a proper flat of M because F1 is a proper flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly, F2 is a proper flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As (F1 ∪C∗)∪F2 = E(M), it follows that (F1 ∪ C∗, F2) is a vertical cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we assume that (F1, F2) is a modular cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F2 is a modular flat of M|H so it immediately follows from [10, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7] SUPERSOLVABLE AND SATURATED MATROIDS 9 that F2 is also a modular flat in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It remains only to prove that F1 ∪ C∗ is a modular flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To this end, assume that F is a flat of M that is disjoint from F1 ∪ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus F is a flat of M|(F2 − F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If we can show that there is no circuit of M|((F1 ∪ C∗) ∪ F) containing elements from both F and F1 ∪ C∗, then the result will follow from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that C is such a circuit, chosen so that C ∩ C∗ is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let f be an element of C ∩ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C ∩ C∗ = ∅, then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 implies that F1 is not a modular flat of M|H, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore C ∩C∗ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C ∩C∗ contains a single element, x, then C certifies that x is in cl(H) = H, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we let x and y be distinct elements in C ∩ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let p be in PH(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus {x, y, p} is a circuit and p is in P, and hence in F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We perform strong circuit elimination on C and {x, y, p} to obtain C′ ⊆ (C − x) ∪ {y, p}, a circuit that contains f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It must contain p, since otherwise it is properly contained in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But now C′ is contained in F1 ∪ C∗ ∪ F, and it contains elements from both F and F1 ∪ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since C′ ∩ C∗ is strictly smaller than C ∩ C∗, we have contradicted our choice of C, so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ The following result provides a partial converse to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (F, F ′) be a modular cover of M such that F ′ is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F ′ ̸= H, then (F ∩ H, F ′ ∩ H) is a modular cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that C∗ is contained in F because F ′ contains no element of C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union of PH(x, y) where x and y range over distinct rank-one flats in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since F contains C∗, and P is spanned by C∗ it follows that P is a subset of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now F ∩ H is the intersection of two modular flats, so Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 implies that it is a modular flat of M, and hence of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because F ′ is contained in H it is also true that F ′∩H = F ′ is a modular flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By hypothesis F ′ ∩H is a proper flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore, F ∩H is a proper flat of M|H, or else F contains H ∪ C∗ = E(M), contradicting the fact that F is a proper flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (F ∩H, F ′ ∩H) is a modular cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F is a round flat not contained in H, then F ⊆ cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume this fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F ∩ C∗ does not span F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is also true that F ∩H does not span F, as cl(F ∩H) ⊆ cl(H) = H and F is not contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (F ∩H, F ∩C∗) is a vertical cover of M|F, and this contradicts the fact that M|F is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then cl(C∗) is a rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore, every other rotunda of M is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 10 MAYHEW AND PROBERT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let R be cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that R is not round, and let (F, F ′) be a vertical cover of M|R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union ∪PH(x, y), where x and y range over all distinct rank-one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that P is contained in R∩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10 says that P is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that one of F ∩P or F ′ ∩ P is equal to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Without loss of generality we will assume the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F ′ contains C∗, then it contains R, which is impossible as (F, F ′) is a vertical cover of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we choose x ∈ C∗ − F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The same argument shows we can choose y ∈ C∗ − F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that x and y are not parallel, since x is in F − F ′ and y is in F ′ − F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let p be in PH(x, y), so that p is in P, and hence in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As {x, y, p} is a circuit and both x and p belong to the flat F it follows that y is in F, contrary to assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore R is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z be any flat that properly contains R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that Z ∩H is a flat that does not contain any element of C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (Z ∩H, R) is a vertical cover of Z, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that R is a maximal round flat, which is to say, a rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Finally, let Z be a rotunda that is not contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='13, we see that Z is contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As Z and R are both rotunda it now follows that Z = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then cl(C∗) ∩ H is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let R be cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume for a contradiction that (F, F ′) is a vertical cover of R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union ∪PH(x, y) where x and y range over all distinct rank-one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that P is contained in R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10 says that P is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (F ∩ P, F ′ ∩ P) is not a vertical cover of P, so we can assume without loss of generality that P is contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6, we see that cl(F ∪ C∗) ∩ H is equal to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus cl(C∗) ∩ H = R ∩ H is contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This contradicts the fact that (F, F ′) is a vertical cover of R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Supersolvability and saturation The following definition was introduced by Stanley [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The rank-r matroid M is supersolvable if it has a chain of modular flats F0 ⊆ F1 ⊆ · · · ⊆ Fr, where r(Fi) = i for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can give an equivalent, recursive, definition: if r(M) > 0 then M is supersolvable if it contains a modular hyperplane H such that M|H is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that every rank-zero matroid is trivially supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid is saturated if every round flat is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a flat of the saturated matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M|F is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let R be a round flat of M|F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R is a round flat of M so it is modular in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now [10, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5] implies that R is a modular flat of M|F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ SUPERSOLVABLE AND SATURATED MATROIDS 11 If M is supersolvable and saturated and H is a modular hyperplane such that M|H is supersolvable, then it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 that M|H is supersolvable and saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hy- perplane of M and let C∗ be the complementary cocircuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C∗ is non- spanning, then (H, cl(C∗)) is a modular cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Certainly H is a proper flat of M, and C∗ is non-spanning by hy- pothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (H, cl(C∗)) is a vertical cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have assumed that H is a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14 says that cl(C∗) is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since M is saturated, it follows that cl(C∗) is modular, so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M is supersolvable if and only if each of its connected components is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly M is saturated if and only if each of its connected components is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This result will follow by an easy inductive argument if we can prove it in the case when M has exactly two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will assume that M = M1⊕M2, where M1 and M2 are non-empty connected matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For i = 1, 2, let ri be r(Mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that M1 and M2 are supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For i = 1, 2, let F i 0 ⊆ F i 1 ⊆ · · ⊆ F i ri be a chain of modular flats in Mi such that each F i j has rank j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Using [10, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10] we see that each F 1 j ∪ F 2 k is a modular flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now it is easy to confirm that the chain F 1 0 ⊆ F 1 1 ⊆ · · · ⊆ F 1 r1 ⊆ F 1 r1 ∪ F 2 1 ⊆ F 1 r1 ∪ F 2 2 · · · ⊆ F 1 r1 ∪ F 2 r2 certifies that M is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the other direction, assume that M is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume for a contradiction that either M1 or M2 is not supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will assume that amongst such counterexamples, M is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now M has a modular hyperplane H such that M|H is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The complement of H is a cocircuit, and is therefore contained in either M1 or M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Without loss of generality we assume that H contains E(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now M|H = (M1|H)⊕M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The minimality of M means that M1|H and M2 are both supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But [10, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10] implies that H ∩ E(M1) is a modular flat of M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is the complement of a cocircuit of M1, so H ∩ M1 is a modular hyperplane of M1, and restricting to this hyperplane produces a supersolvable matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that M1 too is supersolvable, so the proof of this direction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From [10, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10] we see that E(M1) and E(M2) are modular flats of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows from [10, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5] that a flat of Mi is modular in M if and only if it is modular in Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F is a round flat of M then F ⊆ E(M1) or F ⊆ E(M2) because otherwise (F ∩ E(M1), F ∩ E(M2)) is a vertical cover of M|F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In fact, the round flats of M are exactly the round flats of M1 along with the round flats of M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From these considerations we can easily see that M is saturated if and only if M1 and M2 are saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 12 MAYHEW AND PROBERT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Chordality for matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We shall start this section by justifying the Venn diagram in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall that if C is a matroid circuit, then a chord of C is an element x /∈ C such that A ∪ z and B ∪ z are both circuits for some partition of C into sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A matroid is C-chordal if every circuit with at least four elements has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As we discussed in the introduction, the matroid in Figure 1 is supersolv- able but not saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To see that it is not C-chordal, note that {3, 5, 6, 7} has no chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because the only round flats of U3,6 are the empty set, the singleton sets, and the entire ground set, we can easily confirm that every round flat is modular, so U3,6 is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It has no modular hyperplane, so it is not supersolvable, and no circuit has a chord so it is not C-chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall that W4 is the rank-three matroid with ground set {a, b, c, d, e, f} and non-spanning circuits {a, b, d}, {b, c, e}, and {a, c, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is easy to confirm that every circuit of size four has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However no line is modular, so W4 is not supersolvable, and it also follows that it is not saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will leave as an exercise the fact that the Fano matroid F7 is supersolv- able, saturated, and C-chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Cordovil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' note that M∗(K3,3) is not su- persolvable [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is an easy exercise to see that it is saturated and C-chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Finally, let M be the rank-three matroid with ground set {p, a, b, c, d, e, f, x} where the non-spanning circuits are {p, a, b, c}, {p, d, e, f}, {a, d, x}, {b, e, x}, and {c, f, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now {p, a, b, c} and {p, d, e, f} are both modular hyperplanes, and we can easily confirm that M is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' On the other hand, {a, d, x} is a round hyperplane that has empty intersection with the rank- two flat {b, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence {a, d, x} is not modular and therefore M is not satu- rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' On the other hand, a simple case-analysis shows that M is C-chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can finish the justification of Figure 2 by proving that every supersolv- able saturated matroid is C-chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In fact, we prove something slightly stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C be a circuit in the supersolvable saturated matroid M and assume that |C| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There exist distinct elements x, y ∈ C and an element z /∈ C such that {x, y, z} and (C − {x, y}) ∪ z are circuits of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a smallest possible counterexample to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If r(M) ≤ 2 then the result holds vacuously, so r(M) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of M such that M|H is supersolvable and saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C∗ be the complement of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Choose C to be an arbitrary circuit of M such that |C| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C is a circuit of M|H, then the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore C ∩C∗ is non- empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because H is a flat it follows that C ∩ C∗ contains distinct elements x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let L be cl({x, y}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that L contains an element in PH(x, y), so that L is a rank-two flat containing at least three rank-one flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now it is easy to confirm that L is a round flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since M is saturated, it follows that L is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that C contains exactly two elements of L because |C| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now r(cl(C − L) ∩ L) = r(C − L) + r(L) − r(C) = (|C| − 2) + 2 − (|C| − 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' SUPERSOLVABLE AND SATURATED MATROIDS 13 Therefore we choose an element z which is in cl(C−L)∩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that neither x nor y is in cl(C − L), or else C properly contains a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore z is in L − {x, y} and {x, y, z} is a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C′ ⊆ (C − L) ∪ z be a circuit that contains z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now (C′ ∪ {x, y}) − z contains a circuit, by circuit elimination with C′ and {x, y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But (C′ ∪ {x, y}) − z is a subset of C, so (C′ ∪ {x, y}) − z = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that C′ = (C − L) ∪ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (C − L) ∪ z and {x, y, z} are both circuits and M is not a counterexample after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ In the next results we justify using supersolvable saturated matroids as analogues for chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph, and let F be a flat of M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then F is round if and only if G[F] is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that G[F] is not a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let u and v be distinct vertices in G[F] that are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let U be the set of edges in F that are incident with u, and let U ′ be F − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If f ∈ F is an edge incident with u, there is no cycle contained in U ′ ∪ f that contains f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that cl(U ′) is a proper flat of M|F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The same argument shows that cl(U) is a proper flat of M|F, so (cl(U), cl(U ′)) is a vertical cover of M|F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus F is not round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the other direction, assume that G[F] is a clique, but that (U, U′) is a vertical cover of G[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We colour the edges of F red if they are in U, and blue if they are in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that an edge may be both red and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let v be an arbitrary vertex of G[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The set of edges incident with v spans F, since G[F] is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If all the edges of F incident with v are red, then U contains F, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By symmetry, we can now let e, f ∈ F be edges incident with v so that e is red but not blue, and f is blue but not red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let g be the edge of F so that {e, f, g} is the edge-set of a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If g is red, then f is also red, and if g is blue, then e is blue, and in either case we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M(G) is a saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7 we see that every round flat of M(G) is a clique of G, and any such flat is modular by [10, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ The next result is a consequence of [11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then G is chordal if and only if M(G) is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The next result implies the known fact [6, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='16] that in a chordal graph the number of maximal cliques does not exceed the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M has at most r(M) rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 14 MAYHEW AND PROBERT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of M such that M|H is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Any rotunda of M that is contained in H is a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But M|H has at most r(M)−1 rotunda by induction, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14 says there is exactly one rotunda of M that is not a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Reduced clique graphs and rotunda graphs Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The clique graph of G, denoted C(G), has the maximal cliques of G as its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Two distinct maximal cliques are adjacent in C(G) if and only if they have at least one vertex in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Our focus will be the reduced clique graph, CR(G), which was introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The vertices of CR(G) are again the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C1 and C2 be distinct maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We say that C1 and C2 are a separating pair if there is at least one vertex in C1 ∩ C2 and any path from a vertex of C1 − C2 to a vertex of C2 − C1 uses a vertex in C1 ∩ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now CR(G) is the subgraph of C(G) where two maximal cliques are adjacent if and only if they form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We now define a matroid analogue of this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall that R(M) is the family of rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The rotunda graph R(M) is the graph with R(M) as its vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The rotunda R1 and R2 are adjacent in R(M) if R1 ∩ R2 ̸= ∅ and there is a modular cover (F1, F2) such that Ri ⊆ Fi for i = 1, 2, and F1 ∩ F2 = R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case we say that the modular cover (F1, F2) certifies the adjacency of R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The next result allows us to prove statements about rotunda graphs in- ductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid and let H be a modular hyperplane of M such that M|H is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C∗ be the complement of H and let R be cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R is a rotunda of M and either: (a) R ∩ H is a rotunda of M|H and R(M) = (R(M|H) − {R ∩ H}) ∪ {R}, or (b) R ∩ H is properly contained in a rotunda of M|H and R(M) = R(M|H) ∪ {R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If case (a) holds then R(M|H) is obtained from R(M) by relabelling R as R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If case (b) holds then R(M|H) is obtained from R(M) by deleting R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that M|H is saturated as well as supersolvable (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14 says that R is a rotunda of M, and that moreover it is the unique rotunda of M that is not contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now it is an easy SUPERSOLVABLE AND SATURATED MATROIDS 15 exercise to prove that every other rotunda of M is a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows R(M) ⊆ R(M|H) ∪ {R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='15 says that R ∩ H is a round flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First assume that R ∩ H is a maximal round flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R ∩ H is a rotunda of M|H but not of M, since R ∩ H is properly contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So in this case R(M) is contained in (R(M|H) − {R ∩ H}) ∪ {R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now let Z be a rotunda of M|H that is not equal to R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will prove that Z is a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because Z is a round flat of M|H, and hence of M, it is properly contained in a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let this rotunda be Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now Z′ is not contained in H, because in this case Z and Z′ would both be rotunda of M|H, and then Z cannot be properly contained in Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So Z′ is a rotunda of M that is not contained in H, and hence Z′ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z is contained in R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because Z is not properly contained in a round flat of M|H we deduce that Z = R ∩ H, contrary to hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z is a rotunda of M and we have shown that when R ∩ H is a rotunda of M|H, the set R(M) is equal to (R(M|H) − {R ∩ H}) ∪ {R} and case (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next we assume that R ∩ H is not a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have already shown that R(M) is contained in R(M|H) ∪ {R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z be a rotunda of M|H and assume that Z is not a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then Z is properly contained in Z′, a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As in the previous paragraph, Z′ = R, so Z is contained in R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Again we deduce that Z = R ∩ H, and we have a contradiction to Z being a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So in the case R(M) is equal to R(M|H) ∪ {R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore, R ∩ H is a round flat of M|H but not a rotunda, so it must be properly contained in a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus case (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume case (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We let Z1 and Z2 be distinct rotunda of M, where Z1 is not equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z1 is a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Either Z2 is equal to R or it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the former case Z2 ∩ H = R ∩ H and in the latter Z2 ∩ H = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In either case Z2 ∩ H is a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will prove that Z1 and Z2 ∩ H are adjacent in R(M|H) if and only if Z1 and Z2 are adjacent in R(M), and this will show that R(M|H) is obtained from R(M) by relabelling R as R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that (F1, F2) is a modular cover of M that certifies the adjacency of Z1 and Z2 in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus F1 and F2 are proper modular flats of M and F1 ∪ F2 = E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover F1 ∩ F2 = Z1 ∩ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that either F1 or F2 contains H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9 implies that neither F1 nor F2 contains Z1 ∪ Z2, we deduce that Z2 = R and F1 = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now R ∩ H ⊆ F1 ∩ F2 = Z1 ∩ Z2 so Z1 contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since Z1 and R ∩ H are both rotunda of M|H, we see that Z1 = R ∩ H, and in this case Z1 is properly contained in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is impossible, so F1 ∩ H or F2 ∩ H are proper flats of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, their union is equal to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since F1 and F2 are modular flats of M it follows that F1 ∩H and F2 ∩H are modular flats of M (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4), and hence modular flats of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 16 MAYHEW AND PROBERT Furthermore, (F1 ∩ H) ∩ (F2 ∩ H) = (F1 ∩ F2) ∩ H = (Z1 ∩ Z2) ∩ H = Z1 ∩ (Z2 ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we see that (F1 ∩ H, F2 ∩ H) is a modular cover of M|H, and that it certifies the adjacency of Z1 and Z2 ∩ H in R(M|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the other direction, assume Z1 and Z2 ∩ H are adjacent in R(M|H), and let (F1, F2) be a modular cover of M|H that certifies their adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union ∪PH(x, y), where x and y range over distinct rank-one flats in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11 and see that P is contained in either F1 or F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that Z2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then P is contained in Z2 ∩ H ⊆ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11 implies that (F1, F2 ∪ C∗) is a modular cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, F1 ∩ (F2 ∪ C∗) = F1 ∩ F2 = Z1 ∩ (Z2 ∩ H) = Z1 ∩ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (F1, F2 ∪ C∗) certifies the adjacency of Z1 and Z2 in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next we assume that Z2 ̸= R, so that Z1 and Z2 are both rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We again apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11 and see that (Fi ∪ C∗, F3−i) is a modular cover of M for some i ∈ {1, 2}, and as before we can see that (Fi ∪ C∗, F3−i) certifies the adjacency of Z1 and Z2 in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus we are now finished with case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume case (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z1 and Z2 be two rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can use exactly the same arguments as in the previous paragraphs to show that Z1 and Z2 are adjacent in R(M|H) if and only if they are adjacent in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus R(M|H) is obtained from R(M) by deleting the rotunda R and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Rotunda graphs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this section we compare rotunda graphs and reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Ultimately we will show that they are identical classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We also consider the connection between the reduced clique graph of G and the rotunda graph of M(G) when G is a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then the maximal cliques of G are the rotunda of M(G), and every edge in R(M(G)) is an edge in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The first statement follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M stand for M(G), so that we identify the vertices of CR(G) and the vertices of R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let R1 and R2 be rotunda that are adjacent in R(M), and let C1 and C2 be the corresponding maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will show that C1 and C2 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (F1, F2) be a modular cover of M certifying the adjacency of R1 and R2, so that Ri ⊆ Fi for i = 1, 2, and F1 ∩F2 = R1 ∩R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because R1 and R2 are adjacent in R(M), they have a non-empty inter- section, which means that C1 and C2 share at least two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let S be the set of vertices in both C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus |S| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C1 and C2 form a separating pair, then there is nothing left for us to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will let P be an S-avoiding path from a1 ∈ C1 − C2 to a2 ∈ C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' SUPERSOLVABLE AND SATURATED MATROIDS 17 Let u be an arbitrary vertex in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that every edge of C1 incident with u is in F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R1 is contained in F2, which contradicts Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we let e1 be an edge of C1 that is incident with u and not in F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By the same reasoning, we can let e2 be an edge of C2 that is incident with u and not in F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that ei joins u to bi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that b1 is in C1 − C2, or else e1 would be in R2 ⊆ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly b2 is in C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We obtain the cycle D from P by appending the edges e1 and e2 as well as a1b1 and a2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (This assumes that a1 ̸= b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' if a1 = b1 then we do not append a1b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The same comment applies if a2 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=') Note that D is not contained in F2, as e1 is not in F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because F2 is a modular flat we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 and deduce that there is an element x ∈ F2 ∩ cl(D − F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus D′ ∪ x is a cycle of G for some subset D′ ⊆ D − F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because (D − F2) ∪ x is a circuit of M it follows that x is in F1 as well as F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore x is in R1 ∩ R2, so x joins two vertices of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let v be a vertex incident with x such that v is not u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus v is in the cycle D, so v is either an internal vertex of P, or is equal to one of a1, b1, a2, or b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But none of the internal vertices of P is in S, and a1, b1 are in C1 −C2 while a2, b2 are in C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we have a contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ From the previous result we know that R(M(G)) is a subgraph of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To see that R(M(G)) and CR(G) need not be equal, we let G be the path with two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus G is a tree and is therefore chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There are two maximal cliques in G, and M(G) has two rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However CR(G) consists of two vertices joined by an edge, whereas R(M(G)) consists of two isolated vertices, since the two rotunda of M(G) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The next result shows that sufficient connectivity prevents this situation from happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a chordal graph that is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then CR(G) = R(M(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We identify the vertices of CR(G) and R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By virtue of Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3, it suffices to show that every edge of CR(G) is also an edge of R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To this end let C1 and C2 be maximal cliques of G that are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Ri be the edge set of Ci for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R1 and R2 are rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will show they are adjacent in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Set S to be the set of vertices in both C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since C1 and C2 are adjacent in CR(G) it follows that S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For each i = 1, 2, let ai be a vertex in Ci − C3−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' R1 ∩ R2 ̸= ∅ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This claim holds if |S| ≥ 2, because then any edge joining two vertices of S is in R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So assume that |S| = 1 and let v be the unique vertex of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now C1 and C2 form a separating pair, so a1 and a2 are in different connected components of G − S = G − v, but this contradicts the fact that G is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 18 MAYHEW AND PROBERT Now we know that R1 and R2 are not disjoint we can complete the proof by constructing a modular cover to certify their adjacency in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let U1 be the set of edges that are contained in S-avoiding paths having a1 as a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Observe that every edge incident with a1 is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let U2 be the set of edges of G not in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (U1, U2) is a partition of the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (U1, U2) is a vertical separation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We must prove that neither U1 nor U2 is spanning in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be any edge incident with a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We claim that e is not in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume otherwise, and let P be an S-avoiding path with a1 as a terminal vertex, where P contains e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since e is incident with a2, we can let P ′ be a subpath of P from a1 to a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As C1 and C2 form a separating pair, it follows that P ′ contains a vertex of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But the end vertices of P ′ are a1 and a2, and neither is in S, so P ′ has an internal vertex in C1 ∩ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus P does as well, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore e is not in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that U1 is spanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be an edge incident with a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then e is in U2 by the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since it is in the closure of U1, we can let D be a cycle containing e such that every other edge of D is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In particular, this means that a2 is incident with an edge of U1, contrary to the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So U1 is not spanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly, if U2 is spanning, then we let e be an edge incident with a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then e is not in U2, so we can let D be a cycle that contains e, where all the other edges of D are in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This implies that an edge incident with a1 is in U2, which contradicts an earlier conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore (U1, U2) is a vertical separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ For i = 1, 2, we let Fi be cl(Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall that Ri is the edge-set of Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' F1 ∩ F2 = R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be an edge that joins vertices u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First assume that e is in R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then u and v are in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means there is no S-avoiding path containing e with a1 as a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence e is not in U1 so it is in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However, a1 is adjacent to u and v, and the edges a1u and a1v are in U1, so e is in cl(U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus e is in F1 ∩ F2 and we have shown that R1 ∩ R2 ⊆ F1 ∩ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the other direction, assume that e is in F1 ∩ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First assume that e is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be an S-avoiding path with a1 as a terminal vertex such that e is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can assume that either u or v is a terminal vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since e is in U1 ∩ cl(U2) we can let D be a cycle such that e is in D, and every other edge of D is in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus both u and v are incident with edges in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e′ be an edge incident with u that is in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume for a contradiction that u is not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If u is a terminal vertex of P then we obtain a new path by adding e′ to the end of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' No internal vertex of this new path is in S, so it implies that e′ is in U1, a contradiction to e′ being in U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore u is not a terminal vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since P contains e, it follows SUPERSOLVABLE AND SATURATED MATROIDS 19 that u is an internal vertex of P, so v is a terminal vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case we can obtain a new path from P by replacing the edge e with e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Again we see that e′ is in U1 and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore u is in S, and by symmetry, so is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence e joins two vertices of S, and is thus in R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We must also consider the case that e is in U2 ∩ cl(U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let D be a cycle that contains e, where every other edge of D is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let x be an edge of D − e that is incident with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus x is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be an S-avoiding path containing x and a1 as a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If u is not in S, then we can either extend P by adding the edge e, or replacing x in P with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In either case, the new path shows that e is in U1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore u, and by symmetry v, is in S, so we again see that e is in R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence F1 ∩ F2 ⊆ R1 ∩ R2 and the claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Recall that a1 is in the clique C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Every edge incident with a1 is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As every edge of C1 − a1 is spanned by two such edges, it follows that R1 is contained in F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We must also show that R2 is spanned by U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be an edge in R2 and assume that it is not in F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In particular, this means that e is not in U2, so it is in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be an S-avoiding path containing e that has a1 as a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let u be the first internal vertex of P that is incident with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then u is not in S, as P is S-avoiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But u is in C2, since e is in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus u is in C2 − C1, and the subpath of P from a1 to u is an S-avoiding path from a vertex of C1 − C2 to a vertex of C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is a contradiction, as C1 and C2 form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that R2 is contained in F2, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we can complete the proof that R1 and R2 are adjacent in R(M) by showing that (F1, F2) is a modular cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that F1 is not a modular flat, so that by utilising Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 we can let F be an arbitrary flat of M that is disjoint from F1 such that some circuit C ⊆ F ∪ F1 contains elements of both F and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If each connected component of G[F] shares at most one vertex with G[F1], then no such cycle can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we let u and v be distinct vertices from the same connected component of G[F] so that both of u and v are incident with edges in F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since u is incident with an edge in F1, it is incident with an edge in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be such an edge, and let P be a shortest-possible S-avoiding path that contains e and has a1 as a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let f be an edge of F that is incident with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If u is not in S, then extending P by adding f shows that f is in U1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore u, and by symmetry v, is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means that there is an edge g of R1 that joins u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus g is in R1 ⊆ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But there is a path of G[F] that joins u to v, so g is in cl(F) = F, and we have contradicted F ∩F1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore F1 is a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Almost exactly the same argument shows that F2 is a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ The previous result shows that when G is a 2-connected chordal graph, CR(G) is isomorphic to the rotunda graph of a supersolvable saturated ma- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In fact, this is true even when G is not 2-connected, as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 20 MAYHEW AND PROBERT First we make a simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 that a matroid is supersolvable and saturated if and only if all its components are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a chordal graph with connected components H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then CR(G) is the disjoint union of CR(H1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , CR(Hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Sim- ilarly, if M is a supersolvable saturated matroid with connected components N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Nk, then R(M) is the disjoint union of R(N1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , R(Nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Maximal cliques in different components of G cannot be adjacent in CR(G) because they have no vertices in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly, rotunda from different components of M are not adjacent in R(M) because they have empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There is a supersolvable saturated matroid M such that CR(G) is isomorphic to R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Hk be the connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If each CR(Hi) is isomorphic to R(Ni) for some supersolvable saturated Ni, then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 implies that CR(G) is isomorphic to R(N1 ⊕ · · · ⊕ Nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In other words, it suffices to prove the lemma when G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case, we will prove that CR(G) is isomorphic to CR(G′), where G′ is a 2-connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 shows that CR(G′) = R(M(G′)), and M(G′) is supersolvable and saturated by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9, so the result will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If G is 2-connected, then there is nothing left to prove, so let v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , vm be the cut-vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We produce G′ by introducing new vertices v′ 1, v′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , v′ m and for each i making v′ i adjacent to vi and all of the neigh- bours of vi in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' G′ is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Certainly G′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that v is a cut-vertex of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that for each i, the graph produced from G′ by deleting v′ i is obtained from G by adding m−1 new vertices and making each of them adjacent to at least one vertex in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since G is connected it follows that G′ − v′ i is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus no vertex v′ i is a cut-vertex of G′ so v is not equal to v′ i for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now v is a vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If v /∈ {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , vm} then G−v is connected and G′ −v is obtained from the connected graph G − v by adding m new vertices and making each of them adjacent to at least one vertex in G−v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus G′ −v is connected, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore v = vi for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But in G′ the vertices vi and v′ i are adjacent to exactly the same vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore G′ − vi is obtained from G′ − v′ i by relabelling v′ i as vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means that G′ − vi is connected, and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' G′ is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We rely on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , un be a perfect elimination order of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We produce an ordering of the vertices of G′ by inserting each v′ i into the order u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , un immediately after vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is easy to verify that this produces a perfect elimination order for G′ and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ SUPERSOLVABLE AND SATURATED MATROIDS 21 We can complete the proof by showing that CR(G) is isomorphic to CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is clear that any maximal clique of G′ contains one of the ver- tices {vi, v′ i} if and only if it contains both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we can easily verify that there is a bijective correspondence between the maximal cliques of G and the maximal cliques of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C is a maximal clique of G, then we obtain the corresponding maximal clique of G′ by adding each vertex v′ i such that vi is in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C1 and C2 be distinct maximal cliques of G, and let C′ 1 and C′ 2 be the corresponding maximal cliques of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will prove that C1 and C2 are adjacent in CR(G) if and only if C′ 1 and C′ 2 are adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First note that C1 ∩ C2 is non-empty if and only if C′ 1 ∩ C′ 2 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C1 and C2 are not adjacent in CR(G), then either C1 ∩ C2 = ∅, or P is a (C1 ∩ C2)-avoiding path of G from a vertex of C1 − C2 to a vertex in C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the first case C′ 1 ∩ C′ 2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the second case, it is obvious that P is a (C′ 1 ∩ C′ 2)-avoiding path of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In either case C′ 1 and C′ 2 are not adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next assume that C′ 1 and C′ 2 are not adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C′ 1 ∩ C′ 2 = ∅ then C1 ∩C2 = ∅ so we have nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will assume that P is a (C′ 1∩C′ 2)-avoiding path in G′, and that P joins a vertex in C′ 1−C′ 2 to a vertex in C′ 2 − C′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If a vertex v′ i appears anywhere in P, then we may replace it with vi, since these two vertices have the same neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that the resulting path is still (C′ 1∩C′ 2)-avoiding, and still joins a vertex of C′ 1 − C′ 2 to a vertex of C′ 2 − C′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus we can assume that P is a path of G, and is consequently a (C1 ∩ C2)-avoiding path of G from a vertex of C1 −C2 to a vertex of C2 −C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that C1 and C2 are not adjacent in CR(G) so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ We have established that every reduced clique graph is isomorphic to a rotunda graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next we start moving towards proving the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a connected supersolvable and saturated matroid, and let G be a 2-connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that θ is a function from E(M) to the powerset of V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If U is a subset of vertices in G, then let θ−1(U) be {x ∈ E(M): θ(x) ⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For any subset R ⊆ E(M), let θ(R) stand for ∪x∈Rθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus we can think of θ as being a function from P(E(M)) to P(V (G)) such that R ⊆ R′ if and only if θ(R) ⊆ θ(R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that the following properties hold: (i) |θ(x)| = 2 for every x ∈ E(M), and (ii) for any vertex v ∈ V (G) there exists exactly one element x ∈ E(M) such that v is in θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' (iii) if R is a non-empty round flat of M, then θ(R) is a clique, (iv) if F is a modular flat of M and U is a union of connected components of G − θ(F), then F ∪ θ−1(U) is a modular flat of M, and (v) the restriction of θ to R(M) is a bijection from R(M) to the maximal cliques of CR(G) and this bijection is an isomorphism between R(M) and CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 22 MAYHEW AND PROBERT If all these conditions hold, then we will say that (G, θ) is compliant with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a connected supersolvable and saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There exists a 2-connected chordal graph G and a function θ: E(M) → P(V (G)) such that (G, θ) is compliant with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The proof is a straightforward induction, although the technical de- tails are require some work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If M has rank at most one, then we can simply make G a clique of the appropriate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we are going to choose C∗ to be the complement of a modular hyperplane, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then inductively M|H has a compliant graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The intersection of H with cl(C∗) is a round flat, and therefore corresponds to a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We create a new maximal clique by adding new vertices and making them adjacent to each other and to the clique corresponding to H ∩ cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The rest of the proof involves nothing more than checking that this construction does indeed satisfy the conditions for compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To implement this strategy, we let M be a supersolvable saturated ma- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume r(M) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can easily see that the only rotunda of M is E(M) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We let G be isomorphic to K2|E(M)|, and we consider an arbitrary partition of V (G) into blocks of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We then set θ to be an arbitrary bijection from E(M) to the blocks of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is not hard to verify that (G, θ) is compliant with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we assume that r(M) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a modular hyperplane of M such that M|H is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then M|H is also saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7 says that M|H is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we can apply the obvious inductive hypothesis and let G′ be a 2-connected chordal graph with a function θ′ : H → P(V (G′)) such that (G′, θ′) is compliant with M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let C∗ be the complementary cocircuit of H, and let R be the closure of C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14 says that R is a rotunda, and furthermore it is the only rotunda of M that is not contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Certainly C∗ is non-empty, and r(H) = r(M) − 1 > 0, so H is non-empty also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But (H, C∗) is not a separation of M, since M is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As H is modular, we deduce that r(R ∩ H) = r(cl(C∗) ∩ H) = r(C∗) + r(H) − r(M) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore R ∩ H is non-empty and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='15 tells us that R ∩ H is round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let W be θ′(R ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since (G′, θ′) is compliant with M|H, we see that W is the set of vertices of a clique in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note also that |W| = 2|R∩H| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We produce G from G′ by adding Y , a set of 2|C∗| new vertices, and making each of them adjacent to all the vertices of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that W ∪Y is a maximal clique of G and G′ = G − Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because W has at least two vertices it is easy to see that G is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The neighbours of any vertex in Y form a clique in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we can construct a perfect elimination order for G by SUPERSOLVABLE AND SATURATED MATROIDS 23 prepending the vertices of Y to a perfect elimination order for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that G is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Consider an arbitrary partition of Y into pairs of vertices, and let φ be an arbitrary bijection from C∗ to the blocks of this partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then we define θ to be the union of θ′ and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that |θ(x)| = 2 for any x ∈ E(M), and for any vertex v of G, there is exactly one element x ∈ E(M) such that v is in θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore the remainder of the proof consists in showing that θ satisfies conditions (iii), (iv), and (v) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='13 tells us that if Z is a round flat of M then either Z ⊆ H or Z ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the former case, Z is a round flat of M|H, and θ(Z) = θ′(Z) is a clique of G, since θ′ satisfies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the latter case θ(Z) is a subset of W ∪ Y , and again θ(Z) is a clique of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So condition (iii) holds for (G, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Condition (iv) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='7 holds for (G, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be a modular flat of M, and let U be a union of connected components of G − θ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let D be F ∪ θ−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus our aim is to show that D is a flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that U is the empty union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case D = F ∪ θ−1(U) = F and since F is a modular flat there is nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we assume that U contains at least one connected component of G − θ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that D is disjoint with C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means that θ(F) ∪ U is disjoint with Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus F is a modular flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If U is not a union of connected components in G′ −θ′(F) then there is a connected component of this graph that contains vertices u ∈ U and v /∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There is a path of G′ − θ′(F) = G − (θ(F) ∪ Y ) from u to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence u and v are in the same component of G − θ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This contradicts the fact that U is a union of components in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence U is a union of components of G′ − θ′(F), so we can apply the inductive assumption and see that D = F ∪(θ′)−1(U) = F ∪θ−1(U) is a modular flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore D is a modular flat of M and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence we assume that D contains at least one element of C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now θ(F) ∪ U contains at least two vertices from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since any such vertex is adjacent to every vertex in W ∪ Y , and U is a non-empty union of connected components, it now follows that θ(F) ∪ U contains W ∪ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that D contains θ−1(W ∪ Y ) = R = cl(C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that U − Y is not a union of connected components in G′ − θ(F ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then there is a connected component of G′ − θ(F ∩ H) that contains vertices u ∈ U − Y and v /∈ U − Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' There is a path from u to v in G′ − θ(F ∩ H) = G − (θ(F) ∪ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus u and v are in the same connected component of G − θ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means that u and v are both in U, since U is a union of connected components in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since v is not in U − Y this means that v is in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But this is impossible, since v is a vertex of G′, which is equal to G − Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that U − Y is a union of connected components in G′ − θ(F ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 24 MAYHEW AND PROBERT We note that F ∩ H is a modular flat of M|H since both F and H are modular in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The inductive hypothesis now tells us that (F ∩ H) ∪ θ−1(U − Y ) is a modular flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let this flat be D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that because C∗ ⊆ D we have D = F ∪ θ−1(U) = (F ∩ H) ∪ θ−1(U − Y ) ∪ C∗ = D′ ∪ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be the union ∪PH(x, y), where x and y range over all distinct rank- one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus P is a subset of R ∩H ⊆ D ∩H = D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that cl(D) = (cl(D) ∩ H) ∪ C∗ = (cl(D′ ∪ C∗) ∩ H) ∪ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since P ⊆ D′ ⊆ H we see that (cl(D′ ∪ C∗) ∩ H) ∪ C∗ = cl(D′) ∪ C∗ = D′ ∪ C∗ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus D is a flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that D is not a modular flat of M, and let F ′ be a flat of M that is disjoint with D, chosen so that C ⊆ D ∪ F ′ is a circuit that contains elements of both D and F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Choose C so that |C∩C∗| is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Exactly as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11 we can prove that C∩C∗ contains distinct elements x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We choose p to be an element in PH(x, y), and we perform strong circuit elimination on C and {x, y, p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this way we find a circuit contained in D ∪ F ′ that contains elements of both sets, and contains fewer elements of C∗ than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This contradiction shows that D is a modular flat of M so condition (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The restriction of θ to R(M) is a bijection between R(M) and the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The inductive hypothesis means that θ′ induces a bijection between the rotunda of M|H and the maximal cliques of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First assume that R ∩ H is a rotunda of M|H, so that W = θ′(R ∩ H) is a maximal clique of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2 shows that the rotunda of M are the rotunda of M|H, except that R ∩ H has been replaced by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is easy to see that he maximal cliques of G are the maximal cliques of G′, except that W has been replaced by W ∪ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We observe that θ(R) = W ∪ Y and now it follows that θ|R(M) is a bijection between the rotunda of M and the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next we assume that R ∩ H is not a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2 implies that every rotunda of M|H is also a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore R is the only rotunda of M that is not a rotunda of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because R ∩ H is not a rotunda of M|H, we can let Z be a rotunda of M|H that properly contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now W = θ′(R ∩ H) is properly contained in θ′(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since Z is round, we see that θ(Z) = θ′(Z) is a clique that properly contains W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore W is not a maximal clique of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now it is easy to see that every maximal clique of G′ is a maximal clique of G, and that W ∪ Y is SUPERSOLVABLE AND SATURATED MATROIDS 25 the only maximal clique of G that is not a maximal clique of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ We can complete the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 by proving that the restriction of θ to R(M) is an isomorphism from R(M) to CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z and Z′ be distinct rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will show that they are adjacent in R(M) if and only if θ(Z) and θ(Z′) are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Neither Z nor Z′ is equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case both Z and Z′ are rotunda of M|H, and θ(Z) and θ(Z′) are maximal cliques of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that θ(Z) and θ(Z′) are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then these maximal cliques have at least one vertex in common, and there is no (θ(Z) ∩ θ(Z′))-avoiding path in G from a vertex of θ(Z)−θ(Z′) to a vertex of θ(Z′)−θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Exactly the same statements apply to θ′(Z) and θ′(Z′) in G′, so θ′(Z) and θ′(Z′) are adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The inductive assumption implies that Z and Z′ are adjacent in R(M|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2 now implies that they are also adjacent in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the converse, assume that Z and Z′ are adjacent in R(M), and let (F, F ′) be a modular cover of M that certifies the adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We assume that Z ⊆ F and Z′ ⊆ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let us assume that both F ∩ C∗ and F ′ ∩ C∗ are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' No element of F ∩ C∗ is in F ′, because any such element would be in F ∩ F ′ = Z ∩ Z′, and this is not possible since Z and Z′ are subsets of H = E(M) − C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Symmetrically, no element of F ′ ∩ C∗ is in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So F ∩ R does not contain any element of F ′ ∩ C∗ and F ′ ∩ R does not contain any element of F ∩ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that (F ∩ R, F ′ ∩ R) is a vertical cover of R, which is impossible as R is a round flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore either F ∩ C∗ = ∅ or F ′ ∩C∗ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We assume the latter, so C∗ is a subset of F and F ′ is a subset of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9 says that F ′ does not contain Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It therefore does not contain H, so we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='12 and deduce that (F ∩ H, F ′ ∩ H) = (F ∩ H, F ′) is a modular cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that (F ∩ H) ∩ F ′ = F ∩ F ′ = Z ∩ Z′ so Z and Z′ are adjacent in R(M|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By the inductive hypothesis, θ′(Z) = θ(Z) and θ′(Z′) = θ(Z′) are adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since Z and Z′ are rotunda of M, neither is equal to R∩H, which is properly contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore neither θ(Z) nor θ(Z′) is equal to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because θ(Z) and θ(Z′) are adjacent in CR(G′) they have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that θ(Z) and θ(Z′) are not adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be a (θ(Z)∩θ(Z′))-avoiding path of G from a vertex a ∈ θ(Z)−θ(Z′) to a vertex b ∈ θ(Z′) − θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because no such path can exist in G′ = G − Y , it follows that P contains a vertex in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let y and y′, respectively, be the first and last vertices of P that are in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that y and y′ are not equal to a or b, which are vertices of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let w be the neighbour of y in the subpath of P from y to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Similarly let w′ be the neighbour of y′ in the subpath from y′ 26 MAYHEW AND PROBERT to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because w and w′ are adjacent to vertices in Y , but are not in Y , they must be in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus w and w′ are adjacent, so there is a path of G′ from a to b that avoids any vertex in θ(Z) ∩ θ(Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is a contradiction, so we conclude that θ(Z) and θ(Z′) are adjacent in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have now completed the case that neither Z nor Z′ is equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' One of Z and Z′ is equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We let Z be a rotunda of M that is distinct from R, and we will prove that Z and R are adjacent in R(M) if and only if θ(Z) and θ(R) = W ∪ Y are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Observe that Z is contained in H by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' First assume that θ(Z) and θ(R) are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because θ sends distinct elements of E(M) to distinct pairs of vertices, it cannot be the case that Z ∩ R = ∅, or else θ(Z) and θ(R) would have no vertices in common, contradicting their adjacency in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z and R are non-disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume Z contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If C∗ is spanning in M, then R ∩ H = H, so θ(H) = W is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case G = W ∪ Y is a clique, but we have assumed that M has at least two distinct rotunda, so G has at least two distinct maximal cliques by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus C∗ is not spanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 says that (H, R) is a modular cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now R ∩ H = R ∩ Z so (H, R) certifies that Z and R are adjacent in R(M) and we have nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will assume that Z does not contain R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence Z ∩ R is a proper and non-empty subset of R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows that θ(Z) contains some, but not all, of the vertices of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='15 we know that R ∩ H is a round flat of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Z0 be a rotunda of M|H that contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z0 is not equal to Z, but it may be equal to R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now θ′(Z0) = θ(Z0) is a maximal clique of G′ that contains W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that θ(Z) and θ(Z0) are not adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because these cliques have at least one vertex of W in common, we can let P be a (θ(Z) ∩ θ(Z0))-avoiding path of G′ from a vertex a ∈ θ(Z) − θ(Z0) to a vertex b ∈ θ(Z0) − θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that P contains no vertex of θ(Z) ∩ θ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But P is also a path of G, and b is adjacent to any vertex of W −θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus, if necessary, we can adjoin an edge to P from b to a vertex of W − θ(Z), and certify that θ(Z) and θ(R) are not adjacent in CR(G), contrary to hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore θ(Z) and θ(Z0) are adjacent in CR(G′), so by induction Z and Z0 are adjacent in R(M|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because Z0 contains R ∩ H, the intersection of Z and Z0 contains Z ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume this containment is proper, and let e be an element of Z ∩ Z0 that is not in Z ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let v be a vertex in θ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus v is in θ(Z) − θ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Choose w, an arbitrary vertex in W − θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because v is in θ(Z0), which contains W, it follows that v and w are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since w is in θ(R) − θ(Z), we now see that θ(Z) and θ(R) are not adjacent in CR(G), contrary to hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We conclude that Z ∩ Z0 = Z ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since Z and Z0 are adjacent in R(M|H), we can let (F, F ′) be a modular cover of M|H that certifies this adjacency, where Z ⊆ F and Z0 ⊆ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because Z0 contains R ∩ H, it follows that F ′ contains ∪PH(x, y), where x and y range over distinct rank-one flats contained in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition SUPERSOLVABLE AND SATURATED MATROIDS 27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='11 says that (F, F ′ ∪ C∗) is a modular cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Certainly Z ⊆ F and Z0 ⊆ F ′ ∪ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Furthermore, F ∩ (F ′ ∪ C∗) = F ∩ F ′ = Z ∩ Z0 = Z ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (F, F ′ ∪ C∗) certifies that Z and R are adjacent in R(M), exactly as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the converse, we assume that Z and R are adjacent in R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z ∩ R is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that Z contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then θ(Z) contains θ(R ∩ H) = W, so θ(Z) ∩ θ(R) = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In G − W there is no path from a vertex of θ(R)−θ(Z) = Y to a vertex not in Y , and in particular there is no path to a vertex in θ(Z) − θ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So in this case θ(Z) and θ(R) are adjacent in CR(G) and we have nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore we will assume that Z does not contain R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence Z ∩ R is a non-empty proper subset of R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since R ∩ H is a round flat of M|H by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='15, we can let Z0 be a rotunda of M|H that contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus Z0 may be equal to R ∩ H, but it is not equal to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (F, F ′) be a modular cover of M that certifies the adjacency of R and Z in R(M), where R ⊆ F and Z ⊆ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because F ∩ F ′ = R ∩ Z and Z is contained in H it follows that F ′ is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If F ′ = H, then F ∩ F ′ contains R ∩ H, which properly contains Z ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This contradicts F ∩ F ′ = R ∩ Z, so F ′ does not contain H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='12, we see that (F ∩ H, F ′ ∩ H) = (F ∩ H, F ′) is a modular cover of M|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because Z0 is round, one of F ∩ Z0 and F ′ ∩ Z0 is not a proper flat of M|Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' That is, Z0 is contained in either F or F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume Z0 is contained in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R ∩ H ⊆ Z0 ⊆ F ′ and R ⊆ F so F ∩ F ′ contains R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This is a contradiction as F ∩ F ′ = R ∩ Z, which is a non-empty proper subset of R ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore Z0 is contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We observe that (F ∩ H) ∩ F ′ = (F ∩ F ′) ∩ H = (R ∩ Z) ∩ H = R ∩ Z = F ∩ F ′ ⊇ Z0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that F ∩ F ′ properly contains Z0 ∩ Z and let e be an element of (F ∩ F ′) − (Z0 ∩ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since F ∩ F ′ = R ∩ Z it follows that e is in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But we also have e ∈ F ∩ F ′ = R ∩ Z ⊂ R ∩ H ⊆ Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus e is in Z0∩Z after all and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (F ∩H)∩F ′ = Z0 ∩ Z = R ∩ Z and the modular cover (F ∩ H, F ′) of M|H certifies that Z0 and Z are adjacent in R(M|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Induction now tells us that θ(Z0) and θ(Z) are adjacent in CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that θ(Z) and θ(R) = W ∪ Y are not adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' These cliques certainly have common vertices, so we can let P be a path from a ∈ θ(Z)−θ(R) to b ∈ θ(R)−θ(Z) such that P contains no vertex of θ(Z)∩ θ(R) = θ(Z)∩θ(Z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If P is a path of G′ then it certifies that θ(Z) and θ(Z0) are not adjacent in R(M|H), contrary to our earlier conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore P contains at least one vertex in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Consider the maximal subpath of P from a to vertex not in Y , and let this vertex be w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that w is in W − θ(Z) ⊆ θ(Z0) − θ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So this subpath certifies that θ(Z) and θ(Z0) are 28 MAYHEW AND PROBERT not adjacent in R(M|H), and we have another contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6 shows that every reduced clique graph is isomorphic to a rotunda graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' On the other hand, if M is a supersolv- able saturated matroid with connected components M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Mn, then R(M) is the disjoint union of R(M1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , R(Mn), as we observed in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 shows that each R(Mi) is isomorphic to CR(Gi) for some 2-connected chordal graph Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If G is the disjoint union of G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Gn, then CR(G) is the disjoint union of CR(G1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , CR(Gn), and is thus isomorphic to R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So any rotunda graph is isomorphic to a reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then R(M) is connected if and only if M is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5 we noted that if N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Nk are the connected components of M, then R(M) is the disjoint union of R(N1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , R(Nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So if M is not connected then neither is R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For the converse, we let M be a connected supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 shows that R(M) is isomorphic to CR(G) where G is a 2-connected chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1 in [9] we see that CR(G), and hence R(M), is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Clique trees and rotunda trees Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid and let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let τ be a function from V (T) to P(E(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Assume that for every element x ∈ E(M) there is at least one vertex v ∈ V (T) such that x ∈ τ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case we say that (T, τ) is a tree-decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If for every element x ∈ E(M) there is exactly one vertex v ∈ V (T) such that x ∈ τ(v) then the tree-decomposition is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In other words, the tree-decomposition is strict if {τ(t)}t∈V (T) is a parti- tion of E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' A clique tree of G is a pair (T, ρ) where T is a tree and ρ is a bijection from V (T) to the set of maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We insist that for any v ∈ V (G), the set {t ∈ V (T): v ∈ ρ(t)} induces a subtree of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Clique trees were introduced by Gavril [5], who showed that a graph has a clique tree if and only if it is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Our next step is to define a matroid analogue of a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid, and let (T, τ) be a tree-decomposition of M such that τ is a bijection from V (T) to R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If, for every x ∈ E(M), the set {t ∈ V (T): x ∈ τ(t)} induces a subtree of T, then (T, τ) is a rotunda tree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the following material we must apply weights to the edges of reduced clique graphs and rotunda graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let σ be a function which takes the set {∅} ∪ {C ∩ C′ : C and C′ are distinct maximal cliques of G} SUPERSOLVABLE AND SATURATED MATROIDS 29 to non-negative integers, and where the following conditions hold: (i) σ(∅) = 0, (ii) if X and X′ are in the domain of σ and X is a proper subset of X′, then σ(X) < σ(X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In this case σ is a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The function σ applies a weight to each edge of CR(G), where the weight of the edge between C and C′ is σ(C ∩ C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The following result is the main theorem of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a connected chordal graph and let σ be a legitimate weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Every clique tree is a spanning tree of CR(G) and every edge of CR(G) is contained in a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, a spanning tree of CR(G) is a clique tree if and only if it has maximum weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Galinier, Habib, and Paul [4] prove the special case of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 where σ(C ∩ C′) = |C ∩ C′|, but their proof contains a flaw which is explained in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Next we consider the matroid analogue of legitimate weightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let σ be a function taking {∅} ∪ {R ∩ R′ : R, R′ ∈ R(M), R ̸= R′} to non-negative integers, where: (i) σ(∅) = 0, (ii) if X and X′ are in the domain of σ and X is a proper subset of X′, then σ(X) < σ(X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then σ is a legitimate weighting of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For examples of legitimate weightings, we may set σ(R ∩ R′) to be either the rank or the size of R ∩ R′, for each pair of rotunda R and R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In the case where we use rank, the legitimacy of the weighting relies on the fact that the intersection of two rotunda is a flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we are able to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2, which we restate in a more general form here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a connected supersolvable and saturated matroid and let σ be a legitimate weighting of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Every rotunda tree of M is a spanning tree of R(M) and every edge of R(M) is contained in a rotunda tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, a spanning tree of R(M) is a rotunda tree if and only if it has maximum weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 and let G be a 2-connected chordal graph and let θ: E(M) → P(V (G)) be a function such that (G, θ) is compliant with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let H be a graph that is isomorphic to both CR(G) and R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let πG be a bijection from V (H) to the family of maximal cliques of G, and let πM be a bijection from V (H) to R(M), such that πG and πM are both isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 30 MAYHEW AND PROBERT Let (T, τ) be a rotunda tree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Define ρ to be the composition θ|R(M)◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This means that ρ is a bijection from V (T) to the set of maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let v be an arbitrary vertex of G, and let x be the unique element of E(M) such that v is in θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now (1) {t ∈ V (T): v ∈ ρ(t)} = {t ∈ V (T): x ∈ τ(M)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because the latter set induces a connected subgraph of T, so does the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that (T, ρ) is a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore T is (isomorphic to) a spanning tree of H by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have now shown that any rotunda tree of M is a spanning tree of R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Moreover, if e is an arbitrary edge of H, then there is some spanning tree T of H such that T contains e and (T, ρ) is a clique tree of G for some bijection ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let τ be the composition (θ|R(M))−1 ◦ ρ, so that τ is a bijection from V (T) to R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If x is an arbitrary element of E(M) and v is a vertex in θ(x), then Equation (1) still holds and we see that (T, τ) is a rotunda tree of M that contains the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus any edge of R(M) is contained in a rotunda tree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We apply weights to the edges of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If u and u′ are adjacent in H, then we weight the edge between them with σ(πM(u)∩πM(u′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is not difficult to see that this weighting of H is also a legitimate weighting of CR(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' that is, if C and C′ are maximal cliques of G that are adjacent in CR(G), and σG applies the weight σ(θ−1(C) ∩ θ−1(C′)) to the edge between C and C′, then σG is a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let T be a maximum-weight spanning tree of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then (T, πG) is a clique tree of G, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Exactly as before, we see that (T, (θ|R(M))−1◦πG) is a rotunda tree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' On the other hand, if T is a spanning tree of H and (T, τ) is a rotunda tree of M, then (T, θ|R(M) ◦ τ) is a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence T is a maximum-weight spanning tree of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have now proved that the rotunda trees of M are exactly the maximum-weight spanning trees of R(M), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ It follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2 that R(M) is the exactly the union of all rotunda trees of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Tree-decompositions We recall the definition of graph tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let T be a tree and let ρ be a function from V (T) to P(V (G)) such that for every v ∈ V (G) the set {t ∈ V (T): v ∈ ρ(t)} is non-empty and induces a subtree of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We further insist that if u and v are adjacent vertices of G, then u, v ∈ ρ(t) for some t ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then (T, ρ) is a tree-decomposition of G, and the sets ρ(t) are the bags of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The width of (T, ρ) is the maximum size of a bag, and the tree-width of G is the minimum width taken over all tree-decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Any clique tree of a chordal graph is a tree-decomposition of optimal width, where the bags of the tree-decomposition are exactly the maximal cliques [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We now move towards a matroid analogue of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' SUPERSOLVABLE AND SATURATED MATROIDS 31 We first introduce the notion of matroid tree-width, as developed by Hlinˇen´y and Whittle [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Recall that a tree-decomposition of a matroid M is a tree T along with a function τ : V (T) → P(E(M)) such that every element x ∈ E(M) is in at least one set τ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid and let (T, τ) be a tree-decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let t be a node of T and let T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Td be the connected components of T − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For each i let Fi be ∪s∈V (Ti)τ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We define the node-width of t to be � � � � d � i=1 r � � � �τ(t) ∪ d� k=1 k̸=i Fk � � � � � � � � − (d − 1)r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The width of (T, τ) is the maximum node-width of any node in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The tree- width of M (denoted tw(M)) is the smallest width of any tree-decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that this definition is not exactly that used by Hlinˇen´y and Whittle because in their definition the minimum ranges over strict tree- decompositions, rather than all tree-decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' To see that this makes no difference to the definition, assume that the element x ∈ E(M) is con- tained in both τ(u) and τ(v), where u and v are distinct vertices of the tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We redefine τ by removing x from τ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It is easy to confirm that the width of no node is increased by this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By repeating this process we can produce a strict tree-decomposition with width no greater than the width of our original decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This argument shows that there ex- ists a strict tree-decomposition whose width is as small as possible amongst all tree-decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus extending Hlinˇen´y and Whittle’s definition to include non-strict tree-decompositions makes no difference to the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We can always let T be a tree with a single node, and let τ take every element of E(M) to this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows from the definition that the width of (T, τ) is r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This shows that the tree-width of any matroid M is bounded above by r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a round matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then tw(M) = r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let E be the ground set of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (T, τ) be any strict tree- decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We direct each edge of T in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be an arbitrary edge of T and assume that e joins u1 to u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For each i let Ti be the connected component of T\\e that contains ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Ui = ∪s∈V (Ti)τ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus (U1, U2) is a partition of E (since the tree-decomposition is strict), and because M is round, either U1 or U2 is spanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' If Ui is spanning then we direct e from u3−i to ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that it is possible for an edge to have two directions applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let P be a maximum length directed path in T, and assume that t is the final node in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Td be the connected components of T − t and let Fi = ∪s∈V (Ti)τ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because the edges incident with t are all directed 32 MAYHEW AND PROBERT towards t, it follows that E − Fi is spanning for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Fd are pairwise disjoint, the width of t is r(M) − d � i=1 (r(M) − r(E − Fi)) = r(M) − d � i=1 (r(M) − r(M)) = r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence the node-width of t is equal to r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus tw(M) ≥ r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We have already observed that tw(M) ≤ r(M) so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Hlinˇen´y and Whittle show that if N is a minor of the matroid M, then tw(N) ≤ tw(M) [8, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' The next result follows from this observation and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a matroid and let R be a round flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then tw(M) ≥ r(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let (T, τ) be a rotunda tree of M, a supersolvable sat- urated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let e be an edge of T that joins vertices u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For i = 1, 2, let Ti be the connected component of T\\e that contains ui and let Fi be ∪t∈V (Ti)τ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then (F1, F2) is a modular cover of M and F1 ∩ F2 = τ(u1) ∩ τ(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that every element of E(M) is contained in a round flat, and hence in a rotunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From this it follows that E(M) = F1 ∪ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8 and we let G be a 2-connected chordal graph with a function θ: E(M) → P(V (G)) such that (G, θ) is compliant with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let ρ be the composition θ|R(M) ◦ τ so that ρ is a bijection between V (T) and the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Exactly as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2 we can show that (T, ρ) is a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Define Ri to be the rotunda τ(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let F be the flat R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that because R1 and R2 are adjacent in a rotunda tree of M, they are adjacent in R(M) by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This implies that F is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let Ci = θ(Ri) for i = 1, 2, so that C1 and C2 are the corresponding maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Define S to be θ(F) = C1 ∩ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Note that if D is a maximal clique of G, then D − S is contained in a connected component of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For i = 1, 2, let vi be an arbitrary vertex of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then the path of T from v1 to v2 contains u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It follows from [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='8] that ρ(v1) − S and ρ(v2) − S are contained in different connected components of G−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now we let U be the union of all connected components of G − S that contains ρ(v) − S for some v in V (T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From the observations in this paragraph we see that F ∪ θ−1(U) is equal to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because (G, θ) is compliant with M this means that F1 is a modular flat of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Symmetrically, F2 is a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let x be an arbitrary element in F1 ∩ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let v ∈ V (T1) and v′ ∈ V (T2) be chosen so that x is in τ(v) ∩ τ(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because (T, τ) is a rotunda tree it follows that x is in τ(w) whenever w is in the path of T from v to v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In particular, x is in τ(u) ∩ τ(u′) = R ∩ R′ = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Thus F1 ∩ F2 ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because SUPERSOLVABLE AND SATURATED MATROIDS 33 ui is in Ti for each i it follows that Ri ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore F = R1 ∩ R2 is a subset of F1 ∩ F2, and now τ(u1) ∩ τ(u2) = R1 ∩ R2 = F = F1 ∩ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From this it follows that F1 ∩ F2 does not contain R1 or R2, so neither F1 nor F2 is equal to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Since F1 and F2 are proper modular flats of M and E(M) = F1 ∪ F2 we see that (F1, F2) is a modular cover and the result is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Let M be a connected supersolvable and saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will now show that a rotunda tree of M has the properties of an optimal tree- decomposition as per Hlinˇen´y and Whittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid and let (T, τ) be a rotunda tree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then the width of (T, τ) is equal to tw(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We will show that the node-width of any t ∈ V (T) is r(τ(t)), so that the width of (T, τ) is the maximum rank of a rotunda of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' From Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='3 we see that tw(M) is bounded below by this rank, so having completed this task, we will have shown that (T, τ) is a tree-decomposition of lowest-possible rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' It will then follow that tw(M) is equal to the width of (T, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So let t be an arbitrary vertex in T and let T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , Td be the connected components of T − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For each i let ti be the vertex of Ti that is adjacent to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Define F to be τ(t), and let Fi be ∪s∈V (Ti)τ(s) for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We define F i to be F ∪ d� k=1 k̸=i Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Therefore the node-width of t is (2) r(F 1) + · · · + r(F d) − (d − 1)r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' In addition, we define F >i to be F ∪ d� k=i+1 Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Notice that F >1 = F 1 and that F >d = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' For any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' , d−1}, the intersection of F >i and F i+1 is F >i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We note that F >i ∩ F i+1 = (F ∪ Fi+1 ∪ · · · ∪ Fd) ∩ (F ∪ F1 ∪ · · · ∪ Fi ∪ Fi+2 ∪ · · · Fd) = (Fi+1 ∩ (F1 ∪ · · · ∪ Fi)) ∪ (F ∪ Fi+2 ∪ · · · ∪ Fd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now Fi+1 ∩ (F1 ∪ · · · ∪ Fi) is contained in Fi+1 ∩ F i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' But Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 tells us that Fi+1 ∩ F i+1 is equal to τ(ti+1) ∩ τ(t), which is therefore contained in τ(t) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Hence we can remove Fi+1 ∩ (F1 ∪ · · · ∪ Fi) from the 34 MAYHEW AND PROBERT equation above and conclude that F >i ∩F i+1 is F ∪Fi+2 ∪· · ·∪Fd = F >i+1, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='4 implies that (Fi, F i) is a modular cover for each i, so that in particular F 2 is a modular flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Now Equation (2) reduces to r(F 1 ∩ F 2) + r(F 1 ∪ F 2) + r(F 3) + · · · + r(F d) − (d − 1)r(M) = r(F >1 ∩ F 2) + r(E(M)) + r(F 3) + · · · + r(F d) − (d − 1)r(M) = r(F >2) + r(F 3) + · · · + r(F d) − (d − 2)r(M) where we have applied 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1 in the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because F 3 is a modular flat, we can again apply 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='1 and reduce to r(F >3) + r(F 4) + · · · + r(F d) − (d − 3)r(M) By continuing this process, we find that Equation (2) is equal to r(F >d) − (d − d)r(M) = r(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' So the node-width of t is r(τ(t)) = r(F), exactly as we claimed, and the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' □ Now we present the central theorem for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Because a rotunda tree is a tree-decomposition of optimal width for a supersolvable saturated matroid M, we can treat it as a canonical tree decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let M be a supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Then tw(M) = max{r(R): R ∈ R(M)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Further observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Let bw(M) denote the branch-width of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' By [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content='2] we see that bw(M) − 1 ≤ tw(M) ≤ max{2 bw(M) − 2, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We see therefore that given a supersolvable saturated matroid M of branch- width k there must be a rotunda tree of M where the rank of the largest maximal rotunda is bounded by a function of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' As a result, we can conclude that supersolvable saturated matroids have canonical tree decompositions of optimal tree-width in much the same way as chordal graphs have canonical tree decompositions where each bag is a clique of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' This theorem has algorithmic implications for how we can efficiently find the tree-width of a supersolvable saturated matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' However, for this to work we would need an efficient method for constructing the rotunda graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Acknowledgements We thank Geoff Whittle, who supervised the thesis of the second author (which includes much of the material in this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' We also thank a referee of an earlier draft for numerous helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' SUPERSOLVABLE 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Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' 21, Oxford University Press, Oxford, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' [12] Geoff Whittle, Some Aspects of the Critical Problem for Matroids, University of Tas- mania, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} +page_content=' PhD Thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E2T4oBgHgl3EQfRAZd/content/2301.03776v1.pdf'} diff --git a/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/2301.08651v1.pdf.txt b/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/2301.08651v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..93ef3e6dd2274813e5b14e0fe4f3b70ac62ef962 --- /dev/null +++ b/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/2301.08651v1.pdf.txt @@ -0,0 +1,2924 @@ +arXiv:2301.08651v1 [math.CA] 20 Jan 2023 +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON +THE PARABOLA +DONGGEUN RYOU +Abstract. For any 0 < α < 1, we construct Cantor sets on the parabola of Hausdorff +dimension α such that they are Salem sets and each associated measure ν satisfies the +estimate ∥� +fdν∥Lp(R2) ≤ Cp∥f∥L2(ν) for all p > 6/α and for some constant Cp > 0 which +may depend on p and ν. The range p > 6/α is optimal except for the endpoint. This is an +analogue of works of Shmerkin-Suomala and �Laba-Wang. They considered fractal subsets +of Rd, whereas we consider fractal subsets of the parabola. +1. Introduction +Let µ is a Borel probability measure in Rd and assume that it satisfies the estimate +∥� +fdµ∥Lp(R2) ≤ Cp∥f∥L2(µ) +∀f ∈ L2(µ) +(1.1) +for some constant Cp which depends on p. It is easy to show that (1.1) holds when p = ∞. +However, we can say more if we focus on more specific measure µ. One typical example +is a surface carried measure on the sphere or on the paraboloid. Then, (1.1) holds when +p ≥ 2(d + 1)/(d − 1) and the range is sharp. This is the result of Tomas-Stein theorem (for +example, see [20]). +Let B(x, r) is a closed ball of radius r centered at x. If we have assumptions on the Fourier +decay of �µ and the upper bound of µ(B(x, r)), the following result generalizes Tomas-Stein +theorem. +Theorem 1.1. Assume that there exist a, b ∈ (0, d) and C1, C2 ≥ 0 such that +µ(B(x, r)) ≤ C1ra +∀x ∈ Rd, r > 0, +(1.2) +|�µ(ξ)| ≤ C2(1 + |ξ|)−b/2 +∀ξ ∈ Rd. +(1.3) +Then, the estimate (1.1) holds when p ≥ (4d − 4a + 2b)/b. +Mockenhaupt [17] and Mitsis [16] proved that (1.1) holds when p > (4d − 4a + 2b)/b and +the endpoint result was proved by Bak and Seeger [1]. If µ is a surface carried measure +on the sphere or the paraboloid, a = b = d − 1. +Thus, Theorem 1.1 recovers the range +p ≥ 2(d + 1)/(d − 1). +Define the critical exponent pc of the measure µ by +pc = inf{p : (1.1) holds}. +Theorem 1.1 means pc ≤ (4d − 4a + 2b)/b. This upper bound of pc is known to be optimal +when d − 1 < b ≤ a < d. When d = 1, Hambrook and �Laba [10] constructed a measure such +that (1.3) holds for every β < α and pc = (4−2a)/a. Chen [6] extended this result to general +0 ≤ b ≤ a ≤ 1. In higher dimensions, it was done in [11] by Hambrook and �Laba. +Date: January 23, 2023. +2020 Mathematics Subject Classification. 42B10 (primary) 28A80 (secondary). +Key words and phrases. Random Cantor sets, Restriction estimate, Salem set. +1 + +2 +DONGGEUN RYOU +However, there exists some measures µ such that pc is strictly smaller than (4d − 4a + 2b)/b. +In [10], Hambrook and �Laba showed that pc ≥ 2d/α if µ is supported on a set of Hausdorff +dimension α. Note that b ≤ a ≤ α. If (1.2) and (1.3) holds for values arbitrarily close to α, +the support of µ is called a Salem set. If 0 < α < d, we have the inequality +2d +α < 4d − 2α +α +≤ 4d − 4a + 2b +b +. +Thus, 2d/α is smaller than (4d − 4a + 2b)/b even when µ is supported on a Salemt set. +Examples such that pc = 2d/α were provided by Chen [5], Chen and Seeger [7], Shmerkin +and Suomala [19] (see also [18]) and �Laba and Wang [13]. Therefore, the lower bound of pc +is optimal for all 0 < α < d. +In [5] and [7], Chen and Seeger constructed measures supported on a Salem set of Hausdorff +dimension α such that pc = 2d/α where α = d/k and k is an integer. They considered k-fold +self convolution of µ. Shmerkin and Suomala [19] constructed the example through random +fractal measures. Their result covers when d = 1 and 1/2 < α < 1. �Laba and Wang [13] +constructed measures of Hausdorff dimension α such that (1.3) holds for β < min(α/2, 1) +and pc = 2d/α. They used Λ(p)-set and decoupling and this construction works for all α +such that 0 < α < d. However, the estimate (1.1) holds when p > 2d/α, while results of +Chen, Chen-Seeger and Shmerkin-Suomala works even at the endpoint. Thus, it is still open +whether, for any 0 < α < d, there exists a measure µ such that its support has Hausdorff +dimension α and (1.1) holds for p = 2d/α. +Now, let us turn to our setting. Earlier works constructed fractal measures on Rd, but we +consider fractal measures on the parabola P1 := {(x, x2) : 0 ≤ x ≤ 1}. Throughout the +paper, we denote by X ≲ Y when X ≤ CY for some constant C > 0 and we write X ≈ Y +to denote that X ≲ Y and Y ≲ X. If the constant C depends on parameters such as ǫ, we +write X(ǫ) ≲ǫ Y (ǫ) instead of X(ǫ) ≤ C(ǫ)Y (ǫ) where the constant C(ǫ) depends on ǫ. +Our main results are as follows. +Theorem 1.2. Let 0 < α < 1. There exists a probability measure ν supported on a subset of +P1 which satisfies the followings. +(1) The support of the measure ν has Hausdorff dimension α. +(2) For any 0 < ǫ < α, we have +ν(B(x, r)) ≲α,ǫ rα−ǫ +∀x ∈ R2, ∀r > 0. +(1.4) +(3) For any ǫ > 0, +|�ν(ξ)| ≲α,ǫ (1 + |ξ|)−α/2+ǫ +∀ξ ∈ R2. +(1.5) +(4) For every p > 6/α, we have the estimate +∥� +fdν∥Lp(R2) ≲p ∥f∥L2(ν) +∀f ∈ L2(ν). +(1.6) +Equivalently, for any 1 ≤ q < 6/(6 − α) we have +∥ �f∥L2(ν) ≲q ∥f∥Lq(R2) +∀f ∈ Lq(R2). +Note that the support of ν is a Salem set and it is a subset of the parabola, which is also +a Salem set. Theorem 1.2 is sharp except the endpoint in the following sense. +Theorem 1.3. Let Pd−1 := {(x, |x|2 : x ∈ [0, 1]d−1} and 0 < α < d − 1. Assume that ν is a +probability measure supported on a subset of Pd−1 whose Hausdorff dimension is α. For any +0 < ǫ < α, assume that +ν(B(x, r)) ≲α,ǫ rα−ǫ +∀x ∈ Pd−1, ∀r > 0. +(1.7) + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +3 +Then, the estimate +∥� +fdν∥Lp(R2) ≲p,q ∥f∥Lq(ν) +∀f ∈ Lq(ν) +(1.8) +cannot hold if p < q′(d + 1)/α where 1/q + 1/q′ = 1. +When d = 2 and q = 2, Theorem 1.3 implies that pc ≥ 6/α. Since ν is on the parabola, +the lower bound of pc increased from 4/α to 6/α when compared to the result by Hambrook +and �Laba [10]. If 0 < α < 1, then 6/α < (8 − 2α)/α. Thus, it is still smaller than the upper +bound of pc obtained from Theorem 1.1. +1.1. Outline of the paper. We will follow the proof of [13], but we cannot apply it directly. +Main obstacle is that we cannot make use of dual boxes as in [13]. +For example, they +decompose the support of �f into cubes of side length R−1 and integrate f over the cube of +side length R. This duality was used to establish decoupling estimate and localized restriction +estimate which are proposition 1 and corollary 2 in [13] respectively. However, P1 can be +covered by rectangles of dimensions R−1/2, R−1 and each rectangles corresponds to dual +rectangles of dimensions R1/2, R. Since these dual rectangles have their long side in different +directions, union of them is a square of side length R. Therefore, we decompose the support +of �f into rectangles of dimensions R−1/2, R−1 while we integrate f over the cube of side +length R. Thus, we should use dual rectangles in a different way. +In Section 2, we construct Cantor sets on [0, 1] by using Λ(p)-sets and consider associated +measures which is supported on a subset of parabola above these Cantor sets. Let ν be a +measure constructed in Section 2. Then ν is supported on a set of Hausdorff dimension α +and satisfies (1.4). In section 3, we obtain decoupling inequality for the support of ν. We +use Λ(p)-sets and the result of [4]. In Section 4, we obtain local restriction estimate of ν +which is an analogue of Corollary 2 in [13]. However, we use mixed norm interpolation [2] in +addition to the argument in [13]. In Section 5, we obtain the Fourier decay of ν. We only +need negative exponent in order to derive (1.6), but it turned out that �ν has decay arbitrarily +close to optimal almost surely as in (1.5). We use the argument in [18], but modified their +method in a way that we can use oscillatory integral. The local restriction estimate and the +Fourier decay leads to global restriction estimate in Section 6, which is (1.6). Tao’s epsilon +removal lemma [23] is used as in [13], but we simplify the proof of Lemma 9 in [13] in order +not to use dual rectangles. Thus, we prove that Theorem 1.2 happens almost surely if ν is a +random measure constructed in Section 6. Lastly, we prove Theorem 1.3 in Section 7. +Acknowledgements. The author would like to thank his advisor Alex Iosevich for many +discussions of this work and encouragement. The author would also like to thank Shaoming +Guo, Zane Kun Li for helpful conversations. +2. The construction of the Cantor set +Our proof uses the following construction of Cantor measures. Let {nj}j∈N∪{0} be a se- +quence of positive integers and let Nj = n0 · · · nj. We assume the following conditions on +nj. +n0 = 1 +and +2 ≤ n1 ≤ n2 ≤ · · · ≤ nj ≤ · · · , +(2.1) +∀a > 0, aj ≲a Nj, +(2.2) +∀ǫ > 0 and ∀j ∈ N, nj+1 ≲ǫ N ǫ +j, +(2.3) +∃B such that ∀j ∈ N, N 1/2 +2j N −1 +j +≤ Bj. +(2.4) + +4 +DONGGEUN RYOU +For example, if nj ≈ j, all conditions from (2.1) to (2.4) are satisfied by Stirling’s formula +(see [9, p. 98]). Condition (2.4) was not assumed in [13], but we need it additionally in +Section 3 and 4. +We need the following theorem in order to use Λ(p)-sets. +Theorem 2.1 (Existence of the Λ(p)-set). Let e(x) denote e2πix and p > 2. For every N ∈ N, +there exists a set S ⊂ {0, 2, · · · , N − 1} such that N 2/p ≤ |S| < N 2/p + 1 and +∥ +� +a∈S +cae(ax)∥Lp([0,1]) ≤ Cp +� � +a∈S +|ca|2 +�1/2 +∀{ca}a∈S ∈ ℓ2 +(2.5) +for some constant Cp > 0 which depends only on p, but not on N. +Theorem 2.1 was first proved by Bourgain [3] and another proof was given by Talagrand +[21], see also [22, Section 19.3] for simpler proof. +Let 0 < α < 1 and for each j ∈ N, let {tj}N be a sequence of integers such that +nα/2 +j +≤ tj < nα/2 +j ++ 1. +Define +Σj := {S ⊂ [0, n1/2 +j +] ∩ Z : |S| = tj and (2.5) holds for p = 2/α, N = n1/2 +j +}. +The set Σj is non-empty because of Theorem 2.1. +Let E0 = [0, 1] and for S1 ∈ Σ1, let +A1 = S1, +E1 = N −1/2 +1 +(A1 + [0, 1]). +For each a ∈ Aj, choose a set Sj+1,a ∈ Σj+1 and let +Aj+1,a = n1/2 +j+1a + Sj+1,a +Aj+1 = ∪a∈AjAj+1,a +Ej+1 = N −1/2 +j+1 (Aj+1 + [0, 1]). +The set Aj is a subset of {0, 2, · · · , N 1/2 +j +− 1} and [0, 1] ⊇ E1 ⊇ · · · ⊇ Ej ⊇ Ej+1. Let +E∞ := ∩∞ +j=1Ej. Similarly, let Pj := {(x1, x2 +1) : x1 ∈ Ej} and P∞ := ∩∞ +j=1Pj. +For each j ∈ N ∪ {0}, we define +µj := +1 +|Ej|1Ej(x1), +x1 ∈ R. +We identify the function µk with the absolutely continuous measures µjdξ1 so that +� +gdµj := +� +gµjdx1. +From µj, we defined the measure νj as follows. +� +fdνj := +� +f(x1, x2 +1)dµj. +For simplicity, let ∥µj∥ := ∥µj∥L1(R) and ∥νj∥ := ∥νj∥L1(R2). The measures µj and νj converge +weakly as j → ∞ to a probability measure µ and ν supported on E∞ and P∞ respectively. +Lemma 2.2 (Lemma 6, [13]). The measure µ and the set E∞ constructed above satisfies the +following. +(1) The set E∞ has Hausdorff dimension α. +(2) For any 0 < ǫ < α, we have that +µ(B(x, r)) ≲α,ǫ rα−ǫ +∀ξ ∈ R, ∀r > 0. + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +5 +It is easy to show that the same are true for ν. +Lemma 2.3. The measure ν and the set P∞ constructed above satisfies the following. +(1) The set P∞ has Hausdorff dimension α. +(2) For any 0 < ǫ < α, we have that +ν(B(x, r)) ≲α,ǫ rα−ǫ +∀ξ ∈ R2, ∀r > 0. +3. Decoupling inequalities +In this section, we will derive the decoupling estimates for E∞ and P∞. We define decou- +pling constants and discrete restriction constants and we will verify relations between them. +Since we need to look into Ej in different scales, we define the following. +For 0 ≤ i < j, write Nj,i = NjN −1 +i +. For Si+1 ∈ Σi+1, let +Ai+1,i = Si+1 +Ei+1,i = n−1/2 +i+1 (Ai+1,i + [0, 1]). +For each a ∈ Aj,i, choose a set Sj+1,a ∈ Σj+1 and let +Aj+1,i,a = n1/2 +j+1a + Sj+1,a +Aj+1,i = ∪a∈Aj+1,iAj+1,i,a +Ej+1,i = N −1/2 +j+1,i(Aj+1,i + [0, 1]). +Let Pj,i(Ej,i) denote the partition of Ej,i into intervals of length N −1/2 +j,i +. If I ∈ Pj(Ej,i), there +exists an unique element a ∈ Aj,i such that I = N −1/2 +j,i +(a + [0, 1]). +For p ≥ 2, let Dp(Ej,i) denotes the best constant such that +∥ +� +I∈Pj,i(Ej,i) +fI∥Lp(R) ≤ Dp(Ej,i) +� +� +I∈Pj,i(Ej,i) +∥fI∥2 +Lp(R) +�1/2 +(3.1) +holds for any choices of Ej,i where �fI(ξ1) = �f(ξ1)1I(ξ1) for ξ1 ∈ R. +Let Dp(Pj,i) be the best constant such that +∥ +� +I∈Pj,i(Ej,i) +fΩI∥Lp(R2) ≤ Dp(Pj,i) +� +� +I∈Pj,i(Ej,i) +∥fΩI∥2 +Lp(R2) +�1/2 +(3.2) +holds for any choices of Ej,i where fΩI = �f(ξ)1ΩI(ξ) and 1ΩI(ξ) is a characteristic function +supported on +ΩI = {ξ ∈ R2 :aN −1/2 +j,i +≤ ξ1 ≤ (a + 1)N −1/2 +j,i +, +|ξ2 − (2a + 1)N −1/2 +j,i +(ξ1 − aN −1/2 +j,i +) − a2N −1 +j,i | ≤ N −1 +j,i } +for a that corresponds to I. +If i = 0, Aj,0 and Ej,0 are same with Aj and Ej defined in section 2 respectively. +Let Kp(Ej,i) be the best constant such that +∥ +� +a∈Aj,i +cae(ax)∥Lp([0,1]) ≤ Kp(Ej,i) +� � +a∈Aj,i +|ca|2 +�1/2 +holds for any choices of Aj,i. Let Kp(Pj,i) be the best constant such that +∥ +� +a∈Aj,i +cae(ax1 + a2x2)∥Lp([0,1]2) ≤ Kp(Pj,i) +� � +a∈Aj,i +|ca|2 +�1/2 + +6 +DONGGEUN RYOU +holds for any choices of Aj,i. +In short, Dp(Ej,i) and Dp(Pj,i) are decoupling constants for Ej,i and Pj,i respectively and +Kp(Ej,i) and Kp(Pj,i) are discrete restriction constants for Ej,i and Pj,i respectively. We will +show that K6/α(Pj) ≲ǫ N ǫ +j through the following inequalities: +K3p(Pj) ≲p D3p(Pj) ≲ǫ N ǫ +jDp(Ej) +and +Dp(Ej) ≈p Kp(Ej) ≲ǫ Cj +p +where p = 2/α and Cp is the constant in (2.5). Note that the inequality Kp(Pj) ≲p Dp(Pj) +is well known, see for example [8, Theorem 13.1]. +3.1. From Λ(p)-sets to decoupling for Ej. We need the following lemmas to use Λ(p)-sets +in multiscale. +Lemma 3.1. For p ≥ 2, let S1 be a subset of Zd. For a ∈ S1 and k ∈ N, let S2,a be subsets +of [0, k − 1] ∩ Zd. Assume that the sets S1 and S2,a satisfy +∥ +� +a∈S1 +cae(a · x)∥Lp([0,1]d) ≤ C1 +� � +a∈S1 +|ca|2 +�1/2 +∀{ca}a∈S1 ∈ ℓ2 +(3.3) +and +∥ +� +b∈S2,a +cbe(b · x)∥Lp([0,1]d) ≤ C2 +� � +b∈S2 +|cb|2 +�1/2 +∀{cb}b∈S2 ∈ ℓ2. +(3.4) +where C2 is uniform over a. Then, we have +∥ +� +a∈S1 +� +b∈S2,a +ca,be((ka + b) · x)∥Lp([0,1]d) ≲p C1C2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�1/2 +(3.5) +for all {ca,b}a∈S1,b∈S2,a ∈ ℓ2. +Lemma 3.2. For p ≥ 2, Dp(Ej,i) ≈p Kp(Ej,i). +Lemma 3.1 and 3.2 follows from the proof of Proposition 1 in [13]. Duality of Lp played a +key role in the proof. In Section 8, we provide alternative proofs of them which do not rely +on duality. +Remark 3.3. Readers may want to compare Lemma 3.1 with Proposition 3.5 in [4]. They +are similar but there is a trade-off. Lemma 3.1 can cover more general cases, because p does +not need to be an even integer and there is no assumption on carryover. However, the implicit +constant of the inequality is larger than Proposition 3.5 in [4] because the constant in (3.5) +is not exactly C1C2, but C′ +pC1C2 where C′ +p > 1. +Lemma 3.4. For p = 2/α where 0 < α < 1, Dp(Ej,i) ≲p Cj−i +p +where the Cp is the constant +in (2.5). +Proof. Combining Lemma 3.1 and 3.2 and (2.5), we obtain that +Dp(Ej,i) ≈ Kp(Ej,i) ≲ Cj−i +p +. +□ + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +7 +3.2. From decoupling for Ej to decoupling for Pj. In [4], they proved that decoupling +for a Cantor set on the parabola can be derived from decoupling for a Cantor set on the line. +We adapt their argument to our setting. +Proposition 3.5. For p = 2/α where 0 < α < 1, we have that D3p(Pj) ≲ǫ N ǫ +j . +Lemma 3.6 (Parabolic rescaling). Suppose that j ≥ i and I ∈ Pi(Ei). Then, +∥ +� +J∈Pj(I∩Ej) +fΩJ∥Lp(R2) ≤ Dp(Pj,i) +� +� +J∈Pj(I∩Ej) +∥fΩJ∥2 +Lp(R2) +�1/2 +. +Lemma 3.7 (Almost multiplicativity). We have +Dp(Pj+i) ≤ Dp(Pj+i,i)Dp(Pi). +For j ≤ i1, i2 ≤ k, we define the bilinear constant Mp(j, k, i1, i2) which is the smallest +constant such that +� +R2 +���� +� +J1∈Pj(I1∩Ej) +fΩJ1 +���� +p���� +� +J2∈Pj(I2∩Ej) +gΩJ2 +���� +2p +≤ Mp(j, k, i1, i2)3p +� +� +J1∈Pj(I1∩Ej) +∥fJ1∥2 +L3p(R2) +�p/2� +� +J2∈Pj(I2∩Ej) +∥gJ2∥2 +3p +�p +for all I1 ∈ Pi1(Ei1) and I2 ∈ Pi2(Ei2) such that d(I1, I2) ≥ N −1/2 +k +and for any choices of Ej. +Lemma 3.8 (Bilinear reduction). If i ≤ j, then +D3p(Pj) ≲ D3p(Pj,i) + N O(1) +i +Mp(j, i, i, i). +(3.6) +Proofs of Lemma 3.6, 3.7 and 3.8 are same as in [4]. +Lemma 3.9 (Key estimate in [4]). Assume that p = 2/α where 0 < α < 1. If 0 ≤ k ≤ +i1, i2, 4i1 ≤ j with 2i1 ≤ i2, then for any ǫ > 0, +Mp(j, k, i1, i2) ≲p,ǫ N O(1) +k +(CpB)i1Mp(j, k, 4i1, i2). +(3.7) +where B is the constant in (2.4) and Cp is the constant in (2.5). +Proof. We follow the proof of Lemma 2.4 in [4] with modifications. The condition (2.4) comes +into play since nk is nondecreasing. Without loss of generality, we can assume that I2 is on +the left of I1 and I2 is centered at the origin. Then, for each J ∈ P4i1(I1 ∩ E4i1), the center +of J is a distance ≳ N −1/2 +k +away from the origin. Let +FJ = +� +J1∈Pj(J∩Ej) +fΩJ1 +and +G = +� +J2∈Pj([0,N−1/2 +i2 +]∩Ej) +gΩJ2. + +8 +DONGGEUN RYOU +As an analogue of (17) in [4], it suffices to prove the following. For fixed x ∈ R, +� +R +���� +� +J∈P4i1(I1∩E4i1) +FJ(x, y)G(x, y)2 +���� +p +dy +≲p,ǫ N O(p) +k +Dp(E4i1,2i1)pDp(E2i1,i1)pB3pi1/2 +� +� +J∈P4i1(I1∩E4i1) +( +� +R +|FJ(x, y)G(x, y)2|pdy)2/p +�p/2 +. +(3.8) +Once we prove (3.8), then (3.7) follows from Lemma 3.4. For J0 ∈ P2i1(I1 ∩ E2i1), let +FJ0 = +� +J∈P4i1(J0∩E4i1) +FJ. +Note that +� +J∈P4i1(I1∩E4i1) +FJ = +� +J0∈P2i1(I1∩E2i1) +FJ0. +The function �G is supported in an O(N −1/2 +i2 +)×O(N −1 +i2 +N −1/2 +j +) rectangle and �FJ0 is supported +on the horizontal strip +{(ξ1, ξ2) : ξ2 = γ2 +J0 + O(N −1/2 +2i1 +)} +where γJ0 is the center of J0. Since 2i1 ≤ i2, � +FJ0G2 is supported in the horizontal strip +{(ξ1, ξ2) : ξ2 = γ2 +J0 + O(N −1/2 +2i1 +)}. +Thus, Fourier transform of FJ0G2 in y for fixed x is also supported on an interval of length +O(N −1/2 +2i1 +) centered at γ2 +J0. +For c ≲ N O(1) +k +N 1/2 +2i1 N −1 +i1 , it is easy to prove that +∥ +� +J∈P2i1,i1(E2i1,i1) +fcJ∥Lp(R) ≲p,ǫ N O(1) +k +Dp(E2i1,i1) +�N 1/2 +2i1 +Ni1 +�� +� +J∈P2i1,i1(E2i1,i1) +∥fcJ∥2 +Lp(R) +�1/2 +(3.9) +where cJ is the interval of length c|J| which has the same center with J. +By (2.4), N 1/2 +2i1 N −1 +i1 +≤ Bi1. It follows from Lemma 3.6 that +∥ +� +J0∈P2i1(I1∩E2i1) +fcJ0∥Lp(R) ≲p,ǫ N O(1) +k +Dp(E2i1,i1)Bi1 +� +� +J0∈P2i1(I1∩E2i1) +∥fcJ0∥2 +Lp(R) +�1/2 +. +(3.10) +Let γm be the left endpoint of I1 and let us consider +T(x) = (2γm + N −1/2 +i1 +)(x − γm) + γ2 +m. + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +9 +Since γm ≥ N −1/2 +k +and N −1/2 +2i1 +≤ N −1 +i1 , we get γ2 +J0 + O(N −1/2 +2i1 +) ⊆ T(cJ0). Therefore, (3.10) +implies that +� +R +���� +� +J∈P4i1(I1∩Ej) +FJ(x, y)G(x, y)2 +���� +p +dy += +� +R +���� +� +J0∈P2i1(I1∩E2i1) +FJ0(x, y)G(x, y)2 +���� +p +dy +≲p N O(p) +k +Dp(E2i1,i1)pBpi1 +� +� +J0∈P2i1(I1∩E2i1) +( +� +R +|FJ0(x, y)G(x, y)2|pdy)2/p +�p/2 +. +(3.11) +Similarly, �FJ is supported on the horizontal strip +{(ξ1, ξ2) : ξ2 = γ2 +J + O(N −1/2 +4i1 +)} +where γJ is the center of J. Since N −1/2 +4i1 +, N −1 +i2 +≤ N −1 +2i1 , Fourier transform of FJG2 in y for +fixed x is supported on an interval of length O(N −1 +2i1 ) centered at γ2 +J. +For c ≲ N O(1) +i +N 1/2 +4i1 N −1 +2i1 , we can prove that +∥ +� +J∈P4i1,2i1(E4i1,2i1) +fcJ∥Lp(R) ≲p,ǫ N O(1) +k +Dp(E4i1,2i1) +�N 1/2 +4i1 +N2i1 +�� +� +J∈P4i1,2i1(E4i1,2i1) +∥fcJ∥2 +Lp(R) +�1/2 +. +(3.12) +As before, this implies that +� +R +���� +� +J∈P4ai(J0∩E4i1) +FJ(x, y)G(x, y)2 +���� +p +dy +≲p N O(p) +k +Dp(E4i1,2i1)pB2pi1 +� +� +J∈P4i1(J0∩E4i1) +( +� +R +|FJ(x, y)G(x, y)2|pdy)2/p +�p/2 +. +(3.13) +Combining (3.11) and (3.13), we get (3.8). +□ +Lemma 3.10. Let k ≤ i2 ≤ i1 ≤ j. Then, +Mp(j, k, i1, i2) ≤ Mp(j, k, i2, i1)1/2D3p(Pj,i2)1/2. +The proof of Lemma 3.10 is same as in [4]. +Proof of Proposition 3.5. Assume that λ is the smallest exponent such that +D3p(Pj,i) ≲ǫ N λ+ǫ +j,i +(3.14) +for all 0 ≤ i < j and for sufficiently small 0 < ǫ < 1. Then, it suffices to show that λ = 0. +We can run the iteration as in [4]. But, we should be careful since nk is nondecreasing. First, +assume that λ > 0. By Lemma 3.9 and 3.10 and (3.14), for any positive integer a such that +1 ≤ a ≤ j +4i, we obtain that +Mp(j, i, 2ai, ai) ≤ Mp(j, i, ai, 2ai)1/2D3p(Pj,ai)1/2 +≲ǫ N O(1) +i +Mp(j, i, 4ai, 2ai)1/2(CpB)ai/2D3p(Pj,ai)1/2. +(3.15) + +10 +DONGGEUN RYOU +It suffices to consider the case j = 2k+1i. By iterating (3.15), it follows from (3.14) that +Mp(j, i, 2i, i) ≲ǫ N O(1) +i +(CpB)ki/2N λ+ǫ(1−1/2k) +j +(NiN 1/2 +2i +· · · N 1/2k−1 +2k−1i )−(λ+ǫ)/2Mp(j, i, 2k+1i, 2ki)1/2k. +Since j = 2k+1i, we have +Mp(j, i, 2k+1i, 2ki) ≤ D3p(Pj,2ki)2/3 ≲ +�N2k+1i +N2ki +�2(λ+ǫ)/3 +. +By (2.2) and (2.4), we obtain that +Mp(j, i, 2k+1i, 2ki)1/2k ≲ (B2kiN 1/2 +j +)(λ+ǫ)/3·2k−1 +≤ B2(λ+ǫ)i/3N (λ+ǫ)/3·2k +j +≲ N O(1) +i +N (λ+ǫ)/2k +j +. +Therefore, we arrive at +Mp(j, i, 2i, i) ≲ǫ N O(1) +i +(CpB)ki/2N λ+ǫ +j +(NiN 1/2 +2i +· · · N 1/2k−1 +2k−1i )−(λ+ǫ)/2. +Since N 2s +i +≤ N2si, by (2.3) we obtain that +Mp(j, i, 2i, i) ≲ǫ N O(1) +i +(CpB)ki/2N λ+ǫ +j +N −kλ/2 +i +. +Conditions (2.1) and (2.2) imply that limj→∞ nj = ∞. Thus, (CpB)i/2 ≤ N λ/4 +i +for sufficiently +large i. +By Lemma 3.8, for sufficiently large k and i, we have +D3p(Pj) ≲ǫ (NjNi−1)λ+ǫ + N O(1) +i +N λ+ǫ +j +N −λk/4 +i +≲ǫ N λ+ǫ +j +N −λ +i +. +Since i = 2−k−1j, (2.2) and (2.4) implies that +N −1 +j/2k+1 ≤ Bj/2k+1N −1/2 +j/2k ≤ · · · ≤ Bj(k+1)/2k+1N −1/2k+1 +j +≲ǫ′ N −(1−(k+1)ǫ′/2)/2k+1 +j +for sufficiently small ǫ′ > 0. If we choose ǫ′ small enough such that (k + 1)ǫ′/2 < 1, then +D3p(Pj) ≲ǫ,ǫ′ N λ+ǫ−(1−(k+1)ǫ′/2)/2k+1 +j +. +The same iteration argument also works for D3p(Pj,i) and decoupling constants D3p(Pj,i′) +where i′ < i are not used there. This contradicts the assumption that λ > 0 is the smallest +which satisfies (3.14). Therefore, λ = 0. +□ +4. Local restriction estimate +We denote Bd(R) by a cube of side length R in Rd centered at the origin. We have the +following result. +Proposition 4.1. Let p = 6/α where 0 < α < 1 and ν be the measure constructed in Section +2. Then, we have +∥� +fdν∥Lp(B2(R)) ≲p,ǫ Rǫ∥f∥L2(ν). +(4.1) +Equivalently, we have +∥ �f∥L2(ν) ≲q,ǫ Rǫ∥f∥Lq(R2) +(4.2) + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +11 +where 1/q + 1/p = 1 and f is supported on B2(R). +For a ∈ A2j and ℓ ≥ 2j, let Aℓ,a be a subset of Aℓ such that +N −1/2 +ℓ +(b + [0, 1]) ⊆ N −1/2 +2j +(a + [0, 1]) +if b ∈ Aℓ,a. +In short, Aℓ,a is the set of ℓ-th level descendants of a. +Lemma 4.2. For ℓ ≥ 2j, we have the estimate +∥ +� +a∈A2j +fΩa∥Lp(B2(Nj)) ≲ǫ,p Kp(P2j)N +ǫ+ 3 +2p − α +4 +2j +N −3/4 +ℓ +|Aℓ|1/2∥ +� +a∈A2j +fΩa∥L2(R2) +(4.3) +where +�fΩa(ξ) = �f(ξ) +� +b∈Aℓ,a +1[0,1]×[−1,1](N 1/2 +ℓ +ξ1 − b, Nℓξ2 − (2b + 1)N 1/2 +ℓ +ξ1 + b2 + b). +Once we prove Lemma 4.2, we can prove Proposition 4.1 as follows. +Proof of Proposition 4.1. Let η is a Schwartz function on R such that |η| ≥ 1 on [−1, 1] and +�η is supported on [0, 1]. Let +�F(ξ) := f(ξ1, ξ2 +1) +� +a∈A2j +� +b∈Aℓ,a +1[0,1](N 1/2 +ℓ +ξ1 − b)�ηℓ(ξ2 − ξ2 +1)|Eℓ|−1 +where ηℓ(x) = η(x/N 1/2 +ℓ +) for x ∈ R. Observe that Nj ≤ N 1/2 +2j +≤ N 1/2 +ℓ +. If x ∈ B2(Nj), we +have +| � +fdνℓ(x)| ≤ | � +fdνℓ(x)ηℓ(−x2)| = |F(−x)|. +Therefore, (4.3) implies that +∥ � +fdνℓ∥Lp(B2(Nj)) ≲ ∥F∥Lp(B2(Nj)) +≲ǫ Kp(P2j)N +ǫ+ 3 +2p − α +4 +2j +N −3/4 +ℓ +|Aℓ|1/2∥F∥L2(R2) += Kp(P2j)N +ǫ+ 3 +2p − α +4 +2j +N −3/4 +ℓ +|Aℓ|1/2∥ �F∥L2(R2). +Since |Eℓ| = |Aℓ|N −1/2 +ℓ +, we get ∥ �F∥L2(R2) ≲ N 3/4 +ℓ +|Aℓ|−1/2∥f∥L2(νℓ). +Since p = 6/α, it follows from Proposition 3.5 that +∥ � +fdνℓ∥Lp(B2(Nj)) ≲ Kp(P2j)N ǫ +2j∥f∥L2(νℓ) +≲ǫ N 2ǫ +2j ∥f∥L2(νℓ). +If Nj−1 ≤ R ≤ Nj, conditions (2.2), (2.3) and (2.4) implies that +N2j ≤ B2jN 2 +j = B2jn2 +jN 2 +j−1 ≲ǫ N 2+ǫ +j−1. +Therefore, +∥ � +fdνℓ∥Lp(B2(R)) ≤ ∥ � +fdνℓ∥Lp(B2(Nj)) +≲ N 2ǫ +2j ∥f∥L2(νℓ) +≲ǫ N O(ǫ) +j−1 ∥f∥L2(νℓ) +≲ǫ RO(ǫ)∥f∥L2(νℓ). +When we take the limit ℓ → ∞, we get (4.1) and by duality, (4.2) follows. +□ + +12 +DONGGEUN RYOU +Now let us turn to the proof of Lemma 4.2. +Proof of Lemma 4.2. Let η is a Schwartz function on R2 such that |η| ≥ 1 on [−1, 1]2 and �η +is supported on [0, 1]2 and write η2j(x) = η(x/N 1/2 +2j ). Since Nj ≤ N 1/2 +2j , we have +∥ +� +a∈A2j +fΩa∥Lp(B2(Nj)) ≤ ∥ +� +a∈A2j +fΩaη2j∥Lp(R2) += +sup +∥g∥Lq(R2)≤1 +� +� +a∈A2j +fΩa(x)η2j(x)g(x)dx += +sup +∥g∥Lq(R2)≤1 +� +� +a∈A2j +�fΩa ∗ �η2j(ξ)�g(ξ)dξ +where 1/p + 1/q = 1. +The function �fΩa is supported on a rectangle of dimensions O(N −1/2 +2j +) × O(N −1 +2j ) centered +at ((a + 1/2)N −1/2 +2j +, (a + 1/2)2N −1 +2j ) and the direction that it is pointing depends on a. The +function �η2j is supported on a square with side length O(N −1/2 +2j +). Therefore, �fΩa ∗ �η2j(ξ) is +supported on a square with side length O(N −1/2 +2j +) centered at (aN −1/2 +2j +, a2N −1 +2j ). For a ∈ A2j +and c = (c1, c2) ∈ Z2, let us consider characteristic functions +1Qa,c(ξ) = 1[0,1]2(N 1/2 +2j ξ1 − (a + c1), N 1/2 +2j ξ2 − (a2 + c2)N −1/2 +2j +). +Then, we have +∥ +� +a∈A2j +fΩa∥Lp(B2(Nj)) ≤ +sup +∥g∥Lq(R2)≤1 +� +|c|≲1 +� +� +a∈A2j +�fΩa ∗ �η2j(ξ)�g(ξ)1Qa,c(ξ)dξ. +By letting +z1 = N 1/2 +2j ξ1 − (a + c1) +and +z2 = N 1/2 +2j ξ2 − (a2 + c2)N −1/2 +2j +, +(4.4) +we obtain +sup +∥g∥Lq(R2)≤1 +� +|c|≲1 +� +[0,1]2 +� +a∈A2j +�fΩa ∗ �η2j(N −1/2 +2j +(z1 + a + c1), N −1/2 +2j +z2 + (a2 + c2)N −1 +2j ) +�g(N −1/2 +2j +(z1 + a + c1), N −1/2 +2j +z2 + (a2 + c2)N −1 +2j )dzN −1 +2j . +Let a′ = (a′ +1, a′ +2) ∈ Z2 and a′ · u = a′ +1u1 + a′ +2u2. Then, we have +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +� +[0,1]2 e(au1 + a2u2)e(−a′ · u)du = 1 +(4.5) +if and only if a′ +1 = a and a′ +2 = a2 for |a| ≲ N 1/2 +2j . Let +Fc(u, z) := +� +a∈A2j +�fΩa ∗ �η2j(N −1/2 +2j +(z1 + a + c1), N −1/2 +2j +z2 + (a2 + c2)N −1 +2j )e(au1 + a2u2) +and +Gc(u, z) := +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +�g(N −1/2 +2j +(z1 + a′ +1 + c1), N −1/2 +2j +z2 + (a′ +2 + c2)N −1 +2j )e(−a′ · u). + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +13 +Also, we denote the mixed norm of f by +∥f∥Lp1 +u Lp2 +z = +� � +[0,1]2 +� � +[0,1]2 |f(u, z)|p1du +�p2/p1 +dz +�1/p2 +. +By (4.5) and H¨older inequality, we have +∥ +� +a∈A2j +fΩa∥Lp(B2(Nj)) ≤ +sup +∥g∥Lq(R2)≤1 +� +|c|≲1 +� +[0,1]2 +� +[0,1]2 Fc(u, z)Gc(u, z)dudzN −1 +2j +≤ +sup +∥g∥Lq(R2)≤1 +� +|c|≲1 +∥Fc(u, z)∥Lp +uL2z∥Gc(u, z)∥Lq +uL2zN −1 +2j . +(4.6) +Since Fc(u, z) is a Fourier series with respect to u variable, the definition of Kp(P2j) and +(4.4) implies that +∥Fc∥Lp +uL2z ≤ Kp(P2j)∥Fc∥L2uL2z +≤ Kp(P2j)N 1/2 +2j +� � +� +a∈A2j +| �fΩa ∗ �η2j(ξ)|2dξ +�1/2 +. +(4.7) +In order to obtain an estimate of ∥Gc∥Lq +uL2z, we consider Gc as a linear operator T : Lq(R2) → +Lq +uL2 +z acting on g, which is defined by +Tg(u, z) = +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +� +g(x)e(N −1/2 +2j +z1x1 + N −1/2 +2j +z2x2) +e(N −1/2 +2j +(a′ +1 + c1)x1 + (a′ +2 + c2)N −1 +2k x2)e(−a′ · u)dx. +If we show that +∥Tg∥L1uL2z ≲ǫ N ǫ +2j∥g∥L1(R2) +(4.8) +and +∥Tg∥L2uL2z ≲ N 3/4 +2j ∥g∥L2(R2), +(4.9) +then the mixed norm interpolation theorem (see [2, Theorem 2 in Section 7]) implies that +∥Gc(u, z)∥Lq +uL2z = ∥Tg∥Lq +uL2z ≲ǫ N ǫ+3/2p +2j +∥g∥Lq(R2). +(4.10) +When q = 1, let us consider the case c = 0. We have +Tg(u, z) = +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +� +R2 g(N 1/2 +2j x1, N2jx2)e(x1z1 + N 1/2 +2j x2z2)N 3/2 +2j e(−a′ · (u − x))dx. +Note that � +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j e(−a′ · x) is a product of Dirichlet kernels, since +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +e(−a′ · x) = +� +|a′ +1|≲N1/2 +2j +e(−a′ +1x1) +� +|a′ +2|≲N2j +e(−a′ +2x2). +Thus, we have +∥ +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +e(−a′ · x)∥L1(du) ≲ log(N 1/2 +2j ) log(N2j). + +14 +DONGGEUN RYOU +Therefore, we obtain that +∥Tg∥L1(du) ≲ +� +R2 +� +[0,1] +����g(N 1/2 +2j x1, N2jx2)N 3/2 +2j +���� +���� +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +e(−a′ · (x − u)) +����dudx +≲ log(N2j)2 +� +R2 +����g(N 1/2 +2j x1, N2jx2)N 3/2 +2j +����dx +≲ǫ N ǫ +2j∥g∥L1(R2). +(4.11) +Since the estimate (4.11) holds uniformly on z, we established (4.8). When c ̸= 0, we can +also get (4.8) by the similar argument. +When q = 2, since Tg(u, z) is a Fourier series with respect to u variable, we can use +Plancherel’s theorem. By (4.4), we have +∥Tg∥L2uL2z = +� � +[0,1]2 +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +|�g(N −1/2 +2j +(z1 + a′ +1 + c1), N −1/2 +2j +z2 + (a′ +2 + c2)N −1 +2j )|2dz +�1/2 += N 1/2 +2j +� � +|�g(ξ)|2 +� +|a′ +1|≲N1/2 +2j ,|a′ +2|≲N2j +1Qa′c(ξ)dξ +�1/2 +where +1Qa′,c(ξ) = 1[0,1]2(N 1/2 +2j ξ1 − (a′ +1 + c1), N 1/2 +2j ξ2 − (a′ +2 + c2)N −1/2 +2j +). +The function 1Qa′,c(ξ) is supported on a square with side length O(N −1/2 +2j +) centered at +(N −1/2 +2j +(a′ +1 + c1), N −1 +2j (a′ +2 + c2)). These squares overlap at most O(N 1/2 +2j ) times. Therefore, +we have +∥Tg∥L2uL2z ≲ N 3/4 +2j ∥�g∥L2(R2) ≤ N 3/4 +2j ∥g∥L2(R2). +Therefore, we established (4.9). +Combining (4.6), (4.7) and (4.10), we obtain that +∥ +� +a∈A2j +fΩa∥Lp(B2(Nj)) ≲ǫ Kp(P2j)N +ǫ+ 3 +2p − 1 +2 +2j +� � +� +a∈A2j +| �fΩa ∗ �η2j(ξ)|2dξ +�1/2 +. +(4.12) +We define +1a,b(ξ) := +� +b∈Aℓ,a +1[0,1]×[−1,1](N 1/2 +ℓ +ξ1 − b, Nℓξ2 − (2b + 1)N 1/2 +ℓ +ξ1 + b2 + b) +so that �fΩa(ξ) = �fΩa1a,b(ξ). +By H¨older’s inequality, we have +| �fΩa ∗ �η2j(ξ)|2 ≤ (| �fΩa1a,b| ∗ |�η2j|(ξ))2 +≤ (| �fΩa|2 ∗ |�η2j|(ξ))(1a,b ∗ |�η2j|(ξ)). +Since |�η2j| ≲ N2j, for fixed ξ, we have +(1a,b ∗ |�η2j|(ξ)) ≤ ∥�η2j∥L∞(R2)∥1a,b∥L1(R2) ≲ N2jN −3/2 +ℓ +|Aℓ| +|A2j|. + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +15 +Since N α/2 +2j +≤ |A2j| and ∥�η2j∥L1(R2) ≲ 1, we obtain +� +� +a∈A2j +| �fΩa ∗ �η2j(ξ)|2dξ ≲ N 1−α/2 +2j +N −3/2 +ℓ +|Aℓ| +� +� +a∈A2j +| �fΩa|2 ∗ |�η2j|(ξ)dξ +≲ N 1−α/2 +2j +N −3/2 +ℓ +|Aℓ| +� +a∈A2j +∥ �fΩa∥2 +L2(R2) +≲ N 1−α/2 +2j +N −3/2 +ℓ +|Aℓ|∥ +� +a∈A2j +�fΩa∥2 +L2(R2). +(4.13) +By (4.12) and (4.13), we obtain (4.3). +□ +5. Fourier decay +We abbreviate almost surely to a.s. When we say an inequality holds a.s., the corresponding +implicit constant may depend on the measure ν. Let us consider a nondecreasing sequence +{mj}j∈N such that 2 ≤ mj. Let Mj := mj · · · m1, M0 = 1 and assume that ∀ǫ > 0 and +∀j ∈ N, +mj+1 ≲ǫ Mǫ +j . +(5.1) +Let Ij be the collection of M−1 +j +-intervals such that +Ij = {M−1 +j +(k + [0, 1]) : 0 ≤ k ≤ Mj − 1}. +We consider a sequence of random functions µj which satisfies the following conditions for +some deterministic nondecreasing sequence {βj}j∈N: +(M1) µ0 = 1[0,1]. +(M2) µj = βj1Ej where Ej is a union of intervals in Ij. +(M3) E(µj+1(x)|Ej) = µj(x) for all x ∈ [0, 1]. +(M4) For I ∈ Ij+1, the random variables µj+1(I) are jointly independent, conditioned on +Ej. +We will identify µj with the measures µjdx. Let α0 be a number such that +1 − α0 = lim +j→∞ +log βj +log Mj +. +(5.2) +Shmerkin and Suomala showed that µj converges weakly to a measure µ supported on [0, 1] +and the support of the measure µ is a Salem set a.s. so that it satisfies |�µ(ξ)| ≲σ (1+|ξ|)−σ/2 +for all σ < α0, see [18, Theorem 4.2] and [19, Theorem 14.1]. +Now, let νj be a measure defined as +� +f(x1, x2)dνj := +� +f(x1, x2 +1)µj(x1)dx1. +Similarly, νj converges weakly to a measure ν supported on P1. Since ν is on the parabola, +we will use their proof with estimates for oscillatory integrals. +Proposition 5.1. Suppose that µj is a sequence of random measures satisfies (M1)-(M4). +For any 0 < σ < α0, the limit measure ν satisfies the following inequality a.s. +|�ν(ξ)| ≲σ,ǫ (1 + |ξ|)−σ/2+ǫ +for all ξ ̸= 0. +(5.3) +One of main ingredients of the proof is Hoeffding’s inequality. + +16 +DONGGEUN RYOU +Theorem 5.2 (Hoeffding’s inequality [12]). Let X1, · · · , Xn be be independent random vari- +ables such that ai ≤ Xi ≤ bi and Sn := X1 + · · · + Xn. For t > 0, +P +�����Sn − E(Sn) +���� > t +� +≤ 2 exp +� +−2t2 +�n +i=1(bi − ai)2 +� +. +(5.4) +For fixed ξ such that |ξ| ≥ 1, let x0 be the point in [0, 1] such that ξ1 + 2xξ2 = 0. Then, +we can prove the following lemma. +Lemma 5.3. For fixed ξ, we have the following tail bounds. +(1) If |ξ| ≤ Mj+1, there exists a constant C1 > 0 such that +P +� +|�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 +j +∥µj∥1/2 +����Ej +� +≲ exp(−C1M1−σ +j +βjβ−2 +j+1). +(2) If Mj+1 ≤ |ξ| ≤ mj+1M2 +j , let k be an integer such that 0 ≤ k ≤ j − 1 and +Mj+1Mk ≤ |ξ| ≤ Mj+1Mk+1. +(5.5) +For such k, there exists a constant C2 > 0 such that +P +� +|�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 +j +� +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) +�1/2����Ej +� +≲ exp(−C2M1−σ +j +βjβ−2 +j+1) +where Ii(x0) is an interval of length 2M−1 +i +centered at x0. +Proof. Given I ∈ Ij, let Pj+1(I) is the collection of M−1 +j+1-intervals in Ij+1 that make up I. +For I ∈ Ij, we consider +XI = +� +I +βj+11Ej+1e(−xξ1 − x2ξ2)dx +for ξ = (ξ1, ξ2) ∈ R2. Then, � +I∈Ij XI = �νj+1(ξ) and (M3) implies that +E( +� +I∈Ij +XI|Ej) = E(�νj+1(ξ)|Ej) += +� +e(−xξ1 − x2ξ2)E(µj+1(x)|Ej)dx += +� +e(−xξ1 − x2ξ2)µj(x)dx = �νj(ξ). +Let SI := � +I∈Ij XI, then +|SI − E(SI)| = |�νj+1(ξ) − �νj(ξ)| +conditioned on Ej. +Now, we need to estimate |XI| where I ⊆ Ej. +If |XI| < CI for some constant CI, let +aI = −CI and bI = CI so that � +I⊆Ej(bI − aI)2 ≈ � +I⊆Ej C2 +I . Then, we can plug it in +Hoeffding’s inequality. +First, when |ξ| ≤ Mj+1, we have +|XI| ≤ βj+1M−1 +j +. +(5.6) +Since the number of I ⊆ Ej is Mjβ−1 +j ∥µj∥, we obbtain +� +I⊆Ej +(bI − aI)2 = +β2 +j+1 +βjMj +∥µj∥. + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +17 +If t = M−σ/2 +j +∥µj∥1/2, then Hoeffding’s inequality implies that +P +� +|�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 +j +∥µj∥1/2 +� +≲ exp(−C1M1−σ +j +βjβ−2 +j+1) +for some constant C1 > 0. +Second, let us consider when Mj+1 ≤ |ξ| ≤ mj+1M2 +j . We have +XI = +� +I′⊂I∩Ej+1 +βj+1 +� +I′ e(−xξ1 − x2ξ2)dx +(5.7) +where I′ = M−1 +j +aj + M−1 +j+1(aj+1 + [0, 1]) for some 0 ≤ aj ≤ Mj − 1 and 0 ≤ aj+1 ≤ mj+1 − 1. +Thus, we get +���� +� +I′ e(−xξ1−x2ξ2)dx +���� = +����M−1 +j+1 +� 1 +0 +e(−M−1 +j+1ξ1x−2M−2 +j+1(ajmj+1+aj+1)ξ2x−M−2 +j+1ξ2x2)dx +����. +(5.8) +Let its phase be +Φ(x) := M−1 +j+1ξ1x + 2M−2 +j+1(ajmj+1 + aj+1)ξ2x + M−2 +j+1ξ2x2, +then +Φ′(x) = 2M−2 +j+1ξ2(x + ajmj+1 + aj+1) + M−1 +j+1ξ1. +It suffices to consider when |ξ| ≳ 1. Then, for fixed ξ, Φ′(x) = 0 can happen in at most two +intervals I′ which are in the same interval I or in two neighboring intervals I, since 0 ≤ x ≤ 1 +and ajmj+1 + aj+1 are integers. Without loss of generality, we can assume that ξ1 = 0 so +that Φ′(x) = 0 can happen when aj = aj+1 = 0 and x = 0. Let us write I′ +0 = [0, M−1 +j+1], +I′ +1 = [M−1 +j+1, 2M−1 +j+1] and I0 = [0, M−1 +j +]. +If I′ ̸= I′ +0, I′ +1, then Φ′(x) ̸= 0. Thus, we have +���� +� +I′ e(−xξ1 − x2ξ2)dx +���� ≲ Mj+1|ξ2(ajmj+1 + aj+1)|−1. +(5.9) +Here, we used the principle of non-stationary phase which is same as [20, Chapter 8, Propo- +sition 2.2]. +Therefore, if I ̸= I0, we get +|XI| ≲ βj+1Mj+1 +mj+1−1 +� +aj+1=0 +|(ajmj+1 + aj+1)ξ2|−1 +≲ βj+1Mj+1|ξ|−1 log(1 + a−1 +j ) +≲ βj+1Mj+1|ξ|−1a−1 +j . +(5.10) +When Mj+1 ≤ |ξ| ≤ mj+1M2 +j , the upper bound of |XI| in (5.10) is not always smaller than +the upper bound in (5.6). Thus, we consider Ij+1(ξ) := Mj+1|ξ|−1[−1, 1]. If I ⊆ Ij+1(ξ), we +use (5.6). Otherwise, we use (5.10). Note that if I ⊆ Ij+1(ξ), then +M−1 +j +aj ≤ Mj+1|ξ|−1 +(5.11) +and it implies that +βj+1M−1 +j +≲ βj+1Mj+1|ξ|−1a−1 +j . + +18 +DONGGEUN RYOU +Also, if I ̸⊆ Ij+1(ξ) , then I ̸= I0. Now, we obtain that +� +I⊆Ej +(bI − aI)2 = +� +I⊆Ej∩Ij+1(ξ) +(bI − aI)2 + +� +I⊆Ej\Ij+1(ξ) +(bI − aI)2 +≲ β2 +j+1M−2 +j +� +aj≤mj+1M2 +j |ξ|−1 +1 + β2 +j+1M2 +j+1|ξ|−2 +� +aj≥mj+1M2 +j |ξ|−1 +a−2 +j . +(5.12) +If aj ≤ mj+1M2 +j |ξ|−1, (5.5) implies that aj ≤ M−1 +k Mj. Therefore, we get +� +aj≤mj+1M2 +j |ξ|−1 +1 ≲ β−1 +j Mjµj([−M−1 +k , M−1 +k ]). +(5.13) +If aj ≥ mj+1M2 +j |ξ|−1, (5.5) implies that aj ≥ M−1 +k+1Mj. Therefore, we have +� +aj≥mj+1M2 +j |ξ|−1 +a−2 +j +≤ +k +� +i=0 +� +M−1 +i+1Mj≤aj≤M−1 +i +Mj +a−2 +j +≤ +k +� +i=0 +M2 +i+1 +M2 +j +� +M−1 +i+1Mj≤aj≤M−1 +i +Mj +1 +≲ +k +� +i=0 +M2 +i+1 +M2 +j +β−1 +j Mjµj([−M−1 +i +, M−1 +i +]) +≤ +k +� +i=0 +M2 +i+1 +βjMj +µj([−M−1 +i +, M−1 +i +]). +Since |ξ|−1 ≤ M−1 +j+1M−1 +k , we obtain that +β2 +j+1M2 +j+1|ξ|−2 +� +aj≥mj+1M2 +j |ξ|−1 +a−2 +j +≤ +β2 +j+1 +βjMj +k +� +i=0 +�Mi+1 +Mk +�2 +µj([−M−1 +i +, M−1 +i +]). +(5.14) +Combining (5.12), (5.13) and (5.14), we get +� +I⊆Ej +(bI − aI)2 ≲ +β2 +j+1 +βjMj +k +� +i=0 +�Mi+1 +Mk +�2 +µj([−M−1 +i +, M−1 +i +]). +If we let +t = M−σ/2 +j +� +k +� +i=0 +�Mi+1 +Mk +�2 +µj([−M−1 +i +, M−1 +i +]) +�1/2 +, +Hoeffding’s inequality implies that +P +� +|�νj+1(ξ) − �νj(ξ)| ≥ t +����Ej +� +≲ exp(−C2M1−σ +j +βjβ−2 +j+1) +for some constant C2 > 0. +□ +Lemma 5.4. If |ξ| ≥ mj+1M2 +j , we have the estimate +|�νj+1(ξ) − �νj(ξ)| ≲ βj+1m1/2 +j+1 log(Mj)|ξ|−1/2. +(5.15) + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +19 +Proof. Let us consider (5.7) and (5.8) again. Similarly, we can assume that ξ1 = 0 so that +Φ′(x) = 0 at x = 0 without loss of generality. If I′ = I′ +0, since +Φ′′(x) = 2M−2 +j+1ξ2 ̸= 0, +we obtain that +���� +� +I′ +0 +e(−xξ1 − x2ξ2)dx +���� ≲ |ξ|−1/2. +(5.16) +Here, we used the principle of stationary phase which is same as [20, Chapter 8, Proposition +2.3]. The same inequality holds for I′ +1 also. +Since |ξ| ≥ mj+1M2 +j , by using (5.9) and (5.16), we obtain that +|XI0| ≲ βj+1 +� +|ξ|−1/2 + Mj+1 +mj+1−1 +� +aj+1=2 +|aj+1ξ2|−1 +� +≲ βj+1 +� +|ξ|−1/2 + m1/2 +j+1 log(mj+1)|ξ|−1 +� +≲ βj+1m1/2 +j+1 log(mj+1)|ξ|−1/2. +(5.17) +For I ̸= I0, by (5.10) and the assumption that |ξ| ≥ mj+1M2 +j , we get that +|XI| ≲ βj+1m1/2 +j+1|ξ|−1/2a−1 +j . +(5.18) +By (5.17) and (5.18), we have +|�νj+1(ξ)| ≤ +� +I∈Ij +|XI| ≲ βj+1m1/2 +j+1 log(Mj)|ξ|−1/2. +Similarly, we obtain that +|�νj(ξ)| ≲ βj log(Mj)|ξ|−1/2. +Therefore, (5.15) follows from triangular inequality. +□ +Proof of Lemma 5.1. As in [18] and [19], by Lemma 9A4 in [24], it suffices to consider ξ ∈ Z2. +Let Ωj be the event that there exists ξ ∈ Z2 such that +|�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 +j +∥µj∥1/2 +where |ξ| ≤ Mj+1 +or +|�νj+1(ξ) − �νj(ξ)| ≳ M−σ/2 +j +� +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) +�1/2 +where Mj+1 ≤ |ξ| ≤ mj+1M2 +j +or +|�νj+1(ξ) − �νj(ξ)| ≳ βj+1m1/2 +j+1 log(Mj)|ξ|−1/2 +where |ξ| ≥ mj+1M2 +j . +By Lemma 5.3 and Lemma 5.4, we have +P(Ωj) ≲ m2 +j+1M4 +j exp(−CM1−σ +j +βjβ−2 +j+1) +(5.19) +for some constant C > 0. For sufficiently large j and sufficiently small ǫ > 0, it follows from +(5.2) that +M1−σ +j +βjβ−2 +j+1 ≥ Mα0−σ+3ǫ +j + +20 +DONGGEUN RYOU +where α0 − σ + 3ǫ > 0. Therefore, �∞ +j=1 P(Ωj) < ∞. +By the assumptions on µj, ∥µj∥ is bounded a.s. We also need an upper bound of +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) +which is uniform over k and j. Since E(µj(Ii(x0))) ≲ βiM−1 +i +and βj and mj are nondecreasing, +we obtain that +E +� +sup +0≤k≤j−1 +β−1 +k Mk +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) +� +≤ +j−1 +� +k=0 +k +� +i=0 +M2 +i+1 +βkMk +E(µj(Ii(x0))) +≤ +j−1 +� +k=0 +k +� +i=0 +βimi+1Mi+1 +βkMk +≲ +j−1 +� +k=0 +km2 +k+1 ≤ j2m2 +j+1. +Thus, it follows from Markov’s inequality that +P +� +sup +0≤k≤j−1 +β−1 +k Mk +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) ≥ j4m2 +j+1 +� +≲ j−2. +(5.20) +By Borel-Cantelli lemma, the following inequality holds a.s. +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) ≲ j4m2 +j+1βkM−1 +k . +(5.21) +where the implicit constant may depends on µ but does not depend on j and k. +It follows from (5.2) that +βk ≲ǫ M1−α0+ǫ +k +. +(5.22) +By (5.5), (5.21) and (5.22), a.s. we obtain that +k +� +i=0 +�Mi+1 +Mk +�2 +µj(Ii(x0)) ≲ǫ j4mα0+2+ǫ +j+1 +�Mj+1 +|ξ| +�α0+ǫ +≲ǫ Mα0+2ǫ +j+1 +|ξ|−α0+ǫ. +(5.23) +We apply Borel-Cantelli lemma to Ωj. Then (5.1), (5.22), (5.23) and the fact that ∥µj∥ is +a.s. bounded imply that a.s. we have the inequality +|�νj+1(ξ) − �νj(ξ)| ≲ M−σ/2 +j +if |ξ| ≤ Mj+1, +≲ǫ M(α0−σ)/2+ǫ +j +|ξ|(−α0+ǫ)/2 +if Mj+1 ≤ |ξ| ≤ mj+1M2 +j , +≲ǫ M1−α0+ǫ +j+1 +|ξ|−1/2 +if |ξ| ≥ mj+1M2 +j . +Let j1, j2 be numbers such that Mj2 ≤ |ξ| ≤ Mj2+1 and mj1+1M2 +j ≤ |ξ| ≤ mj1+2M2 +j1+1. +Then, we obtain +|�νn(ξ) − �ν0(ξ)| ≲ǫ +� +0≤j≤j1 +M1−α0+ǫ +j+1 +|ξ|−1/2 + +� +j1≤j≤j2 +M(α0−σ)/2+ǫ +j +|ξ|(−α0+ǫ)/2 ++ +� +j≥j2 +M−σ/2 +j +. + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +21 +For sufficiently small ǫ > 0, we get +|�νn(ξ) − �ν0(ξ)| ≲ǫ M1−α0+ǫ +j1+1 +|ξ|−1/2 + M(−σ+α0)/2+ǫ +j2 +|ξ|(−α0+ǫ)/2 + M−σ/2+ǫ +j2 +≲ǫ |ξ|−σ/2+ǫ. +Since |�ν0(ξ)| ≲ |ξ|−1/2, by letting n → ∞, we have (5.3). +□ +6. Global restriction estimate +As in Theorem 4 in [13], we need to randomize the choice of Sj,a. +Proposition 6.1. Let {nj}j∈N be a sequence of positive numbers which satisfies (2.1), (2.2), +(2.3) and (2.4). Let {µj}j∈N∪{0} be a sequence of random measures on [0, 1] which satisfies +the following. +(1) µ0, µ1, · · · are constructed through the process in section 2. +(2) For each j ∈ N and for each a ∈ Aj, the set Sj,a are chosen randomly and independently +from Σj with probability distribution such that +E(µj(x)|Ej−1) = µj−1(x). +Then, we get the corresponding sequence of random measures {νj}j∈N∩{0} and its limiting +measure ν satisfies all conclusions of Theorem 1.2 almost surely. +As an example of a random measure ν which satisfies conditions Proposition 6.1, we can +consider random translate of Sj,a described in [13, Section 8]. +Once we prove the following lemma, we can prove Proposition 6.1 and it implies Theorem +1.2. +Lemma 6.2. Let p = 6/α where 0 < α < 1 and assume that ν is the measure constructed in +Section 2 and satisfies (1.5). For sufficiently large R > 0, suppose that {Qi}M +i=1 be a sparse +collection of R-cubes in R2, which means that their centers x1, · · · , xM are RBMB-separated +from each other for some sufficiently large constant B which will be determined later. If f is +a function supported on ∪M +j=1Qi, then we have +∥ �f∥L2(dν) ≲q,ǫ Rǫ∥f∥Lq(R2) +(6.1) +where 1/p + 1/q = 1. +Proof. Let f = �M +i=1 fφi where φi is a Schwartz function supported on a ball of radius 2R +centered at xi and |φi(x)| ≥ 1 if x ∈ B2(xi, R). +∥ �f∥2 +L2(dν) ≤ +� +| +M +� +i=1 +� +fφi|2dν += +� +M +� +i=1 +|� +fφi|2dν + +� � +i̸=j +� +fφi� +fφjdν. +It follows from Proposition 4.1 that +M +� +i=1 +� +|� +fφi|2dν ≲q,ǫ Rǫ +M +� +i=1 +∥fφi∥2 +Lq(R2) +≤ Rǫ +� M +� +i=1 +∥fφi∥q +Lq(R2) +�2/q += Rǫ∥f∥2 +Lq(R2). +(6.2) + +22 +DONGGEUN RYOU +In the last inequality, we used that 1 ≤ q ≤ 2. For the second term, Young’s convolution +inequality implies that +� � +i̸=j +� +fφi� +fφjdν = +� +i̸=j +�� +fφi(x)fφj(y)� +dν(y − x)dxdy +≤ +� +i̸=j +∥fφi∥Lq(R2)∥fφj∥Lq(R2)∥� +dν1i,j∥Lr(R2) +(6.3) +where 2 +q + 1 +r = 2, so that r = p +2 ≥ 1 and +1i,j = 1B2(2R)(x − (xi − xj)). +Since xi and xj are separated at least by RBMB, (1.5) implies that +∥� +dν1i,j∥Lr(R2) ≲ǫ (RBMB)−α/2+ǫR2/r. +If we choose B such that +B ≥ +2 +α/2 − ǫ, +then we have +∥� +dν1i,j∥Lr(R2) ≲ 1. +(6.4) +By (6.3) and (6.4), we obtain that +� � +i̸=j +( �f ∗ �φi)( �f ∗ �φj)dν ≲ +� +i̸=j +∥fφi∥Lq(R2)∥fφj∥Lq(R2) +≤ +� M +� +i=1 +∥fφi∥2 +Lq(R2) +�1/2� M +� +j=1 +∥fφj∥2 +Lq(R2) +�1/2 +≤ ∥f∥2 +Lq(R2). +(6.5) +In the last inequality, we used again that 1 ≤ q ≤ 2. Combining (6.2) and (6.5), we have +(6.1). +□ +Let us turn to the proof of Lemma 6.1. +Proof of Proposition 6.1. The inequality (1.4) follows from Lemma 2.3. When we let n1/2 +k += +mk, N 1/2 +k += Mk, βk = |Ek|−1 and α0 = α, then {µk}k∈N∪{0} satisfies all conditions of +Proposition 5.1. Thus, �ν satisfies (1.5) almost surely. Now, we can use Tao’s epsilon removal +argument to prove (1.6). Lemma 6.2 replaces Lemma 3.2 in [23] and rest of the proof (1.6) +is same as in [23, Theorem 1.2]. +□ +7. Proof of Theorem 1.3 +The proof is based on Knapp-type example. +Proof. Since ν is supported on Pd−1, there exists a measure µ on [0, 1]d−1 such that +� +f(x)dν = +� +f(x′, |x′|2)dµ +for all ν-measurable function f on Rd where x′ ∈ Rd−1. +The assumption (1.7) implies that for any 0 < ǫ < α, +µ(B(x′, r)) ≲α,ǫ rα−ǫ +(7.1) + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +23 +for all x′ ∈ Rd−1 and ∀r > 0. +The measure µ is also supported on a set of Hausdorff +dimension α in Rd−1. Therefore, µ(B(x′, r)), ≲α,ǫ rα+ǫ cannot hold by Frostmans lemma +(see, for example [15, Theorem 2.7]). Thus, for any ǫ > 0, there exist sequences of {ai}i∈N +and {ri}i∈N such that limi→∞ ri = 0 and +rα+ǫ +i +≲α,ǫ µ(B(ai, ri)). +(7.2) +Assume that the estimate (1.8) holds for some p and q and let hi(x′) = 1[0,1]d−1(r−1 +i +(x′ −ai)). +Then, we have +� � ���� +� +hi(x′)e(−x′ · ξ′ − |x′|2ξd)dµ(x′) +���� +p +dξ +�1/p +≲p,q +� � +|hi(x′)|qdµ +�1/q +(7.3) +where ξ′ ∈ Rd−1 and ξd ∈ R. +By (7.1), we obtain that +� � +|hi(x′)|qdµ +�1/q +≲α,ǫ r(α−ǫ)/q +i +. +(7.4) +By change of variable, we get the following lower bound of the left-hand side of (7.3). +r−(d+1)/p +i +� � ���� +� +1[0,1]d−1(r−1 +i +(x′ − ai))e(−r−1 +i +x′ · ξ′ − r−2 +i +|x′|2ξd)dµ(x′) +���� +p +dξ +�1/p +≥ r−(d+1)/p +i +� � ���� +� +1[0,1]d−1(r−1 +i +(x′ − ai)) +e(−r−1 +i +(x′ − ai) · (ξ′ + 2ξdr−1 +i +ai) − r−2 +i +ξd|x′ − ai|2)dµ(x′) +���� +p +dξ +�1/p +. +(7.5) +Let us consider when |ξ′ + 2ξdr−1 +i +ai| ≤ 1/100 and |ξd| ≤ 1/100, then (7.2) and (7.5) implies +that +� � ���� +� +hi(x′)e(−x′ · ξ′ − |x′|2ξd)dµ(x′) +���� +p +dξ +�1/p +≳ r−(d+1)/p +i +���� +� +1[0,1]d−1(r−1 +i +(x′ − ai))dµ(x′) +���� ≳α,ǫ r−(d+1)/p+α+ǫ +i +. +(7.6) +Combining (7.3), (7.4) and (7.6), we get +r−(d+1)/p+α+ǫ +i +≲α,ǫ,p,q r(α−ǫ)/q +i +. +Therefore, +α +q − ǫ +q ≤ −d + 1 +p ++ α + ǫ. +Since ǫ is arbitrary, p ≥ q′(d + 1)/α if the estimate (1.8) holds. +□ +8. Appendix +Proof of Lemma 3.1. First, assume that S1 is a subset of [−k0, k0]d ∩ Zd. Let ψ be a non- +negative smooth function supported on [0, 1]d such that ψ ≥ 1 on [1/4, 3/4]d. For x ∈ [−k, k]d, +1 ≤ +� +|n|≲k +|ψ(x − n)|p + +� +|n′|≲1 +� +|n|≲k +|ψ(x − n − n′/2)|p + +24 +DONGGEUN RYOU +where n, n′ ∈ Zd, but n′ ̸= 0. Thus, we have +∥ +� +a∈S1 +� +b∈S2,a +ca,be((ka + b) · x)∥p +Lp([0,1]d) = +� +[0,k]d +���� +� +a∈S1 +� +b∈S2,a +ca,be +�� +a + b +k +� +· x +����� +p +k−ddx +≤ +� +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� +b∈S2,a +ca,be(a · x)e +� b +k · (x + n) +� +ψ(x) +���� +p +dx ++ +� +|n′|≲1 +� +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� +b∈S2,a +ca,be(a · (x + n′/2))e +� b +k · (x + n + n′/2) +� +ψ(x) +���� +p +dx. +We will show that the first term is bounded by a constant multiple of +C1C2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�p/2 +. +(8.1) +Then, the second term will be also bounded by a constant multiple of (8.1) by the same +argument. +Since ψ is supported on [0, 1]d, we have +ψ(x)e(b · x/k) = +� +m∈Zd +�ψb(m)e(m · x) +where �ψb(m) is the m-th Fourier coefficient of ψ(x)e(b · x/k). Since � +m | �ψb(m)| ≲ 1 and S1 +is a subset of [−k0, k0]d, we obtain +� � +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� +b∈S2,a +ca,be(a · x)e +� b +k · (x + n) +� +ψ(x) +���� +p +dx +�1/p +≤ +� +m∈Zd +� � +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m)e((a + m) · x) +���� +p +dx +�1/p +≤ +� +m∈Zd +� � +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� � +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +� +e(a · x) +���� +p +dx +�1/p +≤ C1 +� +m∈Zd +� � +|n|≲k +k−d +� � +a∈S1 +���� +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +���� +2�p/2�1/p +. +(8.2) + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +25 +In the last inequality, we used (3.3). +Now, let us consider when |m| ≤ 10. Since p ≥ 2 and |ψ| ≲ 1, we get +� � +|n|≲k +k−d +� � +a∈S1 +���� +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +���� +2�p/2�1/p +≤ +� � +a∈S1 +� � +|n|≲k +k−d +���� +� +b∈S2,a +ca,be +�b · n +k +� � +[0,1]d ψ(x)e +�� b +k − m +� +· x +� +dx +���� +p�2/p�1/2 +≲ +� � +a∈S1 +� +k−d � +|n|≲k +� +[0,1]d +���� +� +b∈S2,a +ca,be +� b +k · (x + n) +����� +p +dx +�2/p�1/2 +≲ +� � +a∈S1 +∥ +� +b∈S2,a +ca,be(b · x)∥2 +p([0,1]d) +�1/2 +≤ C2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�1/2 +. +(8.3) +If |m| ≥ 10, since ψ is supported on [0, 1]d and decreases rapidly, we can use the principle +of non-stationary phase. Let (b/k − m)i denote the i-th coordinate of (b/k − m) and denote +∇b,k,m by +∇b,k,m = +d +� +i=1 +���� +b +k − m +���� +−2� b +k − m +� +i +∂ +∂xi +. +Note that +�ψb(m) = +� +[0,1]d(∇2 +b,k,mψ(x))e +�� b +k − m +� +x +� +dx. +If we let +˜ca,b,i,j,m = ca,b +���� +b +k − m +���� +−4� b +k − m +� +i +� b +k − m +� +j +, +then +���� +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +���� ≤ +d +� +i,j=1 +� +[0,1]d +���� +� +b∈S2,a +˜ca,b,i,j,me +� b +k · (x + n) +� +∂ijψ(x) +����dx. +Since |∂i,jψ| ≲ 1 uniformly in i and j, we obtain that +� � +|n|≲k +k−d +� � +a∈S1 +���� +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +���� +2�p/2�1/p +≲ +d +� +i,j=1 +� � +a∈S1 +� +k−d � +|n|≲k +� +[0,1]d +���� +� +b∈S2,a +˜ca,b,i,j,me +� b +k · (x + n) +����� +p +dx +�2/p�1/2 +≲ +d +� +i,j=1 +� � +a∈S1 +∥ +� +b∈S2,a +˜ca,b,i,j,me(b · x)∥2 +p([0,1]d) +�1/2 +≤ C2 +d +� +i,j=1 +� � +a∈S1 +� +b∈S2,a +|˜ca,b,i,j,m|2 +�1/2 +≲d C2|m|−2 +� � +a∈S1 +� +b∈S2,a +|cab|2 +�1/2 +. + +26 +DONGGEUN RYOU +In the last inequality, we used that |b/k − m| ≈ |m| since |m| ≥ 10. Therefore, +� +|m|≥10 +� � +|n|≲k +k−d +� � +a∈S1 +���� +� +b∈S2,a +ca,be +�b · n +k +� +�ψb(m) +���� +2�p/2�1/p +≲ C2 +� +|m|≥10 +|m|−2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�1/2 +≲ C2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�1/2 +. +(8.4) +By (8.2), (8.3) and (8.4), it follows that +� � +|n|≲k +k−d +� +[0,1]d +���� +� +a∈S1 +� +b∈S2,a +ca,be(a · x)e +� b +k · (x + n) +� +ψ(x) +���� +p +dx +�1/p +≲ C1C2 +� � +a∈S1 +� +b∈S2,a +|ca,b|2 +�1/2 +. +This proves Lemma 3.1 when S1 is a subset of [−k0, k0]d∩Zd. The estimate holds for arbitrary +k0. Thus, letting k0 → ∞ finishes the proof. +□ +Proof of Lemma 3.2. The inequality Kp(Ej) ≲p Dp(Ej) is well known, for example, see [8, +Theorem 13.1]. Thus, it suffices to prove the converse. Let R be a positive large number and +S be a subset of [−R, R]d ∩ Zd such that +∥ +� +a∈S +cae(a · x)∥Lp([0,1]d) ≤ C +� � +a∈S +|ca|2 +�1/2 +(8.5) +and let f = � +a∈S fa where +�fa(ξ) = �f(ξ)1[0,1]d(Rξ − a). +Let us consider a Schwartz function η such that |η| ≥ 1 on [−1, 1]d and �η is supported on +[0, 1]d. +Recall that Bd(R) is a cube of side length R in Rd centered at the origin. We will show that +∥f∥Lp(Bd(R)) ≲ C +� � +a∈S +∥fa∥2 +p(ηR) +�1/2 +(8.6) +where ηR(x) = η(x/R). +And (8.6) implies that +∥f∥Lp(Rd) ≲ C +� � +a∈S +∥fa∥2 +p(Rd) +�1/2 +. +(8.7) +For the proof of (8.7) from (8.6), see [8, Proposition 9.15] and [14, Chapter 4.1]. +Then, +Lemma 3.2 easily follows. We have +∥f∥Lp(Bd(R)) ≤ ∥fηR∥Lp(Bd(R)) += ∥ +� +a∈S +� +�fa ∗ �ηR(ξ)e(−x · ξ)dξ∥Lp(Bd(R)). + +NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA +27 +The function �fa∗�ηR is supported on a cube of side length O(R−1) centered at a/R. Therefore, +∥f∥Lp(Bd(R)) ≤ ∥fηR∥Lp(Bd(R)) +≲ ∥ +� +|c|≲1 +� +a∈S +� +[0,1]d +�fa ∗ �ηR(R−1(ξ + a + c))e(−x · R−1(ξ + a + c))R−ddξ∥Lp(Bd(R)) +≤ R−d � +|c|≲1 +� +[0,1]d ∥ +� +a∈S +�fa ∗ �ηR(R−1(ξ + a + c))e(−x · R−1(ξ + a + c))∥Lp(Bd(R))dξ +≤ R−d+d/p � +|c|≲1 +� +[0,1]d ∥ +� +a∈S +�fa ∗ �ηR(R−1(ξ + a + c))e(−x · (ξ + a + c))∥Lp(Bd(1))dξ +≲ R−d+d/p � +|c|≲1 +� +[0,1]d ∥ +� +a∈S +�fa ∗ �ηR(R−1(ξ + a + c))e(−x · a)∥Lp(Bd(1))dξ. +By (8.5), we obtain that +∥f∥Lp(Bd(R)) ≲ CR−d+d/p � +|c|≲1 +� +[0,1]d( +� +a∈S +| �fa ∗ �ηR(R−1(ξ + a + c))|2)1/2dξ +≲ CR−d+d/p � +|c|≲1 +� � +a∈S +� +[0,1]d | �fa ∗ �ηR(R−1(ξ + a + c))|2dξ +�1/2 +≲ CR−d/2+d/p +� � +a∈S +∥ �fa ∗ �ηR(ξ)∥2 +L2(Rd) +�1/2 +≤ CR−d/2+d/p +� � +a∈S +∥faηR∥2 +L2(Rd) +�1/2 +≤ CR−d/2+d/p +� � +a∈S +∥fa∥2 +Lp(ηR)∥ηR∥1−2/p +L1(Rd) +�1/2 +. +Since ∥ηR∥L1 ≈ Rd, this completes the proof of (8.6). +□ +References +[1] Jong-Guk Bak and Andreas Seeger. Extensions of the Stein-Tomas theorem. Math. Res. Lett., 18(4):767– +781, 2011. +[2] A. Benedek and R. Panzone. The space Lp, with mixed norm. Duke Math. J., 28:301–324, 1961. +[3] J. Bourgain. Bounded orthogonal systems and the Λ(p)-set problem. Acta Math., 162(3-4):227–245, 1989. +[4] Alan Chang, Jaume de Dios Pont, Rachel Greenfeld, Asgar Jamneshan, Zane Kun Li, and Jos´e Madrid. +Decoupling for fractal subsets of the parabola. Math. Z., 301(2):1851–1879, 2022. +[5] Xianghong Chen. A Fourier restriction theorem based on convolution powers. Proc. Amer. Math. Soc., +142(11):3897–3901, 2014. +[6] Xianghong Chen. Sets of Salem type and sharpness of the L2-Fourier restriction theorem. Trans. Amer. +Math. Soc., 368(3):1959–1977, 2016. +[7] Xianghong Chen and Andreas Seeger. Convolution powers of Salem measures with applications. Canad. +J. Math., 69(2):284–320, 2017. +[8] Ciprian Demeter. Fourier restriction, decoupling, and applications, volume 184 of Cambridge Studies in +Advanced Mathematics. Cambridge University Press, Cambridge, 2020. +[9] Rick Durrett. Probability—theory and examples, volume 49 of Cambridge Series in Statistical and Prob- +abilistic Mathematics. Cambridge University Press, Cambridge, 2019. Fifth edition of [ MR1068527]. + +28 +DONGGEUN RYOU +[10] Kyle Hambrook and Izabella �Laba. On the sharpness of Mockenhaupt’s restriction theorem. Geom. Funct. +Anal., 23(4):1262–1277, 2013. +[11] Kyle Hambrook and Izabella �Laba. Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger restriction theorem +in higher dimensions. Bull. Lond. Math. Soc., 48(5):757–770, 2016. +[12] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. +Assoc., 58:13–30, 1963. +[13] Izabella �Laba and Hong Wang. Decoupling and near-optimal restriction estimates for Cantor sets. Int. +Math. Res. Not. IMRN, 2018(9):2944–2966, 2018. +[14] Zane Kun Li. Decoupling for the parabola and connections to efficient congruencing. University of Cali- +fornia, Los Angeles, 2019. +[15] Pertti Mattila. Fourier analysis and Hausdorff dimension, volume 150 of Cambridge Studies in Advanced +Mathematics. Cambridge University Press, Cambridge, 2015. +[16] Themis Mitsis. A Stein-Tomas restriction theorem for general measures. Publ. Math. Debrecen, 60(1- +2):89–99, 2002. +[17] G. Mockenhaupt. Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal., +10(6):1579–1587, 2000. +[18] Pablo Shmerkin and Ville Suomala. A class of random Cantor measures, with applications. In Recent +developments in fractals and related fields, Trends Math., pages 233–260. Birkh¨auser/Springer, Cham, +2017. +[19] Pablo Shmerkin and Ville Suomala. Spatially independent martingales, intersections, and applications. +Mem. Amer. Math. Soc., 251(1195):v+102, 2018. +[20] Elias M. Stein and Rami Shakarchi. Functional analysis, volume 4 of Princeton Lectures in Analysis. +Princeton University Press, Princeton, NJ, 2011. Introduction to further topics in analysis. +[21] Michel Talagrand. Sections of smooth convex bodies via majorizing measures. Acta Math., 175(2):273–300, +1995. +[22] Michel Talagrand. Upper and lower bounds for stochastic processes. decomposition theorems. Ergebnisse +der Mathematik und ihrer Grenzgebiete, 60, 2021. +[23] Terence Tao. The Bochner-Riesz conjecture implies the restriction conjecture. Duke Math. J., 96(2):363– +375, 1999. +[24] Thomas H. Wolff. Lectures on harmonic analysis, volume 29 of University Lecture Series. American +Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by +Izabella �Laba, Edited by �Laba and Carol Shubin. +Department of Mathematics, University of Rochester, Rochester, NY, USA +Email address: dryou@ur.rochester.edu + diff --git a/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/load_file.txt b/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..334826733552db381bea7db50775d726e14186d9 --- /dev/null +++ b/_9FAT4oBgHgl3EQfrB3j/content/tmp_files/load_file.txt @@ -0,0 +1,953 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf,len=952 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='08651v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='CA] 20 Jan 2023 NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA DONGGEUN RYOU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For any 0 < α < 1, we construct Cantor sets on the parabola of Hausdorff dimension α such that they are Salem sets and each associated measure ν satisfies the estimate ∥� fdν∥Lp(R2) ≤ Cp∥f∥L2(ν) for all p > 6/α and for some constant Cp > 0 which may depend on p and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The range p > 6/α is optimal except for the endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This is an analogue of works of Shmerkin-Suomala and �Laba-Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' They considered fractal subsets of Rd, whereas we consider fractal subsets of the parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Introduction Let µ is a Borel probability measure in Rd and assume that it satisfies the estimate ∥� fdµ∥Lp(R2) ≤ Cp∥f∥L2(µ) ∀f ∈ L2(µ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) for some constant Cp which depends on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' It is easy to show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds when p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' However, we can say more if we focus on more specific measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' One typical example is a surface carried measure on the sphere or on the paraboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds when p ≥ 2(d + 1)/(d − 1) and the range is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This is the result of Tomas-Stein theorem (for example, see [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let B(x, r) is a closed ball of radius r centered at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we have assumptions on the Fourier decay of �µ and the upper bound of µ(B(x, r)), the following result generalizes Tomas-Stein theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Assume that there exist a, b ∈ (0, d) and C1, C2 ≥ 0 such that µ(B(x, r)) ≤ C1ra ∀x ∈ Rd, r > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) |�µ(ξ)| ≤ C2(1 + |ξ|)−b/2 ∀ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) Then, the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds when p ≥ (4d − 4a + 2b)/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Mockenhaupt [17] and Mitsis [16] proved that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds when p > (4d − 4a + 2b)/b and the endpoint result was proved by Bak and Seeger [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If µ is a surface carried measure on the sphere or the paraboloid, a = b = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 recovers the range p ≥ 2(d + 1)/(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Define the critical exponent pc of the measure µ by pc = inf{p : (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 means pc ≤ (4d − 4a + 2b)/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This upper bound of pc is known to be optimal when d − 1 < b ≤ a < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When d = 1, Hambrook and �Laba [10] constructed a measure such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) holds for every β < α and pc = (4−2a)/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Chen [6] extended this result to general 0 ≤ b ≤ a ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In higher dimensions, it was done in [11] by Hambrook and �Laba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 42B10 (primary) 28A80 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Random Cantor sets, Restriction estimate, Salem set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 1 2 DONGGEUN RYOU However, there exists some measures µ such that pc is strictly smaller than (4d − 4a + 2b)/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In [10], Hambrook and �Laba showed that pc ≥ 2d/α if µ is supported on a set of Hausdorff dimension α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that b ≤ a ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) holds for values arbitrarily close to α, the support of µ is called a Salem set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If 0 < α < d, we have the inequality 2d α < 4d − 2α α ≤ 4d − 4a + 2b b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, 2d/α is smaller than (4d − 4a + 2b)/b even when µ is supported on a Salemt set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Examples such that pc = 2d/α were provided by Chen [5], Chen and Seeger [7], Shmerkin and Suomala [19] (see also [18]) and �Laba and Wang [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, the lower bound of pc is optimal for all 0 < α < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In [5] and [7], Chen and Seeger constructed measures supported on a Salem set of Hausdorff dimension α such that pc = 2d/α where α = d/k and k is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' They considered k-fold self convolution of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Shmerkin and Suomala [19] constructed the example through random fractal measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Their result covers when d = 1 and 1/2 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' �Laba and Wang [13] constructed measures of Hausdorff dimension α such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) holds for β < min(α/2, 1) and pc = 2d/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' They used Λ(p)-set and decoupling and this construction works for all α such that 0 < α < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' However, the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds when p > 2d/α, while results of Chen, Chen-Seeger and Shmerkin-Suomala works even at the endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, it is still open whether, for any 0 < α < d, there exists a measure µ such that its support has Hausdorff dimension α and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds for p = 2d/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, let us turn to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Earlier works constructed fractal measures on Rd, but we consider fractal measures on the parabola P1 := {(x, x2) : 0 ≤ x ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Throughout the paper, we denote by X ≲ Y when X ≤ CY for some constant C > 0 and we write X ≈ Y to denote that X ≲ Y and Y ≲ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If the constant C depends on parameters such as ǫ, we write X(ǫ) ≲ǫ Y (ǫ) instead of X(ǫ) ≤ C(ǫ)Y (ǫ) where the constant C(ǫ) depends on ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Our main results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' There exists a probability measure ν supported on a subset of P1 which satisfies the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1) The support of the measure ν has Hausdorff dimension α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2) For any 0 < ǫ < α, we have ν(B(x, r)) ≲α,ǫ rα−ǫ ∀x ∈ R2, ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) (3) For any ǫ > 0, |�ν(ξ)| ≲α,ǫ (1 + |ξ|)−α/2+ǫ ∀ξ ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) (4) For every p > 6/α, we have the estimate ∥� fdν∥Lp(R2) ≲p ∥f∥L2(ν) ∀f ∈ L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) Equivalently, for any 1 ≤ q < 6/(6 − α) we have ∥ �f∥L2(ν) ≲q ∥f∥Lq(R2) ∀f ∈ Lq(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that the support of ν is a Salem set and it is a subset of the parabola, which is also a Salem set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 is sharp except the endpoint in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Pd−1 := {(x, |x|2 : x ∈ [0, 1]d−1} and 0 < α < d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Assume that ν is a probability measure supported on a subset of Pd−1 whose Hausdorff dimension is α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For any 0 < ǫ < α, assume that ν(B(x, r)) ≲α,ǫ rα−ǫ ∀x ∈ Pd−1, ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 3 Then, the estimate ∥� fdν∥Lp(R2) ≲p,q ∥f∥Lq(ν) ∀f ∈ Lq(ν) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) cannot hold if p < q′(d + 1)/α where 1/q + 1/q′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When d = 2 and q = 2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3 implies that pc ≥ 6/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ν is on the parabola, the lower bound of pc increased from 4/α to 6/α when compared to the result by Hambrook and �Laba [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If 0 < α < 1, then 6/α < (8 − 2α)/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, it is still smaller than the upper bound of pc obtained from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We will follow the proof of [13], but we cannot apply it directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Main obstacle is that we cannot make use of dual boxes as in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For example, they decompose the support of �f into cubes of side length R−1 and integrate f over the cube of side length R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This duality was used to establish decoupling estimate and localized restriction estimate which are proposition 1 and corollary 2 in [13] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' However, P1 can be covered by rectangles of dimensions R−1/2, R−1 and each rectangles corresponds to dual rectangles of dimensions R1/2, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since these dual rectangles have their long side in different directions, union of them is a square of side length R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we decompose the support of �f into rectangles of dimensions R−1/2, R−1 while we integrate f over the cube of side length R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we should use dual rectangles in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In Section 2, we construct Cantor sets on [0, 1] by using Λ(p)-sets and consider associated measures which is supported on a subset of parabola above these Cantor sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let ν be a measure constructed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then ν is supported on a set of Hausdorff dimension α and satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In section 3, we obtain decoupling inequality for the support of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We use Λ(p)-sets and the result of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In Section 4, we obtain local restriction estimate of ν which is an analogue of Corollary 2 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' However, we use mixed norm interpolation [2] in addition to the argument in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In Section 5, we obtain the Fourier decay of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We only need negative exponent in order to derive (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6), but it turned out that �ν has decay arbitrarily close to optimal almost surely as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We use the argument in [18], but modified their method in a way that we can use oscillatory integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The local restriction estimate and the Fourier decay leads to global restriction estimate in Section 6, which is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Tao’s epsilon removal lemma [23] is used as in [13], but we simplify the proof of Lemma 9 in [13] in order not to use dual rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we prove that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 happens almost surely if ν is a random measure constructed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lastly, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3 in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The author would like to thank his advisor Alex Iosevich for many discussions of this work and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The author would also like to thank Shaoming Guo, Zane Kun Li for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The construction of the Cantor set Our proof uses the following construction of Cantor measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let {nj}j∈N∪{0} be a se- quence of positive integers and let Nj = n0 · · · nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We assume the following conditions on nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' n0 = 1 and 2 ≤ n1 ≤ n2 ≤ · · · ≤ nj ≤ · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) ∀a > 0, aj ≲a Nj, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) ∀ǫ > 0 and ∀j ∈ N, nj+1 ≲ǫ N ǫ j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) ∃B such that ∀j ∈ N, N 1/2 2j N −1 j ≤ Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) 4 DONGGEUN RYOU For example, if nj ≈ j, all conditions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) are satisfied by Stirling’s formula (see [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) was not assumed in [13], but we need it additionally in Section 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We need the following theorem in order to use Λ(p)-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 (Existence of the Λ(p)-set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let e(x) denote e2πix and p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For every N ∈ N, there exists a set S ⊂ {0, 2, · · · , N − 1} such that N 2/p ≤ |S| < N 2/p + 1 and ∥ � a∈S cae(ax)∥Lp([0,1]) ≤ Cp � � a∈S |ca|2 �1/2 ∀{ca}a∈S ∈ ℓ2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) for some constant Cp > 0 which depends only on p, but not on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 was first proved by Bourgain [3] and another proof was given by Talagrand [21], see also [22, Section 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3] for simpler proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let 0 < α < 1 and for each j ∈ N, let {tj}N be a sequence of integers such that nα/2 j ≤ tj < nα/2 j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Define Σj := {S ⊂ [0, n1/2 j ] ∩ Z : |S| = tj and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) holds for p = 2/α, N = n1/2 j }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The set Σj is non-empty because of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let E0 = [0, 1] and for S1 ∈ Σ1, let A1 = S1, E1 = N −1/2 1 (A1 + [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For each a ∈ Aj, choose a set Sj+1,a ∈ Σj+1 and let Aj+1,a = n1/2 j+1a + Sj+1,a Aj+1 = ∪a∈AjAj+1,a Ej+1 = N −1/2 j+1 (Aj+1 + [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The set Aj is a subset of {0, 2, · · · , N 1/2 j − 1} and [0, 1] ⊇ E1 ⊇ · · · ⊇ Ej ⊇ Ej+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let E∞ := ∩∞ j=1Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Similarly, let Pj := {(x1, x2 1) : x1 ∈ Ej} and P∞ := ∩∞ j=1Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For each j ∈ N ∪ {0}, we define µj := 1 |Ej|1Ej(x1), x1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We identify the function µk with the absolutely continuous measures µjdξ1 so that � gdµj := � gµjdx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' From µj, we defined the measure νj as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' � fdνj := � f(x1, x2 1)dµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For simplicity, let ∥µj∥ := ∥µj∥L1(R) and ∥νj∥ := ∥νj∥L1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The measures µj and νj converge weakly as j → ∞ to a probability measure µ and ν supported on E∞ and P∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 (Lemma 6, [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The measure µ and the set E∞ constructed above satisfies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1) The set E∞ has Hausdorff dimension α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2) For any 0 < ǫ < α, we have that µ(B(x, r)) ≲α,ǫ rα−ǫ ∀ξ ∈ R, ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 5 It is easy to show that the same are true for ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The measure ν and the set P∞ constructed above satisfies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1) The set P∞ has Hausdorff dimension α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2) For any 0 < ǫ < α, we have that ν(B(x, r)) ≲α,ǫ rα−ǫ ∀ξ ∈ R2, ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Decoupling inequalities In this section, we will derive the decoupling estimates for E∞ and P∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We define decou- pling constants and discrete restriction constants and we will verify relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since we need to look into Ej in different scales, we define the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For 0 ≤ i < j, write Nj,i = NjN −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For Si+1 ∈ Σi+1, let Ai+1,i = Si+1 Ei+1,i = n−1/2 i+1 (Ai+1,i + [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For each a ∈ Aj,i, choose a set Sj+1,a ∈ Σj+1 and let Aj+1,i,a = n1/2 j+1a + Sj+1,a Aj+1,i = ∪a∈Aj+1,iAj+1,i,a Ej+1,i = N −1/2 j+1,i(Aj+1,i + [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Pj,i(Ej,i) denote the partition of Ej,i into intervals of length N −1/2 j,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If I ∈ Pj(Ej,i), there exists an unique element a ∈ Aj,i such that I = N −1/2 j,i (a + [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For p ≥ 2, let Dp(Ej,i) denotes the best constant such that ∥ � I∈Pj,i(Ej,i) fI∥Lp(R) ≤ Dp(Ej,i) � � I∈Pj,i(Ej,i) ∥fI∥2 Lp(R) �1/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) holds for any choices of Ej,i where �fI(ξ1) = �f(ξ1)1I(ξ1) for ξ1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Dp(Pj,i) be the best constant such that ∥ � I∈Pj,i(Ej,i) fΩI∥Lp(R2) ≤ Dp(Pj,i) � � I∈Pj,i(Ej,i) ∥fΩI∥2 Lp(R2) �1/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) holds for any choices of Ej,i where fΩI = �f(ξ)1ΩI(ξ) and 1ΩI(ξ) is a characteristic function supported on ΩI = {ξ ∈ R2 :aN −1/2 j,i ≤ ξ1 ≤ (a + 1)N −1/2 j,i , |ξ2 − (2a + 1)N −1/2 j,i (ξ1 − aN −1/2 j,i ) − a2N −1 j,i | ≤ N −1 j,i } for a that corresponds to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If i = 0, Aj,0 and Ej,0 are same with Aj and Ej defined in section 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Kp(Ej,i) be the best constant such that ∥ � a∈Aj,i cae(ax)∥Lp([0,1]) ≤ Kp(Ej,i) � � a∈Aj,i |ca|2 �1/2 holds for any choices of Aj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Kp(Pj,i) be the best constant such that ∥ � a∈Aj,i cae(ax1 + a2x2)∥Lp([0,1]2) ≤ Kp(Pj,i) � � a∈Aj,i |ca|2 �1/2 6 DONGGEUN RYOU holds for any choices of Aj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In short, Dp(Ej,i) and Dp(Pj,i) are decoupling constants for Ej,i and Pj,i respectively and Kp(Ej,i) and Kp(Pj,i) are discrete restriction constants for Ej,i and Pj,i respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We will show that K6/α(Pj) ≲ǫ N ǫ j through the following inequalities: K3p(Pj) ≲p D3p(Pj) ≲ǫ N ǫ jDp(Ej) and Dp(Ej) ≈p Kp(Ej) ≲ǫ Cj p where p = 2/α and Cp is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that the inequality Kp(Pj) ≲p Dp(Pj) is well known, see for example [8, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' From Λ(p)-sets to decoupling for Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We need the following lemmas to use Λ(p)-sets in multiscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For p ≥ 2, let S1 be a subset of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For a ∈ S1 and k ∈ N, let S2,a be subsets of [0, k − 1] ∩ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Assume that the sets S1 and S2,a satisfy ∥ � a∈S1 cae(a · x)∥Lp([0,1]d) ≤ C1 � � a∈S1 |ca|2 �1/2 ∀{ca}a∈S1 ∈ ℓ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) and ∥ � b∈S2,a cbe(b · x)∥Lp([0,1]d) ≤ C2 � � b∈S2 |cb|2 �1/2 ∀{cb}b∈S2 ∈ ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) where C2 is uniform over a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we have ∥ � a∈S1 � b∈S2,a ca,be((ka + b) · x)∥Lp([0,1]d) ≲p C1C2 � � a∈S1 � b∈S2,a |ca,b|2 �1/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) for all {ca,b}a∈S1,b∈S2,a ∈ ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For p ≥ 2, Dp(Ej,i) ≈p Kp(Ej,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 follows from the proof of Proposition 1 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Duality of Lp played a key role in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In Section 8, we provide alternative proofs of them which do not rely on duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Readers may want to compare Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' They are similar but there is a trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 can cover more general cases, because p does not need to be an even integer and there is no assumption on carryover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' However, the implicit constant of the inequality is larger than Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5 in [4] because the constant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) is not exactly C1C2, but C′ pC1C2 where C′ p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For p = 2/α where 0 < α < 1, Dp(Ej,i) ≲p Cj−i p where the Cp is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5), we obtain that Dp(Ej,i) ≈ Kp(Ej,i) ≲ Cj−i p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' From decoupling for Ej to decoupling for Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In [4], they proved that decoupling for a Cantor set on the parabola can be derived from decoupling for a Cantor set on the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We adapt their argument to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For p = 2/α where 0 < α < 1, we have that D3p(Pj) ≲ǫ N ǫ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6 (Parabolic rescaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Suppose that j ≥ i and I ∈ Pi(Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, ∥ � J∈Pj(I∩Ej) fΩJ∥Lp(R2) ≤ Dp(Pj,i) � � J∈Pj(I∩Ej) ∥fΩJ∥2 Lp(R2) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7 (Almost multiplicativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We have Dp(Pj+i) ≤ Dp(Pj+i,i)Dp(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For j ≤ i1, i2 ≤ k, we define the bilinear constant Mp(j, k, i1, i2) which is the smallest constant such that � R2 ���� � J1∈Pj(I1∩Ej) fΩJ1 ���� p���� � J2∈Pj(I2∩Ej) gΩJ2 ���� 2p ≤ Mp(j, k, i1, i2)3p � � J1∈Pj(I1∩Ej) ∥fJ1∥2 L3p(R2) �p/2� � J2∈Pj(I2∩Ej) ∥gJ2∥2 3p �p for all I1 ∈ Pi1(Ei1) and I2 ∈ Pi2(Ei2) such that d(I1, I2) ≥ N −1/2 k and for any choices of Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8 (Bilinear reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If i ≤ j, then D3p(Pj) ≲ D3p(Pj,i) + N O(1) i Mp(j, i, i, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) Proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8 are same as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9 (Key estimate in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Assume that p = 2/α where 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If 0 ≤ k ≤ i1, i2, 4i1 ≤ j with 2i1 ≤ i2, then for any ǫ > 0, Mp(j, k, i1, i2) ≲p,ǫ N O(1) k (CpB)i1Mp(j, k, 4i1, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) where B is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) and Cp is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We follow the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4 in [4] with modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) comes into play since nk is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Without loss of generality, we can assume that I2 is on the left of I1 and I2 is centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, for each J ∈ P4i1(I1 ∩ E4i1), the center of J is a distance ≳ N −1/2 k away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let FJ = � J1∈Pj(J∩Ej) fΩJ1 and G = � J2∈Pj([0,N−1/2 i2 ]∩Ej) gΩJ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 8 DONGGEUN RYOU As an analogue of (17) in [4], it suffices to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For fixed x ∈ R, � R ���� � J∈P4i1(I1∩E4i1) FJ(x, y)G(x, y)2 ���� p dy ≲p,ǫ N O(p) k Dp(E4i1,2i1)pDp(E2i1,i1)pB3pi1/2 � � J∈P4i1(I1∩E4i1) ( � R |FJ(x, y)G(x, y)2|pdy)2/p �p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) Once we prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8), then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For J0 ∈ P2i1(I1 ∩ E2i1), let FJ0 = � J∈P4i1(J0∩E4i1) FJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that � J∈P4i1(I1∩E4i1) FJ = � J0∈P2i1(I1∩E2i1) FJ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The function �G is supported in an O(N −1/2 i2 )×O(N −1 i2 +N −1/2 j ) rectangle and �FJ0 is supported on the horizontal strip {(ξ1, ξ2) : ξ2 = γ2 J0 + O(N −1/2 2i1 )} where γJ0 is the center of J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since 2i1 ≤ i2, � FJ0G2 is supported in the horizontal strip {(ξ1, ξ2) : ξ2 = γ2 J0 + O(N −1/2 2i1 )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, Fourier transform of FJ0G2 in y for fixed x is also supported on an interval of length O(N −1/2 2i1 ) centered at γ2 J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For c ≲ N O(1) k N 1/2 2i1 N −1 i1 , it is easy to prove that ∥ � J∈P2i1,i1(E2i1,i1) fcJ∥Lp(R) ≲p,ǫ N O(1) k Dp(E2i1,i1) �N 1/2 2i1 Ni1 �� � J∈P2i1,i1(E2i1,i1) ∥fcJ∥2 Lp(R) �1/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9) where cJ is the interval of length c|J| which has the same center with J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4), N 1/2 2i1 N −1 i1 ≤ Bi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6 that ∥ � J0∈P2i1(I1∩E2i1) fcJ0∥Lp(R) ≲p,ǫ N O(1) k Dp(E2i1,i1)Bi1 � � J0∈P2i1(I1∩E2i1) ∥fcJ0∥2 Lp(R) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) Let γm be the left endpoint of I1 and let us consider T(x) = (2γm + N −1/2 i1 )(x − γm) + γ2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 9 Since γm ≥ N −1/2 k and N −1/2 2i1 ≤ N −1 i1 , we get γ2 J0 + O(N −1/2 2i1 ) ⊆ T(cJ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) implies that � R ���� � J∈P4i1(I1∩Ej) FJ(x, y)G(x, y)2 ���� p dy = � R ���� � J0∈P2i1(I1∩E2i1) FJ0(x, y)G(x, y)2 ���� p dy ≲p N O(p) k Dp(E2i1,i1)pBpi1 � � J0∈P2i1(I1∩E2i1) ( � R |FJ0(x, y)G(x, y)2|pdy)2/p �p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='11) Similarly, �FJ is supported on the horizontal strip {(ξ1, ξ2) : ξ2 = γ2 J + O(N −1/2 4i1 )} where γJ is the center of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since N −1/2 4i1 , N −1 i2 ≤ N −1 2i1 , Fourier transform of FJG2 in y for fixed x is supported on an interval of length O(N −1 2i1 ) centered at γ2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For c ≲ N O(1) i N 1/2 4i1 N −1 2i1 , we can prove that ∥ � J∈P4i1,2i1(E4i1,2i1) fcJ∥Lp(R) ≲p,ǫ N O(1) k Dp(E4i1,2i1) �N 1/2 4i1 N2i1 �� � J∈P4i1,2i1(E4i1,2i1) ∥fcJ∥2 Lp(R) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='12) As before, this implies that � R ���� � J∈P4ai(J0∩E4i1) FJ(x, y)G(x, y)2 ���� p dy ≲p N O(p) k Dp(E4i1,2i1)pB2pi1 � � J∈P4i1(J0∩E4i1) ( � R |FJ(x, y)G(x, y)2|pdy)2/p �p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let k ≤ i2 ≤ i1 ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, Mp(j, k, i1, i2) ≤ Mp(j, k, i2, i1)1/2D3p(Pj,i2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10 is same as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Assume that λ is the smallest exponent such that D3p(Pj,i) ≲ǫ N λ+ǫ j,i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14) for all 0 ≤ i < j and for sufficiently small 0 < ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, it suffices to show that λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We can run the iteration as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' But, we should be careful since nk is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' First, assume that λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14), for any positive integer a such that 1 ≤ a ≤ j 4i, we obtain that Mp(j, i, 2ai, ai) ≤ Mp(j, i, ai, 2ai)1/2D3p(Pj,ai)1/2 ≲ǫ N O(1) i Mp(j, i, 4ai, 2ai)1/2(CpB)ai/2D3p(Pj,ai)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='15) 10 DONGGEUN RYOU It suffices to consider the case j = 2k+1i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By iterating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='15), it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14) that Mp(j, i, 2i, i) ≲ǫ N O(1) i (CpB)ki/2N λ+ǫ(1−1/2k) j (NiN 1/2 2i · · N 1/2k−1 2k−1i )−(λ+ǫ)/2Mp(j, i, 2k+1i, 2ki)1/2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since j = 2k+1i, we have Mp(j, i, 2k+1i, 2ki) ≤ D3p(Pj,2ki)2/3 ≲ �N2k+1i N2ki �2(λ+ǫ)/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4), we obtain that Mp(j, i, 2k+1i, 2ki)1/2k ≲ (B2kiN 1/2 j )(λ+ǫ)/3·2k−1 ≤ B2(λ+ǫ)i/3N (λ+ǫ)/3·2k j ≲ N O(1) i N (λ+ǫ)/2k j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we arrive at Mp(j, i, 2i, i) ≲ǫ N O(1) i (CpB)ki/2N λ+ǫ j (NiN 1/2 2i · · N 1/2k−1 2k−1i )−(λ+ǫ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since N 2s i ≤ N2si, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) we obtain that Mp(j, i, 2i, i) ≲ǫ N O(1) i (CpB)ki/2N λ+ǫ j N −kλ/2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) imply that limj→∞ nj = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, (CpB)i/2 ≤ N λ/4 i for sufficiently large i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8, for sufficiently large k and i, we have D3p(Pj) ≲ǫ (NjNi−1)λ+ǫ + N O(1) i N λ+ǫ j N −λk/4 i ≲ǫ N λ+ǫ j N −λ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since i = 2−k−1j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) implies that N −1 j/2k+1 ≤ Bj/2k+1N −1/2 j/2k ≤ · · · ≤ Bj(k+1)/2k+1N −1/2k+1 j ≲ǫ′ N −(1−(k+1)ǫ′/2)/2k+1 j for sufficiently small ǫ′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we choose ǫ′ small enough such that (k + 1)ǫ′/2 < 1, then D3p(Pj) ≲ǫ,ǫ′ N λ+ǫ−(1−(k+1)ǫ′/2)/2k+1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The same iteration argument also works for D3p(Pj,i) and decoupling constants D3p(Pj,i′) where i′ < i are not used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This contradicts the assumption that λ > 0 is the smallest which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Local restriction estimate We denote Bd(R) by a cube of side length R in Rd centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let p = 6/α where 0 < α < 1 and ν be the measure constructed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we have ∥� fdν∥Lp(B2(R)) ≲p,ǫ Rǫ∥f∥L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) Equivalently, we have ∥ �f∥L2(ν) ≲q,ǫ Rǫ∥f∥Lq(R2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 11 where 1/q + 1/p = 1 and f is supported on B2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For a ∈ A2j and ℓ ≥ 2j, let Aℓ,a be a subset of Aℓ such that N −1/2 ℓ (b + [0, 1]) ⊆ N −1/2 2j (a + [0, 1]) if b ∈ Aℓ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' In short, Aℓ,a is the set of ℓ-th level descendants of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For ℓ ≥ 2j, we have the estimate ∥ � a∈A2j fΩa∥Lp(B2(Nj)) ≲ǫ,p Kp(P2j)N ǫ+ 3 2p − α 4 2j N −3/4 ℓ |Aℓ|1/2∥ � a∈A2j fΩa∥L2(R2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) where �fΩa(ξ) = �f(ξ) � b∈Aℓ,a 1[0,1]×[−1,1](N 1/2 ℓ ξ1 − b, Nℓξ2 − (2b + 1)N 1/2 ℓ ξ1 + b2 + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Once we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2, we can prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let η is a Schwartz function on R such that |η| ≥ 1 on [−1, 1] and �η is supported on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let �F(ξ) := f(ξ1, ξ2 1) � a∈A2j � b∈Aℓ,a 1[0,1](N 1/2 ℓ ξ1 − b)�ηℓ(ξ2 − ξ2 1)|Eℓ|−1 where ηℓ(x) = η(x/N 1/2 ℓ ) for x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Observe that Nj ≤ N 1/2 2j ≤ N 1/2 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If x ∈ B2(Nj), we have | � fdνℓ(x)| ≤ | � fdνℓ(x)ηℓ(−x2)| = |F(−x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) implies that ∥ � fdνℓ∥Lp(B2(Nj)) ≲ ∥F∥Lp(B2(Nj)) ≲ǫ Kp(P2j)N ǫ+ 3 2p − α 4 2j N −3/4 ℓ |Aℓ|1/2∥F∥L2(R2) = Kp(P2j)N ǫ+ 3 2p − α 4 2j N −3/4 ℓ |Aℓ|1/2∥ �F∥L2(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |Eℓ| = |Aℓ|N −1/2 ℓ , we get ∥ �F∥L2(R2) ≲ N 3/4 ℓ |Aℓ|−1/2∥f∥L2(νℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since p = 6/α, it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5 that ∥ � fdνℓ∥Lp(B2(Nj)) ≲ Kp(P2j)N ǫ 2j∥f∥L2(νℓ) ≲ǫ N 2ǫ 2j ∥f∥L2(νℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If Nj−1 ≤ R ≤ Nj, conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) implies that N2j ≤ B2jN 2 j = B2jn2 jN 2 j−1 ≲ǫ N 2+ǫ j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, ∥ � fdνℓ∥Lp(B2(R)) ≤ ∥ � fdνℓ∥Lp(B2(Nj)) ≲ N 2ǫ 2j ∥f∥L2(νℓ) ≲ǫ N O(ǫ) j−1 ∥f∥L2(νℓ) ≲ǫ RO(ǫ)∥f∥L2(νℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When we take the limit ℓ → ∞, we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) and by duality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 12 DONGGEUN RYOU Now let us turn to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let η is a Schwartz function on R2 such that |η| ≥ 1 on [−1, 1]2 and �η is supported on [0, 1]2 and write η2j(x) = η(x/N 1/2 2j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since Nj ≤ N 1/2 2j , we have ∥ � a∈A2j fΩa∥Lp(B2(Nj)) ≤ ∥ � a∈A2j fΩaη2j∥Lp(R2) = sup ∥g∥Lq(R2)≤1 � � a∈A2j fΩa(x)η2j(x)g(x)dx = sup ∥g∥Lq(R2)≤1 � � a∈A2j �fΩa ∗ �η2j(ξ)�g(ξ)dξ where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The function �fΩa is supported on a rectangle of dimensions O(N −1/2 2j ) × O(N −1 2j ) centered at ((a + 1/2)N −1/2 2j , (a + 1/2)2N −1 2j ) and the direction that it is pointing depends on a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The function �η2j is supported on a square with side length O(N −1/2 2j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, �fΩa ∗ �η2j(ξ) is supported on a square with side length O(N −1/2 2j ) centered at (aN −1/2 2j , a2N −1 2j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For a ∈ A2j and c = (c1, c2) ∈ Z2, let us consider characteristic functions 1Qa,c(ξ) = 1[0,1]2(N 1/2 2j ξ1 − (a + c1), N 1/2 2j ξ2 − (a2 + c2)N −1/2 2j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we have ∥ � a∈A2j fΩa∥Lp(B2(Nj)) ≤ sup ∥g∥Lq(R2)≤1 � |c|≲1 � � a∈A2j �fΩa ∗ �η2j(ξ)�g(ξ)1Qa,c(ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By letting z1 = N 1/2 2j ξ1 − (a + c1) and z2 = N 1/2 2j ξ2 − (a2 + c2)N −1/2 2j , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) we obtain sup ∥g∥Lq(R2)≤1 � |c|≲1 � [0,1]2 � a∈A2j �fΩa ∗ �η2j(N −1/2 2j (z1 + a + c1), N −1/2 2j z2 + (a2 + c2)N −1 2j ) �g(N −1/2 2j (z1 + a + c1), N −1/2 2j z2 + (a2 + c2)N −1 2j )dzN −1 2j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let a′ = (a′ 1, a′ 2) ∈ Z2 and a′ · u = a′ 1u1 + a′ 2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we have � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j � [0,1]2 e(au1 + a2u2)e(−a′ · u)du = 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) if and only if a′ 1 = a and a′ 2 = a2 for |a| ≲ N 1/2 2j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Fc(u, z) := � a∈A2j �fΩa ∗ �η2j(N −1/2 2j (z1 + a + c1), N −1/2 2j z2 + (a2 + c2)N −1 2j )e(au1 + a2u2) and Gc(u, z) := � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j �g(N −1/2 2j (z1 + a′ 1 + c1), N −1/2 2j z2 + (a′ 2 + c2)N −1 2j )e(−a′ · u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 13 Also, we denote the mixed norm of f by ∥f∥Lp1 u Lp2 z = � � [0,1]2 � � [0,1]2 |f(u, z)|p1du �p2/p1 dz �1/p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) and H¨older inequality, we have ∥ � a∈A2j fΩa∥Lp(B2(Nj)) ≤ sup ∥g∥Lq(R2)≤1 � |c|≲1 � [0,1]2 � [0,1]2 Fc(u, z)Gc(u, z)dudzN −1 2j ≤ sup ∥g∥Lq(R2)≤1 � |c|≲1 ∥Fc(u, z)∥Lp uL2z∥Gc(u, z)∥Lq uL2zN −1 2j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) Since Fc(u, z) is a Fourier series with respect to u variable, the definition of Kp(P2j) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) implies that ∥Fc∥Lp uL2z ≤ Kp(P2j)∥Fc∥L2uL2z ≤ Kp(P2j)N 1/2 2j � � � a∈A2j | �fΩa ∗ �η2j(ξ)|2dξ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) In order to obtain an estimate of ∥Gc∥Lq uL2z, we consider Gc as a linear operator T : Lq(R2) → Lq uL2 z acting on g, which is defined by Tg(u, z) = � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j � g(x)e(N −1/2 2j z1x1 + N −1/2 2j z2x2) e(N −1/2 2j (a′ 1 + c1)x1 + (a′ 2 + c2)N −1 2k x2)e(−a′ · u)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we show that ∥Tg∥L1uL2z ≲ǫ N ǫ 2j∥g∥L1(R2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) and ∥Tg∥L2uL2z ≲ N 3/4 2j ∥g∥L2(R2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9) then the mixed norm interpolation theorem (see [2, Theorem 2 in Section 7]) implies that ∥Gc(u, z)∥Lq uL2z = ∥Tg∥Lq uL2z ≲ǫ N ǫ+3/2p 2j ∥g∥Lq(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) When q = 1, let us consider the case c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We have Tg(u, z) = � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j � R2 g(N 1/2 2j x1, N2jx2)e(x1z1 + N 1/2 2j x2z2)N 3/2 2j e(−a′ · (u − x))dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j e(−a′ · x) is a product of Dirichlet kernels, since � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j e(−a′ · x) = � |a′ 1|≲N1/2 2j e(−a′ 1x1) � |a′ 2|≲N2j e(−a′ 2x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we have ∥ � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j e(−a′ · x)∥L1(du) ≲ log(N 1/2 2j ) log(N2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 14 DONGGEUN RYOU Therefore, we obtain that ∥Tg∥L1(du) ≲ � R2 � [0,1] ����g(N 1/2 2j x1, N2jx2)N 3/2 2j ���� ���� � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j e(−a′ · (x − u)) ����dudx ≲ log(N2j)2 � R2 ����g(N 1/2 2j x1, N2jx2)N 3/2 2j ����dx ≲ǫ N ǫ 2j∥g∥L1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='11) Since the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='11) holds uniformly on z, we established (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When c ̸= 0, we can also get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) by the similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When q = 2, since Tg(u, z) is a Fourier series with respect to u variable, we can use Plancherel’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4), we have ∥Tg∥L2uL2z = � � [0,1]2 � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j |�g(N −1/2 2j (z1 + a′ 1 + c1), N −1/2 2j z2 + (a′ 2 + c2)N −1 2j )|2dz �1/2 = N 1/2 2j � � |�g(ξ)|2 � |a′ 1|≲N1/2 2j ,|a′ 2|≲N2j 1Qa′c(ξ)dξ �1/2 where 1Qa′,c(ξ) = 1[0,1]2(N 1/2 2j ξ1 − (a′ 1 + c1), N 1/2 2j ξ2 − (a′ 2 + c2)N −1/2 2j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The function 1Qa′,c(ξ) is supported on a square with side length O(N −1/2 2j ) centered at (N −1/2 2j (a′ 1 + c1), N −1 2j (a′ 2 + c2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' These squares overlap at most O(N 1/2 2j ) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we have ∥Tg∥L2uL2z ≲ N 3/4 2j ∥�g∥L2(R2) ≤ N 3/4 2j ∥g∥L2(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we established (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10), we obtain that ∥ � a∈A2j fΩa∥Lp(B2(Nj)) ≲ǫ Kp(P2j)N ǫ+ 3 2p − 1 2 2j � � � a∈A2j | �fΩa ∗ �η2j(ξ)|2dξ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='12) We define 1a,b(ξ) := � b∈Aℓ,a 1[0,1]×[−1,1](N 1/2 ℓ ξ1 − b, Nℓξ2 − (2b + 1)N 1/2 ℓ ξ1 + b2 + b) so that �fΩa(ξ) = �fΩa1a,b(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By H¨older’s inequality, we have | �fΩa ∗ �η2j(ξ)|2 ≤ (| �fΩa1a,b| ∗ |�η2j|(ξ))2 ≤ (| �fΩa|2 ∗ |�η2j|(ξ))(1a,b ∗ |�η2j|(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |�η2j| ≲ N2j, for fixed ξ, we have (1a,b ∗ |�η2j|(ξ)) ≤ ∥�η2j∥L∞(R2)∥1a,b∥L1(R2) ≲ N2jN −3/2 ℓ |Aℓ| |A2j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 15 Since N α/2 2j ≤ |A2j| and ∥�η2j∥L1(R2) ≲ 1, we obtain � � a∈A2j | �fΩa ∗ �η2j(ξ)|2dξ ≲ N 1−α/2 2j N −3/2 ℓ |Aℓ| � � a∈A2j | �fΩa|2 ∗ |�η2j|(ξ)dξ ≲ N 1−α/2 2j N −3/2 ℓ |Aℓ| � a∈A2j ∥ �fΩa∥2 L2(R2) ≲ N 1−α/2 2j N −3/2 ℓ |Aℓ|∥ � a∈A2j �fΩa∥2 L2(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Fourier decay We abbreviate almost surely to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When we say an inequality holds a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=', the corresponding implicit constant may depend on the measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let us consider a nondecreasing sequence {mj}j∈N such that 2 ≤ mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Mj := mj · · · m1, M0 = 1 and assume that ∀ǫ > 0 and ∀j ∈ N, mj+1 ≲ǫ Mǫ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) Let Ij be the collection of M−1 j intervals such that Ij = {M−1 j (k + [0, 1]) : 0 ≤ k ≤ Mj − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We consider a sequence of random functions µj which satisfies the following conditions for some deterministic nondecreasing sequence {βj}j∈N: (M1) µ0 = 1[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (M2) µj = βj1Ej where Ej is a union of intervals in Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (M3) E(µj+1(x)|Ej) = µj(x) for all x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (M4) For I ∈ Ij+1, the random variables µj+1(I) are jointly independent, conditioned on Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We will identify µj with the measures µjdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let α0 be a number such that 1 − α0 = lim j→∞ log βj log Mj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) Shmerkin and Suomala showed that µj converges weakly to a measure µ supported on [0, 1] and the support of the measure µ is a Salem set a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' so that it satisfies |�µ(ξ)| ≲σ (1+|ξ|)−σ/2 for all σ < α0, see [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2] and [19, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, let νj be a measure defined as � f(x1, x2)dνj := � f(x1, x2 1)µj(x1)dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Similarly, νj converges weakly to a measure ν supported on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ν is on the parabola, we will use their proof with estimates for oscillatory integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Suppose that µj is a sequence of random measures satisfies (M1)-(M4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For any 0 < σ < α0, the limit measure ν satisfies the following inequality a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' |�ν(ξ)| ≲σ,ǫ (1 + |ξ|)−σ/2+ǫ for all ξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) One of main ingredients of the proof is Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 16 DONGGEUN RYOU Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 (Hoeffding’s inequality [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let X1, · · · , Xn be be independent random vari- ables such that ai ≤ Xi ≤ bi and Sn := X1 + · · · + Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For t > 0, P �����Sn − E(Sn) ���� > t � ≤ 2 exp � −2t2 �n i=1(bi − ai)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) For fixed ξ such that |ξ| ≥ 1, let x0 be the point in [0, 1] such that ξ1 + 2xξ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we can prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For fixed ξ, we have the following tail bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1) If |ξ| ≤ Mj+1, there exists a constant C1 > 0 such that P � |�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 j ∥µj∥1/2 ����Ej � ≲ exp(−C1M1−σ j βjβ−2 j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2) If Mj+1 ≤ |ξ| ≤ mj+1M2 j , let k be an integer such that 0 ≤ k ≤ j − 1 and Mj+1Mk ≤ |ξ| ≤ Mj+1Mk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) For such k, there exists a constant C2 > 0 such that P � |�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 j � k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) �1/2����Ej � ≲ exp(−C2M1−σ j βjβ−2 j+1) where Ii(x0) is an interval of length 2M−1 i centered at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Given I ∈ Ij, let Pj+1(I) is the collection of M−1 j+1-intervals in Ij+1 that make up I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For I ∈ Ij, we consider XI = � I βj+11Ej+1e(−xξ1 − x2ξ2)dx for ξ = (ξ1, ξ2) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, � I∈Ij XI = �νj+1(ξ) and (M3) implies that E( � I∈Ij XI|Ej) = E(�νj+1(ξ)|Ej) = � e(−xξ1 − x2ξ2)E(µj+1(x)|Ej)dx = � e(−xξ1 − x2ξ2)µj(x)dx = �νj(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let SI := � I∈Ij XI, then |SI − E(SI)| = |�νj+1(ξ) − �νj(ξ)| conditioned on Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, we need to estimate |XI| where I ⊆ Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If |XI| < CI for some constant CI, let aI = −CI and bI = CI so that � I⊆Ej(bI − aI)2 ≈ � I⊆Ej C2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we can plug it in Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' First, when |ξ| ≤ Mj+1, we have |XI| ≤ βj+1M−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) Since the number of I ⊆ Ej is Mjβ−1 j ∥µj∥, we obbtain � I⊆Ej (bI − aI)2 = β2 j+1 βjMj ∥µj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 17 If t = M−σ/2 j ∥µj∥1/2, then Hoeffding’s inequality implies that P � |�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 j ∥µj∥1/2 � ≲ exp(−C1M1−σ j βjβ−2 j+1) for some constant C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Second, let us consider when Mj+1 ≤ |ξ| ≤ mj+1M2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We have XI = � I′⊂I∩Ej+1 βj+1 � I′ e(−xξ1 − x2ξ2)dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) where I′ = M−1 j aj + M−1 j+1(aj+1 + [0, 1]) for some 0 ≤ aj ≤ Mj − 1 and 0 ≤ aj+1 ≤ mj+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we get ���� � I′ e(−xξ1−x2ξ2)dx ���� = ����M−1 j+1 � 1 0 e(−M−1 j+1ξ1x−2M−2 j+1(ajmj+1+aj+1)ξ2x−M−2 j+1ξ2x2)dx ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) Let its phase be Φ(x) := M−1 j+1ξ1x + 2M−2 j+1(ajmj+1 + aj+1)ξ2x + M−2 j+1ξ2x2, then Φ′(x) = 2M−2 j+1ξ2(x + ajmj+1 + aj+1) + M−1 j+1ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' It suffices to consider when |ξ| ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, for fixed ξ, Φ′(x) = 0 can happen in at most two intervals I′ which are in the same interval I or in two neighboring intervals I, since 0 ≤ x ≤ 1 and ajmj+1 + aj+1 are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Without loss of generality, we can assume that ξ1 = 0 so that Φ′(x) = 0 can happen when aj = aj+1 = 0 and x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let us write I′ 0 = [0, M−1 j+1], I′ 1 = [M−1 j+1, 2M−1 j+1] and I0 = [0, M−1 j ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If I′ ̸= I′ 0, I′ 1, then Φ′(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we have ���� � I′ e(−xξ1 − x2ξ2)dx ���� ≲ Mj+1|ξ2(ajmj+1 + aj+1)|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9) Here, we used the principle of non-stationary phase which is same as [20, Chapter 8, Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, if I ̸= I0, we get |XI| ≲ βj+1Mj+1 mj+1−1 � aj+1=0 |(ajmj+1 + aj+1)ξ2|−1 ≲ βj+1Mj+1|ξ|−1 log(1 + a−1 j ) ≲ βj+1Mj+1|ξ|−1a−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) When Mj+1 ≤ |ξ| ≤ mj+1M2 j , the upper bound of |XI| in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) is not always smaller than the upper bound in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we consider Ij+1(ξ) := Mj+1|ξ|−1[−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If I ⊆ Ij+1(ξ), we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Otherwise, we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that if I ⊆ Ij+1(ξ), then M−1 j aj ≤ Mj+1|ξ|−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='11) and it implies that βj+1M−1 j ≲ βj+1Mj+1|ξ|−1a−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 18 DONGGEUN RYOU Also, if I ̸⊆ Ij+1(ξ) , then I ̸= I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, we obtain that � I⊆Ej (bI − aI)2 = � I⊆Ej∩Ij+1(ξ) (bI − aI)2 + � I⊆Ej\\Ij+1(ξ) (bI − aI)2 ≲ β2 j+1M−2 j � aj≤mj+1M2 j |ξ|−1 1 + β2 j+1M2 j+1|ξ|−2 � aj≥mj+1M2 j |ξ|−1 a−2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='12) If aj ≤ mj+1M2 j |ξ|−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) implies that aj ≤ M−1 k Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we get � aj≤mj+1M2 j |ξ|−1 1 ≲ β−1 j Mjµj([−M−1 k , M−1 k ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13) If aj ≥ mj+1M2 j |ξ|−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) implies that aj ≥ M−1 k+1Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, we have � aj≥mj+1M2 j |ξ|−1 a−2 j ≤ k � i=0 � M−1 i+1Mj≤aj≤M−1 i Mj a−2 j ≤ k � i=0 M2 i+1 M2 j � M−1 i+1Mj≤aj≤M−1 i Mj 1 ≲ k � i=0 M2 i+1 M2 j β−1 j Mjµj([−M−1 i , M−1 i ]) ≤ k � i=0 M2 i+1 βjMj µj([−M−1 i , M−1 i ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |ξ|−1 ≤ M−1 j+1M−1 k , we obtain that β2 j+1M2 j+1|ξ|−2 � aj≥mj+1M2 j |ξ|−1 a−2 j ≤ β2 j+1 βjMj k � i=0 �Mi+1 Mk �2 µj([−M−1 i , M−1 i ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='12), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='14), we get � I⊆Ej (bI − aI)2 ≲ β2 j+1 βjMj k � i=0 �Mi+1 Mk �2 µj([−M−1 i , M−1 i ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we let t = M−σ/2 j � k � i=0 �Mi+1 Mk �2 µj([−M−1 i , M−1 i ]) �1/2 , Hoeffding’s inequality implies that P � |�νj+1(ξ) − �νj(ξ)| ≥ t ����Ej � ≲ exp(−C2M1−σ j βjβ−2 j+1) for some constant C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If |ξ| ≥ mj+1M2 j , we have the estimate |�νj+1(ξ) − �νj(ξ)| ≲ βj+1m1/2 j+1 log(Mj)|ξ|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='15) NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let us consider (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Similarly, we can assume that ξ1 = 0 so that Φ′(x) = 0 at x = 0 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If I′ = I′ 0, since Φ′′(x) = 2M−2 j+1ξ2 ̸= 0, we obtain that ���� � I′ 0 e(−xξ1 − x2ξ2)dx ���� ≲ |ξ|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='16) Here, we used the principle of stationary phase which is same as [20, Chapter 8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The same inequality holds for I′ 1 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |ξ| ≥ mj+1M2 j , by using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='16), we obtain that |XI0| ≲ βj+1 � |ξ|−1/2 + Mj+1 mj+1−1 � aj+1=2 |aj+1ξ2|−1 � ≲ βj+1 � |ξ|−1/2 + m1/2 j+1 log(mj+1)|ξ|−1 � ≲ βj+1m1/2 j+1 log(mj+1)|ξ|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='17) For I ̸= I0, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='10) and the assumption that |ξ| ≥ mj+1M2 j , we get that |XI| ≲ βj+1m1/2 j+1|ξ|−1/2a−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='18) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='18), we have |�νj+1(ξ)| ≤ � I∈Ij |XI| ≲ βj+1m1/2 j+1 log(Mj)|ξ|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Similarly, we obtain that |�νj(ξ)| ≲ βj log(Mj)|ξ|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='15) follows from triangular inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' As in [18] and [19], by Lemma 9A4 in [24], it suffices to consider ξ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let Ωj be the event that there exists ξ ∈ Z2 such that |�νj+1(ξ) − �νj(ξ)| ≥ M−σ/2 j ∥µj∥1/2 where |ξ| ≤ Mj+1 or |�νj+1(ξ) − �νj(ξ)| ≳ M−σ/2 j � k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) �1/2 where Mj+1 ≤ |ξ| ≤ mj+1M2 j or |�νj+1(ξ) − �νj(ξ)| ≳ βj+1m1/2 j+1 log(Mj)|ξ|−1/2 where |ξ| ≥ mj+1M2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4, we have P(Ωj) ≲ m2 j+1M4 j exp(−CM1−σ j βjβ−2 j+1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='19) for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For sufficiently large j and sufficiently small ǫ > 0, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) that M1−σ j βjβ−2 j+1 ≥ Mα0−σ+3ǫ j 20 DONGGEUN RYOU where α0 − σ + 3ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, �∞ j=1 P(Ωj) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By the assumptions on µj, ∥µj∥ is bounded a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We also need an upper bound of k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) which is uniform over k and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since E(µj(Ii(x0))) ≲ βiM−1 i and βj and mj are nondecreasing, we obtain that E � sup 0≤k≤j−1 β−1 k Mk k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) � ≤ j−1 � k=0 k � i=0 M2 i+1 βkMk E(µj(Ii(x0))) ≤ j−1 � k=0 k � i=0 βimi+1Mi+1 βkMk ≲ j−1 � k=0 km2 k+1 ≤ j2m2 j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, it follows from Markov’s inequality that P � sup 0≤k≤j−1 β−1 k Mk k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) ≥ j4m2 j+1 � ≲ j−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='20) By Borel-Cantelli lemma, the following inequality holds a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) ≲ j4m2 j+1βkM−1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='21) where the implicit constant may depends on µ but does not depend on j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) that βk ≲ǫ M1−α0+ǫ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='22) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='22), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' we obtain that k � i=0 �Mi+1 Mk �2 µj(Ii(x0)) ≲ǫ j4mα0+2+ǫ j+1 �Mj+1 |ξ| �α0+ǫ ≲ǫ Mα0+2ǫ j+1 |ξ|−α0+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='23) We apply Borel-Cantelli lemma to Ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='22), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='23) and the fact that ∥µj∥ is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' bounded imply that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' we have the inequality |�νj+1(ξ) − �νj(ξ)| ≲ M−σ/2 j if |ξ| ≤ Mj+1, ≲ǫ M(α0−σ)/2+ǫ j |ξ|(−α0+ǫ)/2 if Mj+1 ≤ |ξ| ≤ mj+1M2 j , ≲ǫ M1−α0+ǫ j+1 |ξ|−1/2 if |ξ| ≥ mj+1M2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let j1, j2 be numbers such that Mj2 ≤ |ξ| ≤ Mj2+1 and mj1+1M2 j ≤ |ξ| ≤ mj1+2M2 j1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we obtain |�νn(ξ) − �ν0(ξ)| ≲ǫ � 0≤j≤j1 M1−α0+ǫ j+1 |ξ|−1/2 + � j1≤j≤j2 M(α0−σ)/2+ǫ j |ξ|(−α0+ǫ)/2 + � j≥j2 M−σ/2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 21 For sufficiently small ǫ > 0, we get |�νn(ξ) − �ν0(ξ)| ≲ǫ M1−α0+ǫ j1+1 |ξ|−1/2 + M(−σ+α0)/2+ǫ j2 |ξ|(−α0+ǫ)/2 + M−σ/2+ǫ j2 ≲ǫ |ξ|−σ/2+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |�ν0(ξ)| ≲ |ξ|−1/2, by letting n → ∞, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Global restriction estimate As in Theorem 4 in [13], we need to randomize the choice of Sj,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let {nj}j∈N be a sequence of positive numbers which satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let {µj}j∈N∪{0} be a sequence of random measures on [0, 1] which satisfies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (1) µ0, µ1, · · · are constructed through the process in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (2) For each j ∈ N and for each a ∈ Aj, the set Sj,a are chosen randomly and independently from Σj with probability distribution such that E(µj(x)|Ej−1) = µj−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we get the corresponding sequence of random measures {νj}j∈N∩{0} and its limiting measure ν satisfies all conclusions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' As an example of a random measure ν which satisfies conditions Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1, we can consider random translate of Sj,a described in [13, Section 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Once we prove the following lemma, we can prove Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 and it implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let p = 6/α where 0 < α < 1 and assume that ν is the measure constructed in Section 2 and satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For sufficiently large R > 0, suppose that {Qi}M i=1 be a sparse collection of R-cubes in R2, which means that their centers x1, · · · , xM are RBMB-separated from each other for some sufficiently large constant B which will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If f is a function supported on ∪M j=1Qi, then we have ∥ �f∥L2(dν) ≲q,ǫ Rǫ∥f∥Lq(R2) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) where 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let f = �M i=1 fφi where φi is a Schwartz function supported on a ball of radius 2R centered at xi and |φi(x)| ≥ 1 if x ∈ B2(xi, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' ∥ �f∥2 L2(dν) ≤ � | M � i=1 � fφi|2dν = � M � i=1 |� fφi|2dν + � � i̸=j � fφi� fφjdν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' It follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 that M � i=1 � |� fφi|2dν ≲q,ǫ Rǫ M � i=1 ∥fφi∥2 Lq(R2) ≤ Rǫ � M � i=1 ∥fφi∥q Lq(R2) �2/q = Rǫ∥f∥2 Lq(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) 22 DONGGEUN RYOU In the last inequality, we used that 1 ≤ q ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For the second term, Young’s convolution inequality implies that � � i̸=j � fφi� fφjdν = � i̸=j �� fφi(x)fφj(y)� dν(y − x)dxdy ≤ � i̸=j ∥fφi∥Lq(R2)∥fφj∥Lq(R2)∥� dν1i,j∥Lr(R2) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) where 2 q + 1 r = 2, so that r = p 2 ≥ 1 and 1i,j = 1B2(2R)(x − (xi − xj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since xi and xj are separated at least by RBMB, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) implies that ∥� dν1i,j∥Lr(R2) ≲ǫ (RBMB)−α/2+ǫR2/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we choose B such that B ≥ 2 α/2 − ǫ, then we have ∥� dν1i,j∥Lr(R2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4), we obtain that � � i̸=j ( �f ∗ �φi)( �f ∗ �φj)dν ≲ � i̸=j ∥fφi∥Lq(R2)∥fφj∥Lq(R2) ≤ � M � i=1 ∥fφi∥2 Lq(R2) �1/2� M � j=1 ∥fφj∥2 Lq(R2) �1/2 ≤ ∥f∥2 Lq(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) In the last inequality, we used again that 1 ≤ q ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5), we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ Let us turn to the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' When we let n1/2 k = mk, N 1/2 k = Mk, βk = |Ek|−1 and α0 = α, then {µk}k∈N∪{0} satisfies all conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, �ν satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, we can use Tao’s epsilon removal argument to prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 replaces Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 in [23] and rest of the proof (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) is same as in [23, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3 The proof is based on Knapp-type example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ν is supported on Pd−1, there exists a measure µ on [0, 1]d−1 such that � f(x)dν = � f(x′, |x′|2)dµ for all ν-measurable function f on Rd where x′ ∈ Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) implies that for any 0 < ǫ < α, µ(B(x′, r)) ≲α,ǫ rα−ǫ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 23 for all x′ ∈ Rd−1 and ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The measure µ is also supported on a set of Hausdorff dimension α in Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, µ(B(x′, r)), ≲α,ǫ rα+ǫ cannot hold by Frostmans lemma (see, for example [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, for any ǫ > 0, there exist sequences of {ai}i∈N and {ri}i∈N such that limi→∞ ri = 0 and rα+ǫ i ≲α,ǫ µ(B(ai, ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) Assume that the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) holds for some p and q and let hi(x′) = 1[0,1]d−1(r−1 i (x′ −ai)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, we have � � ���� � hi(x′)e(−x′ · ξ′ − |x′|2ξd)dµ(x′) ���� p dξ �1/p ≲p,q � � |hi(x′)|qdµ �1/q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) where ξ′ ∈ Rd−1 and ξd ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1), we obtain that � � |hi(x′)|qdµ �1/q ≲α,ǫ r(α−ǫ)/q i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) By change of variable, we get the following lower bound of the left-hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' r−(d+1)/p i � � ���� � 1[0,1]d−1(r−1 i (x′ − ai))e(−r−1 i x′ · ξ′ − r−2 i |x′|2ξd)dµ(x′) ���� p dξ �1/p ≥ r−(d+1)/p i � � ���� � 1[0,1]d−1(r−1 i (x′ − ai)) e(−r−1 i (x′ − ai) · (ξ′ + 2ξdr−1 i ai) − r−2 i ξd|x′ − ai|2)dµ(x′) ���� p dξ �1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) Let us consider when |ξ′ + 2ξdr−1 i ai| ≤ 1/100 and |ξd| ≤ 1/100, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) implies that � � ���� � hi(x′)e(−x′ · ξ′ − |x′|2ξd)dµ(x′) ���� p dξ �1/p ≳ r−(d+1)/p i ���� � 1[0,1]d−1(r−1 i (x′ − ai))dµ(x′) ���� ≳α,ǫ r−(d+1)/p+α+ǫ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6), we get r−(d+1)/p+α+ǫ i ≲α,ǫ,p,q r(α−ǫ)/q i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, α q − ǫ q ≤ −d + 1 p + α + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ǫ is arbitrary, p ≥ q′(d + 1)/α if the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Appendix Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' First, assume that S1 is a subset of [−k0, k0]d ∩ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let ψ be a non- negative smooth function supported on [0, 1]d such that ψ ≥ 1 on [1/4, 3/4]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' For x ∈ [−k, k]d, 1 ≤ � |n|≲k |ψ(x − n)|p + � |n′|≲1 � |n|≲k |ψ(x − n − n′/2)|p 24 DONGGEUN RYOU where n, n′ ∈ Zd, but n′ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, we have ∥ � a∈S1 � b∈S2,a ca,be((ka + b) · x)∥p Lp([0,1]d) = � [0,k]d ���� � a∈S1 � b∈S2,a ca,be �� a + b k � x ����� p k−ddx ≤ � |n|≲k k−d � [0,1]d ���� � a∈S1 � b∈S2,a ca,be(a · x)e � b k · (x + n) � ψ(x) ���� p dx + � |n′|≲1 � |n|≲k k−d � [0,1]d ���� � a∈S1 � b∈S2,a ca,be(a · (x + n′/2))e � b k · (x + n + n′/2) � ψ(x) ���� p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We will show that the first term is bounded by a constant multiple of C1C2 � � a∈S1 � b∈S2,a |ca,b|2 �p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) Then, the second term will be also bounded by a constant multiple of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1) by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ψ is supported on [0, 1]d, we have ψ(x)e(b · x/k) = � m∈Zd �ψb(m)e(m · x) where �ψb(m) is the m-th Fourier coefficient of ψ(x)e(b · x/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since � m | �ψb(m)| ≲ 1 and S1 is a subset of [−k0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' k0]d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' we obtain � � |n|≲k k−d � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1]d ���� � a∈S1 � b∈S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='a ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='be(a · x)e � b k · (x + n) � ψ(x) ���� p dx �1/p ≤ � m∈Zd � � |n|≲k k−d � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1]d ���� � a∈S1 � b∈S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='a ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='be �b · n k � �ψb(m)e((a + m) · x) ���� p dx �1/p ≤ � m∈Zd � � |n|≲k k−d � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1]d ���� � a∈S1 � � b∈S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='a ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='be �b · n k � �ψb(m) � e(a · x) ���� p dx �1/p ≤ C1 � m∈Zd � � |n|≲k k−d � � a∈S1 ���� � b∈S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='a ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='be �b · n k � �ψb(m) ���� 2�p/2�1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2) NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 25 In the last inequality, we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Now, let us consider when |m| ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since p ≥ 2 and |ψ| ≲ 1, we get � � |n|≲k k−d � � a∈S1 ���� � b∈S2,a ca,be �b · n k � �ψb(m) ���� 2�p/2�1/p ≤ � � a∈S1 � � |n|≲k k−d ���� � b∈S2,a ca,be �b · n k � � [0,1]d ψ(x)e �� b k − m � x � dx ���� p�2/p�1/2 ≲ � � a∈S1 � k−d � |n|≲k � [0,1]d ���� � b∈S2,a ca,be � b k · (x + n) ����� p dx �2/p�1/2 ≲ � � a∈S1 ∥ � b∈S2,a ca,be(b · x)∥2 p([0,1]d) �1/2 ≤ C2 � � a∈S1 � b∈S2,a |ca,b|2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) If |m| ≥ 10, since ψ is supported on [0, 1]d and decreases rapidly, we can use the principle of non-stationary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let (b/k − m)i denote the i-th coordinate of (b/k − m) and denote ∇b,k,m by ∇b,k,m = d � i=1 ���� b k − m ���� −2� b k − m � i ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Note that �ψb(m) = � [0,1]d(∇2 b,k,mψ(x))e �� b k − m � x � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' If we let ˜ca,b,i,j,m = ca,b ���� b k − m ���� −4� b k − m � i � b k − m � j , then ���� � b∈S2,a ca,be �b · n k � �ψb(m) ���� ≤ d � i,j=1 � [0,1]d ���� � b∈S2,a ˜ca,b,i,j,me � b k · (x + n) � ∂ijψ(x) ����dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since |∂i,jψ| ≲ 1 uniformly in i and j, we obtain that � � |n|≲k k−d � � a∈S1 ���� � b∈S2,a ca,be �b · n k � �ψb(m) ���� 2�p/2�1/p ≲ d � i,j=1 � � a∈S1 � k−d � |n|≲k � [0,1]d ���� � b∈S2,a ˜ca,b,i,j,me � b k · (x + n) ����� p dx �2/p�1/2 ≲ d � i,j=1 � � a∈S1 ∥ � b∈S2,a ˜ca,b,i,j,me(b · x)∥2 p([0,1]d) �1/2 ≤ C2 d � i,j=1 � � a∈S1 � b∈S2,a |˜ca,b,i,j,m|2 �1/2 ≲d C2|m|−2 � � a∈S1 � b∈S2,a |cab|2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' 26 DONGGEUN RYOU In the last inequality, we used that |b/k − m| ≈ |m| since |m| ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, � |m|≥10 � � |n|≲k k−d � � a∈S1 ���� � b∈S2,a ca,be �b · n k � �ψb(m) ���� 2�p/2�1/p ≲ C2 � |m|≥10 |m|−2 � � a∈S1 � b∈S2,a |ca,b|2 �1/2 ≲ C2 � � a∈S1 � b∈S2,a |ca,b|2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4) By (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='3) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='4), it follows that � � |n|≲k k−d � [0,1]d ���� � a∈S1 � b∈S2,a ca,be(a · x)e � b k · (x + n) � ψ(x) ���� p dx �1/p ≲ C1C2 � � a∈S1 � b∈S2,a |ca,b|2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' This proves Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1 when S1 is a subset of [−k0, k0]d∩Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The estimate holds for arbitrary k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, letting k0 → ∞ finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' The inequality Kp(Ej) ≲p Dp(Ej) is well known, for example, see [8, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Thus, it suffices to prove the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let R be a positive large number and S be a subset of [−R, R]d ∩ Zd such that ∥ � a∈S cae(a · x)∥Lp([0,1]d) ≤ C � � a∈S |ca|2 �1/2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5) and let f = � a∈S fa where �fa(ξ) = �f(ξ)1[0,1]d(Rξ − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Let us consider a Schwartz function η such that |η| ≥ 1 on [−1, 1]d and �η is supported on [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Recall that Bd(R) is a cube of side length R in Rd centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We will show that ∥f∥Lp(Bd(R)) ≲ C � � a∈S ∥fa∥2 p(ηR) �1/2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) where ηR(x) = η(x/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' And (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6) implies that ∥f∥Lp(Rd) ≲ C � � a∈S ∥fa∥2 p(Rd) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) For the proof of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='7) from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6), see [8, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='15] and [14, Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Then, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='2 easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' We have ∥f∥Lp(Bd(R)) ≤ ∥fηR∥Lp(Bd(R)) = ∥ � a∈S � �fa ∗ �ηR(ξ)e(−x · ξ)dξ∥Lp(Bd(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS ON THE PARABOLA 27 The function �fa∗�ηR is supported on a cube of side length O(R−1) centered at a/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Therefore, ∥f∥Lp(Bd(R)) ≤ ∥fηR∥Lp(Bd(R)) ≲ ∥ � |c|≲1 � a∈S � [0,1]d �fa ∗ �ηR(R−1(ξ + a + c))e(−x · R−1(ξ + a + c))R−ddξ∥Lp(Bd(R)) ≤ R−d � |c|≲1 � [0,1]d ∥ � a∈S �fa ∗ �ηR(R−1(ξ + a + c))e(−x · R−1(ξ + a + c))∥Lp(Bd(R))dξ ≤ R−d+d/p � |c|≲1 � [0,1]d ∥ � a∈S �fa ∗ �ηR(R−1(ξ + a + c))e(−x · (ξ + a + c))∥Lp(Bd(1))dξ ≲ R−d+d/p � |c|≲1 � [0,1]d ∥ � a∈S �fa ∗ �ηR(R−1(ξ + a + c))e(−x · a)∥Lp(Bd(1))dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' By (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='5), we obtain that ∥f∥Lp(Bd(R)) ≲ CR−d+d/p � |c|≲1 � [0,1]d( � a∈S | �fa ∗ �ηR(R−1(ξ + a + c))|2)1/2dξ ≲ CR−d+d/p � |c|≲1 � � a∈S � [0,1]d | �fa ∗ �ηR(R−1(ξ + a + c))|2dξ �1/2 ≲ CR−d/2+d/p � � a∈S ∥ �fa ∗ �ηR(ξ)∥2 L2(Rd) �1/2 ≤ CR−d/2+d/p � � a∈S ∥faηR∥2 L2(Rd) �1/2 ≤ CR−d/2+d/p � � a∈S ∥fa∥2 Lp(ηR)∥ηR∥1−2/p L1(Rd) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Since ∥ηR∥L1 ≈ Rd, this completes the proof of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' □ References [1] Jong-Guk Bak and Andreas Seeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Extensions of the Stein-Tomas theorem.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Wolff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Lectures on harmonic analysis, volume 29 of University Lecture Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' With a foreword by Charles Fefferman and a preface by Izabella �Laba, Edited by �Laba and Carol Shubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FAT4oBgHgl3EQfrB3j/content/2301.08651v1.pdf'} +page_content=' Department of Mathematics, University of Rochester, Rochester, NY, USA Email address: dryou@ur.' metadata={'source': 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Mikhaela, Jeremy P. +Voiseya, Keith Goatmana, Alexander Weira, Nicolas Pugeaultb, Sotirios A. Tsaftarisc,d, Alison Q. O’Neila,c +aCanon Medical Research Europe, Bonnington Bond, 2 Anderson Pl, Edinburgh EH6 5NP, United Kingdom +bUniversity of Glasgow, Glasgow G12 8QQ, United Kingdom +cUniversity of Edinburgh, Kings Buildings, Edinburgh EH9 3FG, United Kingdom +dThe Alan Turing Institute, London, United Kingdom +A R T I C L E I N F O +2000 MSC: 68T99, 92C55, 68U10 +Keywords: Anomaly detection, Unsuper- +vised learning, Autoencoder, Denoising, +Diffusion +A B S T R A C T +Pathological brain lesions exhibit diverse appearance in brain images, in terms of in- +tensity, texture, shape, size, and location. Comprehensive sets of data and annotations +are difficult to acquire. Therefore, unsupervised anomaly detection approaches have +been proposed using only normal data for training, with the aim of detecting outlier +anomalous voxels at test time. Denoising methods, for instance classical denoising +autoencoders (DAEs) and more recently emerging diffusion models, are a promising +approach, however naive application of pixelwise noise leads to poor anomaly detec- +tion performance. We show that optimization of the spatial resolution and magnitude +of the noise improves the performance of different model training regimes, with simi- +lar noise parameter adjustments giving good performance for both DAEs and diffusion +models. Visual inspection of the reconstructions suggests that the training noise influ- +ences the trade-off between the extent of the detail that is reconstructed and the extent +of erasure of anomalies, both of which contribute to better anomaly detection perfor- +mance. We validate our findings on two real-world datasets (tumor detection in brain +MRI and hemorrhage/ischemia/tumor detection in brain CT), showing good detection +on diverse anomaly appearances. Overall, we find that a DAE trained with coarse noise +is a fast and simple method that gives state-of-the-art accuracy. Diffusion models ap- +plied to anomaly detection are as yet in their infancy and provide a promising avenue +for further research. +Code for our DAE model and coarse noise is provided at: https://github.com/A +ntanasKascenas/DenoisingAE. +1. Introduction +Anomaly detection is a fundamental task in medical image +analysis, mimicking the initial review that a radiologist per- +forms of imaging studies to identify abnormal regions which +should be reviewed and characterized further. Supervised ma- +chine learning methods have shown promising results, however +comprehensive supervised pathology detection methods require +extensive and heterogeneous training sets which are costly to +annotate and difficult to acquire. +Conversely, unsupervised +anomaly detection (UAD) methods require only identification +e-mail: antanas.kascenas@mre.medical.canon (Antanas Kascenas) +of a healthy cohort of patients for training (therefore these meth- +ods are sometimes regarded as semi-supervised), after which +they may be applied to detect out-of-distribution anomalous re- +gions in test data. +Autoencoder deep learning methods have been commonly +used for UAD in brain scans (Baur et al., 2021), relying on the +assumption that normal data as seen during training will be re- +constructed better than unseen anomalous – potentially patho- +logical – regions. A classical approach is denoising autoen- +coders (DAEs) (Vincent et al., 2008) in which corrupting noise +is added to the input and the network must learn to remove the +noise in order to reconstruct the original image. This training +task of removing noise can be regarded as a proxy for the test +time task of removing anomalies in order to reconstruct an im- +arXiv:2301.08330v1 [eess.IV] 19 Jan 2023 + +2 +Antanas Kascenas et al. / Preprint (2023) +age of normal appearance. It was shown in Kascenas et al. +(2021) that for detection of brain tumors in MRI data, train- +ing with naive pixelwise noise gave poor anomaly detection +performance, while training with coarse noise (see Algorithm +1) gave good performance. Following simple optimization of +noise resolution and magnitude, a classical DAE outperformed +more complex previous state-of-the-art models. +Our contributions are as follows: +1. We take the simple and effective DAE that was proposed +by Kascenas et al. (2021) for brain anomaly detection in +medical 2D MRI images, and investigate its application to +3D CT images with a range of anomalies, showing that +optimal noise resolution and magnitude parameters are +largely transferable between modalities and anomalies. +2. We analyze noise type in the alternative denoising model +paradigm of diffusion models, showing that similar adjust- +ment of the type of noise gives accuracy gains also for this +alternative denoising approach. +3. We additionally analyze an alternative noise type (Simplex +noise) which has been recently advocated by Wyatt et al. +(2022), showing our proposed coarse noise to be superior +in most anomaly detection setups. +4. Finally, since we consider training an anomaly detection +algorithm in a practical setting where a large uncurated +dataset of scans is available, we demonstrate that NLP +analysis of radiology reports can be effectively used to se- +lect the training cohort of normal scans. +2. Related Work +Anomaly detection is an open-ended task for which a variety +of approaches have been proposed. +2.1. Autoencoder approaches to anomaly detection +Many modifications to the standard autoencoder pipeline +have been proposed to improve performance on the task of +anomaly detection, which has the potentially conflicting twin +goals of reconstructing normal regions of the original brain scan +with high fidelity, while reconstructing any anomalous regions +with poor fidelity (in order to distinguish them). +Variational autoencoders (VAEs) (Zimmerer et al., 2019) +constrain the latent bottleneck representation to follow a pa- +rameterized multivariate Gaussian distribution. Zimmerer et al. +(2021) further add a context-encoding task and combine recon- +struction error with density-based scoring to obtain the anomaly +scores, while Chen et al. (2020) use an iterative gradient descent +restoration process at test time in restoration-VAE, replacing the +reconstruction error with a restoration error to estimate anomaly +scores. +Architectural changes have also been proposed. +Atlason +et al. (2019); Baur et al. (2018) introduce convolutional autoen- +coders and higher capacity spatial bottlenecks instead of fully- +connected (dense) bottlenecks to achieve better reconstruction. +Chen and Konukoglu (2018) use constrained autoencoders to +improve latent representation consistency in anomalous images +at test time. Bayesian skip-autoencoders Baur et al. (2020a) +use skip connections with dropout to improve reconstruction +and allow uncertainty to be measured via dropout stochasticity. +Baur et al. (2020b) use scale-space autoencoders to compress +and reconstruct different frequency bands of brain MRI using +the Laplacian pyramid to achieve higher reconstruction fidelity. +The UAD autoencoder framework of encoder-decoder com- +ponents and reconstruction error for anomaly scores has fea- +tured in more complex approaches. Schlegl et al. (2019) train a +generative adversarial network called f-AnoGAN which reuses +the generator and discriminator to train an autoencoder with +both reconstruction and adversarial losses for the anomaly de- +tection task. Pinaya et al. (2022a) combine a vector quantized +VAE (VQ-VAE) to encode an image with a transformer model +to resample low-likelihood latent variables in order to produce +reconstructions with fewer reproduced anomalies. +Baur et al. (2021) have performed an evaluation of some +common autoencoder methods for anomaly detection in brain +MRI, finding restoration-VAE Chen et al. (2020) and f- +AnoGAN Schlegl et al. (2019) to be among the best. How- +ever, more recently Meissen et al. (2021a) showed that most +autoencoder-based MRI UAD methods can be outperformed by +a simple thresholding baseline, applied to the FLAIR sequence +after histogram equalization preprocessing. This training-free +approach detected hyperintense brain tumor and multiple scle- +rosis lesions better than most UAD approaches that require +healthy data to train. +Our work relies on the same principle of using reconstruction +error for anomaly detection as most autoencoding methods but +we use noise instead of architectural constraints to make the +autoencoding training task non-trivial. +2.2. Denoising methods +The above evaluations of medical anomaly detection meth- +ods largely omitted consideration of classical denoising autoen- +coders (DAEs) (Vincent et al., 2008) and other methods exploit- +ing noise, however a few approaches have shown promise. Alex +et al. (2017) applied DAEs as pretraining for brain lesion detec- +tion with limited labels and for simple novelty detection using +patch-based masking. Collin and De Vleeschouwer (2021) use +a DAE for anomaly detection in industrial vision with a stain +noise model with randomized shape, color, size and location. +Bengs et al. (2021) use 3D VAEs with spatial patches replaced +with voxelwise noise to train an inpainting model for anomaly +detection. Generative diffusion models (Ho et al., 2020), in +which noise is added and removed over many iterations, have +been used in the context of anomaly detection (Pinaya et al., +2022b; Wyatt et al., 2022) by assuming that models trained on +only healthy data will fail during reconstruction of anomalous +features. +Recently, Kascenas et al. (2021) showed that when noise +coarseness and intensity are adjusted, a DAE can achieve com- +petitive results for the detection of tumors in brain MRI images. +Further, (Daras et al., 2022; Wyatt et al., 2022) have shown +recently that diffusion models can be trained with degradation +functions other than Gaussian noise. In fact, Wyatt et al. (2022) +showed that using Simplex noise in diffusion models can signif- +icantly improve anomaly detection performance over traditional +Gaussian noise. + +Antanas Kascenas et al. / Preprint (2023) +3 +Test time +Input +Reconstruction +Reconstruction +Residuals +Anomaly scores +Ground truth +upsample +& +mask +Normal image +Noisy input +post-process +residuals +Noise +Denoising +model +Reconstruction +procedure +Noise generation +diffusion models input +additional parameter +“t”, corresponding to +noise magnitude +Corruption +process C +Training time +- +| +| +Fig. 1. Workflow for denoising anomaly detection methods. During training (top), noise is added to the normal image, and the network is trained to +reconstruct the original image. At test time (bottom), different methods are applied to reconstruct and post-process the potentially anomalous input image +to produce the anomaly score. For the simple denoising autoencoder (DAE) approach, the denoising model is applied once to the input and the anomaly +score is simply the reconstruction residuals followed by median filtering. However, the diffusion models apply more complex iterative noise addition +followed by iterative denoising to obtain the reconstruction. +In this paper, we examine this theme of the role of noise in +anomaly detection, investigating and comparing types of noise +and denoising methods in a common setting. +3. Method +In summary, we employ one of three types of noise (Gaus- +sian, Simplex or coarse) to train neural network models to de- +noise healthy images. At test time, anomalies are detected via +reconstruction error (see Figure 1). Below we describe this pro- +cess in more detail. +3.1. Denoising Models for Anomaly Detection +Denoising neural networks ϵθ receive corrupted data ˜x as +input and are trained to recover original (uncorrupted) data +ˆx = ϵθ (˜x). We consider the corruption process to have a con- +ditional distribution C (˜x | x, n), degrading x into ˜x with the in- +jection of some noise n. Training a denoising neural network ϵθ +with parameters θ can then be written as: +θ∗ = arg min +θ +Ex∼pdata, ˜x∼C(x) +� +∥ϵθ (˜x) − x∥2� +. +(1) +The resulting network learns to reconstruct samples x that be- +long to pdata. +However, we consider a distribution panomaly +which is similar to pdata but contains features (anomalies) that +are not present in pdata. As shown by Kascenas et al. (2021), an +anomalous sample x′ ∼ panomaly will not be reconstructed ap- +propriately by the denoising network ϵθ. The training and test +pipelines are visualized in Figure 1. +The anomalies are detected by taking the absolute difference +between the input data and the resulting reconstruction |x′ − ˆx′|. +3.2. Denoising Autoencoder (DAE) approach +We implement a simple denoising deep autoencoder neural +network, and use reconstruction error to detect and localize +anomalies at test time. The network has a U-Net (Ronneberger +et al., 2015) style architecture with skip connections which en- +ables significantly better image reconstructions compared to +bottleneck architectures such as the VAE (see Appendix A). We +note that any neural network architecture yielding dense predic- +tions (e.g. segmentations) could be trained as a DAE. Details of +the network architecture and training procedure can be found in +Section 6 and Appendix B. During training, we corrupt images +according to: +C (˜x | x) =⇒ ˜x = x + σn, +(2) +where σ is the standard deviation which controls the inten- +sity magnitude and n is noise. Classically, n is sampled from +a Gaussian distribution N(0, I), but in Section 3.4 we explore +more efficient techniques in the context of anomaly detection. +At inference time, the DAE is used to localize anomalies by +calculating pixelwise/voxelwise anomaly scores A(x). If we de- +note the input image as x, the number of image channels as M +(e.g. for multiple imaging sequences or imaging modalities), + +n~XxX'A(x')4 +Antanas Kascenas et al. / Preprint (2023) +t = 0.1T +t = 0.3T +t = 0.6T +Fig. 2. Diffusion model input at different timesteps t. Noise component is +larger at further timesteps. +a background mask of pixels with x intensities across channels +equal to 0 as B, the median filtering operation as f, and the +reconstruction as ˆx, then the anomaly score can be defined as: +A(x) = f +�������(1 − B) ⊙ +M +� +m +|xm − ˆxm| +M +������� +(3) +No noise is used at test time. +3.3. Diffusion model approach +We next explore the diffusion model methods developed in +Pinaya et al. (2022b) and Wyatt et al. (2022). Both methods fol- +low a training strategy that was initially proposed by Ho et al. +(2020). In contrast to the denoising model used in the DAE, dif- +fusion denoising models are trained to predict the noise rather +than to reconstruct the original image itself. In particular, con- +sider a model trained to find optimal parameters as +θ∗ = arg min +θ +Ex∼pdata, t∼U(0,T), ˜x∼C(x,t) +� +λ(t) ∥ϵθ (˜x, t) − n∥2� +, (4) +where the timestep t is sampled from a uniform distribution be- +tween 0 and T (T is a hyperparameter which we set to 1000) +and λ(t) is a loss weighting term. Here the corruption process +C (˜x | x, t) depends also on t which controls the strength of the +corruption through αt as described in Song et al. (2021) , ac- +cording to: +C (˜x | x, t) =⇒ ˜x = √αtx + +� +1 − αtn, +(5) +where the coefficient αt runs from α0 = 1 (original image) +through to αT = 0 (noise). +Figure 2 shows examples of a +corrupted image for different values of t. Training with multi- +ple t values corresponds to training with multiple noise magni- +tudes. Training with C (˜x | x, t) has been extensively studied in +the diffusion probabilistic modeling (DPM) literature (Ho et al., +2020). For example, when using Gaussian noise, adding noise +with high standard deviation causes the network to focus on +coarse features while low standard deviation noise causes focus +on texture and other high frequency detail. Most importantly, +training to denoise at multiple magnitudes enables image gen- +eration; we refer the reader to Ho et al. (2020) for details on the +image generation procedure. The ability of DPMs to denoise at +different noise magnitudes as well as its generative power has +inspired methods for anomaly detection using diffusion models +(Pinaya et al., 2022b; Wyatt et al., 2022; Sanchez et al., 2022). +Once the diffusion denoising model has been trained, we in- +vestigate two inference techniques to detect anomalies: +Algorithm 1 +Generation of noise with spatial resolution α, and output shape +a × b × c +1: procedure Noise(α, a, b, c) +2: +n ∼ N(0, I) ∈ Rα,α,α +3: +n ← upsample(n, (a, b, c)) +▷ Bilinearly upsample +4: +x ∼ U(0, a) +▷ Uniformly sample in range (0, a) +5: +y ∼ U(0, b) +6: +z ∼ U0, c) +7: +n ← translate(n, (x, y, z)) +▷ Randomly translate +8: +return n +▷ The generated noise is n +9: end procedure +1. Reconstruction (Wyatt et al., 2022) - In the AnoDDPM +method, noise is injected at a selected magnitude; we use +t = 0.25T because this was found to be the best in Wyatt +et al. (2022). We then run the DDPM (Ho et al., 2020) +iterative generation from t = 0.25T → 0, using the noisy +image as the starting point (i.e. 250 steps where T=1000). +We follow Wyatt et al. (2022) in averaging the reconstruc- +tions across 5 runs of this generation procedure. +2. KL divergence + inpainting (Pinaya et al., 2022b) - In +this method, noise is injected with different magnitudes +(i.e. different t ∈ [0.4T, 0.6T]) and the difference is com- +puted between the predicted output ϵθ (˜x, t) and the ex- +pected output n. A heatmap is obtained by averaging the +difference images produced by different values of t; since +DPMs have a probabilistic interpretation, Pinaya et al. +term this the Kullback–Leibler divergence. The KL diver- +gence heatmap is binarized at a threshold corresponding +to the 97.5 percentile value of the heatmap, to produce a +mask for the region of interest. The masked region only is +then reconstructed by the model using the DPM i.e. “in- +painted” (Lugmayr et al., 2022) by running iterative gen- +eration from t = 0.5T → 0. The final heatmap is the +difference between the original and inpainted images. +3.4. Coarse noise generation +It was shown in Kascenas et al. (2021) that training with +lower resolution noise leads to better anomaly detection than +naive pixelwise Gaussian noise for a DAE detecting brain tu- +mors in brain MRI data; in this paper we are interested to fur- +ther investigate the impact of the type of noise with different +data and denoising models. We generate the lower resolution +(“coarse”) noise by sampling random pixelwise Gaussian noise +at a low resolution and bilinearly (trilinearly for 3D) upsam- +pling it to the input resolution. We then randomly translate the +generated noise to avoid consistent upsampling patterns. See +Figure 4 for examples of generated noise and Algorithm 1 for +the pseudocode. +To examine the effect of noise, we take the DAE and vary the +noise resolution α and the standard deviation σ used for gener- +ating Gaussian noise before upsampling (see Figure 3) on the +two datasets described later in Section 4. We find that a reason- +ably coarse noise is critical, as DAE models trained using stan- +dard pixel-level noise (e.g. generated at 128 × 128 resolution + +Antanas Kascenas et al. / Preprint (2023) +5 +0.1 +0.2 +0.3 +0.4 +Noise magnitude, σ +0.6 +0.7 +0.8 +AUPRC +MRI +1 +2 +4 +8 +16 +32 +64 +128 +Noise resolution, α +0.25 +0.50 +0.75 +AUPRC +0.02 0.05 +0.10 +0.20 +0.30 +Noise magnitude, σ +0.4 +0.6 +AUPRC +CT +1 +2 +4 +8 +16 +32 +64 +128 +Noise resolution, α +0.2 +0.4 +0.6 +AUPRC +Fig. 3. Noise coarseness and magnitude ablation results on BraTS head MRI (left) and iCAIRD head CT (right) data. Magnitude ablation uses noise +resolution α = 16. Coarseness ablation uses σ = 0.2. Error bars show standard deviation across three runs. +1×1 +2×2 +4×4 +8×8 +16×16 +32×32 +64×64 +128×128 +Fig. 4. Samples of 2D noise generated at different resolutions, from 1×1 +through to 128×128. The background mask B is also visible. +for 128 × 128 pixel 2D head MRI slices) or using the opposite +extreme of image-level noise (i.e. generated at 1 × 1 resolu- +tion) perform significantly worse. DAEs appear to be less sen- +sitive to the magnitude of the noise (σ of the generating Gaus- +sian distribution). The noise parameters are robust and transfer +well between modalities (i.e. from MRI to CT) and patholo- +gies (e.g. tumor to hemorrhage) and 2D to 3D as long as the +image field-of-view and resolution are comparable i.e. we used +1.62mm2/pixel for 2D MRI and 2mm3/voxel for 3D CT. +For diffusion models, we similarly modify the corruption +process C, adopting the noise generation process described in +Algorithm 1 instead of pixel-wise Gaussian noise both during +training and when applying each of the inference techniques. +4. Datasets +We evaluate anomaly detection in 2D brain MRI slices and +3D brain CT volumes using the two datasets described below. +4.1. BraTS challenge dataset: Brain MRI +We evaluate the anomaly detection performance on the sur- +rogate task of brain tumor segmentation using data from the +BraTS 2021 challenge (Menze et al., 2014; Bakas et al., 2017, +2018). +This data comprises native (T1), post-contrast T1- +weighted (T1Gd), T2-weighted (T2), and T2 Fluid Attenuated +Inversion Recovery (FLAIR) modality volumes for each patient +from a variety of institutions and scanners. +4.1.1. Selecting the training and test data +We randomly split the dataset into 938 training, 62 valida- +tion, and 251 test patients. During training, we use only slices +that do not contain any tumor pixels, under the assumption that +these non-tumor slices represent healthy tissue. At test time, +we consider the union of the tumor sub-region labels to be the +anomalous regions. +4.1.2. Preprocessing +The data has already been co-registered, skull-stripped and +interpolated to the same resolution. Labels are provided for +tumor sub-regions: the GD-enhancing tumor, the peritumoral +edema, and the necrotic and non-enhancing tumor. +For the data input to the models, we stack all four modal- +ities at the channel dimension for each patient. We normalize +(rescale) the pixel intensity values in each modality of each scan +by dividing by the 99th percentile foreground voxel intensity. +Values are scaled to a range of [−1, 1] for diffusion methods + +16 +Antanas Kascenas et al. / Preprint (2023) +Table 1. Data filtering steps towards obtaining a healthy training set for +anomaly detection. +Filtering step +Images +Patients +Initial Data cohort +16,559 +7,122 +After filtering on report labels from +Schrempf et al. (2021) +2,350 +1,788 +After filtering out follow-up scans +1,020 +961 +After rapid manual image review +996 +939 +Healthy training set +804 +757 +and [0, 1] otherwise. All slices are downsampled to a resolution +of 128×128 (1.62mm/pixel). +4.2. iCAIRD GG&C NHS dataset: Head CT +We use head CT scans obtained through a collaboration with +the Industrial Centre for Artificial Intelligence Research in Dig- +ital Diagnostics (iCAIRD)1. The data has been sourced from +hospitals in the Greater Glasgow & Clyde (GG&C) area in +Scotland and comprises all patients who were diagnosed with a +stroke in the period 2013-2018. The data is pseudonymised and +we obtain access onsite via the West of Scotland Safe Haven +within NHS Greater Glasgow and Clyde via the Safe Haven +Artificial Intelligence Platform (SHAIP) (Wilde et al., 2022). +We have obtained ethical approval to use this data2. +The data was originally collected by identifying hospital ad- +missions which were assigned International Classification of +Diseases (ICD-10)3 codes relating to stroke diagnoses, and then +selecting medical data from the stroke event hospital admission +as well as the documentation from 18 months prior and all prior +images held at the GG&C. In total, the dataset contains infor- +mation about 15,882 stroke events from 10,143 patients and in- +cludes CT images, radiology reports, clinical documents and +structured clinical data. We use 16,559 head CT images avail- +able from 7,122 patients for the purpose of this work and refer +to this as the iCAIRD dataset. +4.2.1. Radiology report NLP for normal scan selection +Identification of normal scans by manual examination of this +large dataset would be time-consuming. +Fortunately, corre- +sponding free text radiology reports are available for most of +the head CT images in the iCAIRD dataset. The reports vary +in depth and exposition reflecting the style and seniority of the +reporting radiologists, but generally describe the radiographic +findings and clinical impressions in the associated CT images. +We use this information to identify and exclude abnormal scans +from our training set. However, comprehensive manual exam- +ination of radiology reports, while faster than examination of +images, would still be slow. +Therefore, we leverage a pre- +viously developed automatic deep learning model (Schrempf +et al., 2020, 2021) which was trained on 357 manually labeled +1https://icaird.com +2West of Scotland Safe Haven ethical approval number GSH19NE004 +3https://www.who.int/standards/classifications/classifi +cation-of-diseases +non-contrast head CT radiology reports and outputs labels for +14 radiographic findings and 19 clinical impressions (see Ap- +pendix C for the list of labels). Each label is assigned one of +the 4 classes: positive, negative, uncertain or not mentioned. +4.2.2. Selecting the training data +Defining Normal vs Abnormal: We aim to obtain a training +set that is as healthy as possible in order to detect as many +anomalies as possible at test time. However, since the dataset +is from an elderly stroke population (mean age of 72 years), re- +ports without any positive findings (labels) are rare. Therefore, +there is a trade-off between how aggressively we filter versus +the size of the final training set. Hence, we include scans for +which the associated reports contain only findings/impressions +that are commonly found in an elderly population, specifically +calcification, atrophy, cerebral small vessel disease and hypo- +density (the latter is most commonly associated with atrophy +and small vessel disease). Applying this more generous defini- +tion of “Normal” leaves a set of 2350 scans from 1788 patients +(see Table 1). +Filtering out follow-up scans: Upon closer manual inspec- +tion we find that many reports are non-exhaustive (note these +are free text rather than structured reports), appearing not to +list all of the findings present in the scan. +This most com- +monly occurs for follow-up scans where the associated report +assumes knowledge of earlier scan reports, usually not explic- +itly re-listing all findings. An example such report would be +“No progression compared to previous scan from 10/22/2021.”. +Thus, absence of positive or uncertain labels does not necessar- +ily equate to absence of pathology. Therefore we further filter +down the remaining cases using keywords and pattern matching +using spaCy (Honnibal et al., 2020), removing reports which +contain references to previous imaging and comparisons. This +keyword filtering leaves 1020 scans from 961 patients. +Rapid manual image review for obvious anomalies: Finally, +we perform a rapid manual review by non-experts which elim- +inates a further 24 scans mostly containing processing issues +(e.g. bone reconstruction, significant artifact, significantly de- +graded scan quality). We use 804 scans from the remaining 996 +cases as our healthy training data. +4.2.3. Selecting and annotating the test data +In addition to the filtered healthy training data, we selected +and annotated a separate set of scans with hemorrhages, is- +chemia and tumors to quantitatively evaluate the methods. The +annotation workflow consisted of several steps: curation, anno- +tation, review and quality assurance. Further details are pro- +vided in Appendix D. The resulting data was split into Test and +Training sets as described below. +Test set: The test set contains voxelwise annotations for 114 +scans of which 104, 23 and 4 contain hemorrhage, ischemia and +tumor ground truth respectively. We use the union of the three +pathologies for evaluating the anomaly detection methods. +Training data for supervised baselines: We further reserve +129 scans annotated with 116 hemorrhage, 30 ischemia and 6 +tumor annotations for training the supervised baselines. + +Antanas Kascenas et al. / Preprint (2023) +7 +4.2.4. Preprocessing +We rigidly register the CT scans to a reference volume and +crop to a fixed field-of-view which includes only the head re- +gion of the scan. Volumes are then resampled to 2mm3 resolu- +tion and windowed to Hounsfield Unit (HU) values from 0 to +80. As for the MRI data, intensities are rescaled to a range of +[−1, 1] for diffusion methods and [0, 1] otherwise. We use ran- +dom flipping and affine transformation data augmentation for +training of all methods. +4.3. qure.ai CQ500: Head CT +We use the CQ500 dataset from qure.ai (Chilamkurthy et al., +2018) for qualitative evaluation (see Figure 7) of the head CT +methods as the data contains similar pathologies (i.e. hemor- +rhages, ischemia). This dataset does not, however, contain any +voxel-level ground truth and could not be used for quantitative +evaluation. +5. Baselines +We compare against a range of common reconstruction-error +based methods as well as providing supervised segmentation +model results trained using ground truth for context. +5.1. 2D brain MRI baselines +We compare the denoising anomaly detection model perfor- +mance against four methods. +Firstly, we implement a stan- +dard VAE (Zimmerer et al., 2019) and f-AnoGAN (Schlegl +et al., 2019) models with pixelwise reconstruction error as the +anomaly scores. Secondly, we use the same VAE model but im- +plement an iterative gradient-based restoration process (Chen +et al., 2020) to produce restoration images. Finally, we apply +the simple thresholding approach from Meissen et al. (2021a) +modified to use median filtering as proposed by (Kascenas +et al., 2021). We use the hyperparameters from the original +works for the deep learning methods but tune manually where +necessary to improve training stability and anomaly detection +performance. +5.2. 3D brain CT baselines +We compare the denoising anomaly detection model perfor- +mance on 3D head CT data against two reconstruction-error +based methods: VAE reconstruction (Zimmerer et al., 2019) +and VAE restoration (Chen et al., 2020). +6. Implementation details +6.1. Noise +Coarse noise is generated by sampling random Gaussian pix- +elwise noise at resolutions of 16×16 and 16×16×16 for 2D and +3D respectively, before bilinearly/trilinearly upsampling to the +input resolution of 128×128 for 2D brain MRI and 80×112×88 +for 3D head CT. The generated noise is then randomly trans- +lated to randomize the centers of the coarse noise peaks that +may occur due to upsampling from very low resolutions. Noise +is generated independently for each image modality in the case +of 2D MRI. We investigate the parameters of the noise, as re- +ported in Section 3.4 (see Figure 3). +Simplex noise is generated using the implementation pro- +vided by Wyatt et al. (2022)4. For DAE experiments with Sim- +plex noise, we scale the generated noise magnitude by a factor +of 0.2. +6.2. Denoising autoencoder +For 2D MRI data, we use a U-Net (Ronneberger et al., +2015) encoder-decoder architecture with three downsam- +pling/upsampling stages. Each encoder stage consists of two +weight-standardized convolutions (Qiao et al., 2019) with ker- +nel sizes of 3 and 64, 128, 256 output channels for the three +stages respectively followed by Swish activations (Ramachan- +dran et al., 2018) and group normalization (Wu and He, 2018). +Average 2 × 2 pooling is used for downsampling. The decoder +architecture mirrors the encoder in reverse, using transposed +convolutional layers for upsampling. Architecture visualization +and further details can be found in Appendix B. +For 3D head CT data, we use an analogous architecture in 3D +with three downsampling/upsampling stages and 32, 64, 128 +output channels for the three stages respectively. +We use mean L2 reconstruction loss in the foreground as the +training objective. 2D DAE Models are trained for 67,200 iter- +ations with a batch size of 16 slices using Adam (Reddi et al., +2018) with a cosine annealed (Loshchilov and Hutter, 2017) +maximum learning rate of 0.0001 with a period of 200 itera- +tions. 3D DAE Models are trained for 25,600 iterations with a +batch size of 3 volumes using Adam with OneCycleLR learning +rate schedule (Smith and Topin, 2019) with a maximum learn- +ing rate of 0.001. +6.3. Diffusion model +We implement a diffusion model with a U-Net-like architec- +ture based on implementation provided by Dhariwal and Nichol +(2021) which includes residual layers, global attention, dropout +and a projection of the timestep embedding to each residual +block. We use T = 1000 diffusion steps with linear noise sched- +ule training models to predict the noise and optimizing the mean +squared loss between the noise which was used for sampling +and the predicted noise. +For 2D MRI data we use the AdamW optimizer (Loshchilov +and Hutter, 2018) with a learning rate of 0.0001 and weight de- +cay of 0.01 with a batch size of 64. Model weights are averaged +by taking the exponential moving average (EMA) with a rate of +0.9999. The 2D U-Net architecture and diffusion training code +can be found on github 5. +For 3D CT data we use the AdamW optimizer with a OneCy- +cleLR learning rate schedule (Smith and Topin, 2019) with a +maximum learning rate of 0.0001 and batch size of 4. Model +weights are averaged by taking the EMA with a rate of 0.95. +4https://github.com/Julian-Wyatt/AnoDDPM +5https://github.com/vios-s/Diff-SCM + +8 +Antanas Kascenas et al. / Preprint (2023) +Gaussian +Simplex +Coarse +Fig. 5. The three noise types tested to train diffusion models. +6.4. VAE reconstruction +VAE models use a similar architecture to their DAE coun- +terparts. Skip connections are removed and a bottleneck with +dimensionality of 128 is added. +For the training objective, +we compute the sum of mean L2 reconstruction error and KL- +divergence with a weight of β = 0.001. +We use the same +training procedure and anomaly score formula as for their DAE +counterparts. +6.5. VAE restoration +Using the VAE model described above, we implement a +restoration method (Chen et al., 2020) to produce the anomaly +scores. We perform the restoration procedure using 100 iter- +ations on individual slices/volumes basing our implementation +on public source code6. Note that due to the iterative nature of +the restoration procedure it takes significantly longer (approx. +×100) to produce predictions compared to the single inference +iteration DAE/VAE reconstruction. +6.6. f-AnoGAN +We adapt the original public implementation7 for the brain +MR data task as follows. We use an additional generator, dis- +criminator and encoder block to account for the higher resolu- +tion. Strided convolutions and transposed convolutions are used +for downsampling and upsampling respectively. We use a batch +size of 32 and learning rates of 0.001, 0.001, 0.00001 for the +generator, discriminator and encoder respectively. The encoder +was trained using κ = 1 × 10−8. +6.7. Thresholding +We follow Meissen et al. (2021a) to obtain results for the +thresholding baseline but omit the connected component filter- +ing as we have found median filtering to be more effective and +computationally efficient (Kascenas et al., 2021). FLAIR se- +quence volumes are used as the anomaly score volumes, fol- +lowing processing by histogram equalization of the foreground +(i.e. excluding surrounding air) and median filtering. +6https://github.com/yousuhang/Unsupervised-Lesion-Detec +tion-via-Image-Restoration-with-a-Normative-Prior +7https://github.com/tSchlegl/f-AnoGAN +Table 2. Relationship between DAE and different diffusion anomaly detec- +tion inference methods and noise used for model training. For the method +of (Pinaya et al., 2022b) we include also the results of the intermediate KL +step. Numbers show area under the precision-recall curve (AURPC). Mean +results reported across 3 runs ± standard deviation. +Noise +Inference method +Gaussian +Simplex +Coarse +2D Head MRI +Reconstruction +(Wyatt et al., 2022) +0.197 +± 0.032 +0.464 +± 0.048 +0.653 +± 0.063 +KL + inpainting +(Pinaya et al., 2022b) +0.305 +± 0.008 +0.640 +± 0.020 +0.689 +± 0.028 +→ KL step only +(Pinaya et al., 2022b) +0.258 +± 0.009 +0.675 +± 0.035 +0.796 +± 0.022 +DAE +(Kascenas et al., 2021) +0.325 +±0.004 +0.723 +±0.019 +0.833 +±0.005 +3D Head CT +Reconstruction +(Wyatt et al., 2022) +0.312 +±0.027 +0.623 +±0.004 +0.573 +±0.012 +KL + inpainting +(Pinaya et al., 2022b) +0.069 +±0.003 +0.357 +±0.005 +0.512 +±0.005 +→ KL step only +(Pinaya et al., 2022b) +0.098 +±0.005 +0.432 +±0.005 +0.629 +±0.002 +DAE +(Kascenas et al., 2021) +0.233 +±0.084 +0.611 +±0.038 +0.693 +±0.004 +6.8. Supervised segmentation baselines +We train supervised baselines to provide context on the ex- +pected performance range. The supervised head MRI baseline +was trained using a 2D U-Net model with the same architecture +as the DAE using 938 annotated volumes with tumor ground +truth. The supervised head CT baseline was trained using the +nnU-Net package (Isensee et al., 2021) using 129 annotated vol- +umes with hemorrhage, ischemia and tumor ground truth as de- +scribed in Section 4.2.3. +6.9. Postprocessing +We use the same postprocessing in all tested methods. We +apply median filtering with a kernel size of 5 which effectively +reduces the false positives in the anomaly score heatmaps as +shown in Kascenas et al. (2021) by filtering out insignificant +reconstruction noise. +7. Results +We now examine the difference in performance between +noise types (Gaussian, Simplex, Coarse), between models +(VAE, DAE, Diffusion models), across different modalities +(MRI and CT), and between noise resolutions applied to dif- +ferent anomaly sizes. We finally inspect the model outputs to +observe the difference in behavior qualitatively. + +Antanas Kascenas et al. / Preprint (2023) +9 +7.1. Metrics +We evaluate the anomaly detection performance of the meth- +ods with two metrics. Firstly, we measure the area under the +precision-recall curve (AUPRC) at the pixel level computed for +the whole test set. AUPRC evaluates anomaly scores directly +without requiring to set an operating point for each method. +Secondly, we calculate ⌈Dice⌉, a Dice score which measures +the segmentation quality using the optimal threshold for bina- +rization found by sweeping over possible values using the test +ground truth. ⌈Dice⌉ represents the upper bound for the Dice +scores that would be obtainable in a more practical scenario. +7.2. Noise comparison +We reported earlier (see Section 3.4) on experiments with +the noise parameters for training a DAE, showing that the noise +resolution makes a significant difference to the test-time perfor- +mance of a DAE, observing similar effects across two datasets. +Remarkably, the same parameters were optimal in the brain +MRI and in the head CT datasets. +We now compare between noise types, namely Gaussian +noise, Simplex noise (advocated by Wyatt et al. (2022)), and our +proposed coarse noise, using the optimal parameters identified +in Section 3.4 for the latter. We measure the impact on perfor- +mance of the DAE and of two diffusion model inference meth- +ods proposed by Pinaya et al. (2022b) and Wyatt et al. (2022). +The results are shown in Table 2. Our proposed coarse noise +achieves most accurate performance, significantly improving +the results compared to models trained with standard Gaussian +noise, and in most cases improving over models trained with +Simplex noise (Wyatt et al., 2022). Interestingly for the method +of Pinaya et al. (2022b), when the model is trained with Sim- +plex or coarse noise, the intermediate KL step (similar to ap- +plying a DAE) gives better results than the subsequent diffusion +inpainting step. +7.3. Model comparison +We compare models trained with coarse noise on the two +datasets. To put the unsupervised anomaly detection perfor- +mance results in context, we also provide supervised U-Net +baselines, trained on a moderate number of labeled volumes. +Table 3. Pathology detection performance as evaluated on BraTS Head +MRI Tumor test set. Metrics are the test set wide pixel-level area under +the precision-recall curve (AUPRC) and ideal Dice score (⌈Dice⌉). Mean +results reported across 3 runs ± standard deviation. +Method +AUPRC +⌈Dice⌉ +Thresholding +0.798 +0.749 +f-AnoGAN +0.365±0.024 +0.449±0.014 +VAE (reconstruction) +0.555±0.004 +0.548±0.003 +VAE (restoration) +0.750±0.006 +0.689±0.005 +Diffusion (reconstruction) +0.653±0.063 +0.610±0.060 +Diffusion (KL + inpainting) +0.689±0.028 +0.675±0.015 +→ KL step only +0.796±0.022 +0.723±0.013 +DAE (α = 16, σ = 0.2) +0.833±0.005 +0.773±0.004 +U-Net (supervised) +0.972±0.001 +0.914±0.002 +Quantitative results can be seen in Tables 3 and 4. Overall, +denoising methods appear to offer more accurate anomaly de- +tection than other unsupervised methods, with the simple DAE +giving overall best performance. Diffusion models perform bet- +ter than unsupervised baselines except for the simple threshold- +ing baseline (provided for MRI but not possible for the multi- +intensity lesions in CT), but worse than the DAE. As seen in +Table 2, the intermediate KL step of Pinaya et al. (2022b)’s +method outperforms the results of diffusion. +7.4. Relationship between noise resolution and anomaly size +We further examine the relationship between the DAE noise +used at training time and performance on anomalies during test +time. In particular, we investigate whether the coarseness of the +noise (i.e. noise resolution α before upsampling) has a large +impact on the size of test anomalies that the DAE successfully +detects as this could imply the need to tune the noise parameters +for specific anomalies. +In order to isolate the effect, we evaluate DAEs trained with +different noise coarseness on synthetic anomalies in 3D Head +CT scans. We synthesize bright spherical anomalies inside the +brain at random locations, sampling the diameter uniformly +from the range of 5mm to 50mm and then multiplying the nor- +mal tissue intensity within the sphere by a factor of 2. +Full results are shown in Appendix E. We observe no rela- +tionship between the noise resolution and the performance on +anomalies in specific size range. That is, selecting a reason- +ably coarse noise shows consistent best performance across a +wide range of synthetic anomaly sizes with no affinity towards +a specific anomaly size when compared to DAEs trained with +different noise coarseness parameters. +7.5. Visualization of model outputs +Selected examples of image slices and the corresponding de- +noising reconstructions and anomaly detection heatmaps for +different methods are shown in Figure 6 for head MRI and Fig- +ure 7 for head CT. The DAE performs consistently well on eas- +ier tumor cases in MRI and larger hemorrhages in CT, produc- +ing strong signal predictions. Weaker signal predictions some- +times correspond to subtler tumors in MRI and ischemia cases +in CT, and sometimes correspond to false positive detections. +Table 4. Pathology detection performance as evaluated on iCAIRD Head +CT Hemorrhage/Ischaemia/Tumour test set. Metrics are the test set wide +pixel-level area under the precision-recall curve (AUPRC) and ideal Dice +score (⌈Dice⌉). Mean results reported across 3 runs ± standard deviation. +Method +AUPRC +⌈Dice⌉ +VAE (reconstruction) +0.382±0.003 +0.432±0.005 +VAE (restoration) +0.542±0.012 +0.537±0.011 +Diffusion (reconstruction) +0.573±0.012 +0.600±0.013 +Diffusion (KL + inpainting) +0.512±0.005 +0.547±0.008 +→ KL step only +0.629±0.002 +0.608±0.003 +DAE (α = 16, σ = 0.2) +0.693±0.004 +0.674±0.003 +nnU-Net (supervised) +0.817±0.002 +0.786±0.004 + +10 +Antanas Kascenas et al. / Preprint (2023) +FLAIR +T1 +T1Gd +T2 +GT +a) +b) +c) +Fig. 6. Sample anomaly score predictions on 2D head MRI (BraTS2021) data. Columns a), b), c) show diffusion reconstruction (Wyatt et al., 2022), KL +divergence (Pinaya et al., 2022b) and DAE anomaly scores respectively. +When we examine more closely the impact of model choice +and noise type on the reconstruction (see Figure 8), we observe +the trade-off between reconstruction of the image detail vs era- +sure of the anomaly. The diffusion model tends to better erase +the anomaly compared to the DAE, particularly when trained +with Simplex or even Gaussian noise, but also yields poorer re- +construction of the image, erasing also fine details of normal +anatomy such as the folds of the gyri. The DAE achieves best +erasure, or at least suppression, of the anomaly when coarse +noise is used, as well as best retention of the normal anatomical +detail. +8. Discussion +8.1. What type of noise is best? +Our results show that the noise used for training denoising +models has a large impact on anomaly detection performance. +Our proposed coarse noise significantly improves the perfor- +mance of both DAEs and diffusion models, outperforming naive +Gaussian noise. Our coarse noise largely also outperforms the +more complex Simplex noise alternative (Wyatt et al., 2022), +which similarly introduces low frequencies to the noise pattern +(alongside higher frequencies). Visual inspection of the recon- +structions suggests that the noise used for model training im- +pacts the trade-off between the extent of the detail that is recon- +structed and the extent of erasure of anomalies, both of which +contribute to better anomaly detection performance. +Remarkably, we found the same noise parameters to be op- +timal across the two datasets described in this paper. Further, +noise parameters appeared not to be related to anomaly size, as +reported in Section 7.4. Thus, it appears that noise coarseness +parameters are independent of the size of test anomalies and can +be set without knowledge of the nature of the target anomalies +ahead of time. We note however that these datasets were pro- +cessed in a similar way and are of similar anatomy (head/brain), +therefore it would be desirable to do rigorous testing of many +anatomies, modalities, and intensity/resolution pre-processing +in order to come to a general conclusion. +8.2. What type of model is best? +In general, we find that employing denoising as the learn- +ing mechanism enables architectures with skip connections to +be used, leading to higher fidelity reconstructions which are +more effective for anomaly detection than those achieved by +VAE methods with bottleneck architectures. +Considering only the denoising approaches, we find that sim- +ple denoising autoencoder (DAE) models currently outperform +more advanced diffusion models across the two datasets that we +evaluated, when trained with optimal noise. However, diffusion +models show an exciting ability to erase anomalies and generate +convincing high definition reconstructions, with the caveat that + +Antanas Kascenas et al. / Preprint (2023) +11 +Input +Supervised +a) +b) +c) +d) +e) +Fig. 7. Sample anomaly score predictions on contrasting example slices from the 3D head CT (CQ500) data. Columns a) and b) show the show coarse noise +diffusion reconstructions and anomaly scores (Wyatt et al., 2022), column c) shows the coarse noise diffusion KL divergence anomaly scores (Pinaya et al., +2022b), and columns d) and e) show coarse noise DAE reconstructions and the associated anomaly scores. +Input +Gaussian Simplex +Coarse +Diffusion +DAE +Fig. 8. Comparison of reconstruction between DAE and diffusion using re- +construction procedure by Wyatt et al. (2022) across the three different +noise types. +normal anatomical detail may also be erased, limiting their ef- +fectiveness at discriminating normal from abnormal. Therefore, +while diffusion models are producing state-of-the-art results in +image generation, further investigation is needed to find appro- +priate methods to apply diffusion models to anomaly detection +specifically. +In terms of practical use, we note that diffusion methods +(similarly to VAE restoration methods) come at a cost since all +inference methods evaluated in this paper take hundreds of it- +erations to produce final predictions, resulting in much longer +inference times than for the DAE. +8.3. Limitations of our anomaly evaluation +A limitation of the evaluations presented in this paper is that +we have focused on a subset of anomalies which are present in +the datasets, albeit also the anomalies which are of most clinical +interest. This is explicit for the iCAIRD GG&C NHS Head +CT dataset, in which we annotated 3 pathologies (hemorrhage, +ischemia, tumor) but extracted NLP labels for several more (see +Appendix C), filtering on these to obtain a healthy training set. +Consequently, our metrics only approximate general anomaly +detection performance. We leave comprehensive annotation of +anomalies as an important avenue for future work; in fact, the +performance on rarer pathologies for which expert annotations +are not available is potentially more important than on common +pathologies since this is where training traditional supervised +approaches might be infeasible. +8.4. Alternatives to residual error for anomaly detection +Finally, DAEs and the diffusion inference methods we have +applied rely on reconstruction error in order to detect anoma- +lies. +Reconstruction error might be suitable for prominent +anomalies (e.g. large hemorrhages) but struggle with anoma- +lies subtler in intensity contrast (e.g. ischemia). Discriminative + +12 +Antanas Kascenas et al. / Preprint (2023) +methods (e.g. Cho et al. (2021); Tan et al. (2022); Kascenas +et al. (2022)) that infer the anomaly score directly have been +achieving success, notably in the Medical Out-of-Distribution +(MOOD) Analysis MICCAI Challenge (Zimmerer et al., 2022). +They might be more suitable for subtle anomalies, since they do +not use the residual error (which will be small for subtle inten- +sity changes) as the anomaly signal; see Meissen et al. (2021b) +for more in-depth analysis of the pitfalls associated with us- +ing residual error. Differently to reconstruction-based methods, +discriminative methods are typically trained by synthesizing ab- +normal data to discriminate from the healthy distribution. This +has pros and cons, allowing easier application of domain knowl- +edge about the nature of anomalies and explicit control over the +definition of “abnormal”, at the risk of losing generality and +overfitting to the selected synthetic anomalies. +9. Conclusion +In this paper we have demonstrated the effectiveness of a sim- +ple coarse noise model in both simple classical DAEs and more +complex recently proposed diffusion models for anomaly de- +tection across two datasets. We find that the parametrization +of the noise model has a wide tolerance, giving robust transfer +across datasets and denoising methods. As part of this work, +we implemented an anomaly detection pipeline in a real-world +scenario involving the collation of a healthy training set by run- +ning NLP methods on radiology reports, thereby showing that +a largely automated pipeline is possible. +Overall the classical DAE outperforms other methods, in +terms of implementation simplicity, accuracy, and inference +speed. While detection is successful for more obvious instances +of anomalies such as tumors, hemorrhages and ischemia, fur- +ther accuracy improvements are required to achieve reliable +detection of subtle anomalies. +Diffusion models applied to +anomaly detection are as yet in their infancy and provide a +promising avenue for further research. +Declaration of Competing Interest +The authors declare that they have no known competing fi- +nancial interests or personal relationships that could have influ- +enced the work reported in this paper. +Acknowledgements +This work is part of the Industrial Centre for AI Research in +Digital Diagnostics (iCAIRD) which is funded by Innovate UK +on behalf of UK Research and Innovation (UKRI) project num- +ber 104690. We thank the West of Scotland Safe Haven at NHS +Greater Glasgow and Clyde for their assistance in creating this +dataset. We would also like to acknowledge assistance of Canon +Medical Research Europe Limited in providing the Canon Safe +Haven Artificial Intelligence Platform (SHAIP) tool, assisting +with the deidentification of data and the provision of a secure +machine learning workspace. +We acknowledge Engineering and Physical Sciences Re- +search Council (EPSRC) for funding part of this work through +the EPSRC Centre for Doctoral Training in Applied Photonics +(CDTAP) managed by Heriot-Watt University. +This work was supported by the University of Edinburgh, +the Royal Academy of Engineering and Canon Medical Re- +search Europe via PhD studentships of Pedro Sanchez (grant +RCSRF1819\8\25). +S.A. Tsaftaris acknowledges the support of Canon Medi- +cal and the Royal Academy of Engineering and the Research +Chairs and Senior Research Fellowships scheme (grant RC- +SRF1819\8\25). +Many thanks to Sin Yee Foo, Harris Hameed, and Paul Don- +nelly from GG&C NHS for creating the pathology annotations +which we used for our evaluation. +Many thanks to Paul Thomson and Ewan Hemingway for +their help with developing the imaging pre-processing pipeline. +References +Alex, V., Vaidhya, K., Thirunavukkarasu, S., Kesavadas, C., Krishnamurthi, +G., 2017. Semisupervised learning using denoising autoencoders for brain +lesion detection and segmentation. 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Un- +supervised anomaly localization using variational auto-encoders, in: Medi- +cal Image Computing and Computer-Assisted Intervention – MICCAI 2019, +Springer. pp. 289–297. +Zimmerer, D., Kohl, S.A., Petersen, J., Isensee, F., Maier-Hein, K.H., 2021. +Context-encoding variational autoencoder for unsupervised anomaly detec- +tion, in: Medical Imaging with Deep Learning (MIDL). + +14 +Antanas Kascenas et al. / Preprint (2023) +Supplementary Material +Appendix A. DAE vs VAE reconstruction comparison +FLAIR +T1 +T1Gd +T2 +Input +VAE +DAE +Fig. A.9. Sample healthy brain reconstructions from VAE and DAE mod- +els. The DAE gives more precise reconstructions. The VAE reconstruction +quality could be improved by increasing bottleneck dimensionality, how- +ever this would negatively impact anomaly detection performance. +Appendix B. Neural network architectures +GN +in|out +Denoising autoencoder +in|out +64|4 +4|64 +64|128 +125|256 +256|512 +512|256 +128|64 +256|128 +64|4 +4|64 +64|128 +128|256 +64|256 +128|64 +256|128 +256|256 +μ128 +σ128 +~ +Swish +GN +k.s. = 1 +k.s. = 1 +k.s. = 16 +out_channels = 256 +stride = 1 +Variational autoencoder +ConvBlock module +Swish +GN +out|out +Swish +GN +in|out +Weight standardized convolution layer: kernel size (k.s.) = 3, stride=1 unless +otherwise noted. Numbers denote channel numbers +in|out +Convolution layer without weight standardization. +Swish +Swish activation function. +Group normalization. +ConvBlock module. Numbers denote channel numbers. +Average pooling, kernel size=2, stride=2 +Transposed convolution layer, kernel size=2, stride=2 +unless otherwise noted. +U-Net skip connection +Fig. B.10. Architectures of 2D DAE and VAE models used in brain MRI +experiments. + +Antanas Kascenas et al. / Preprint (2023) +15 +Appendix C. Radiology report labels +Table C.5. List of report labels extracted from radiology reports using the +method of Schrempf et al. (2021). We do not exclude scans with associated +positive/uncertain labels which are underlined from our healthy training +set, since we decide that scans with only these labels (and no others) are +“normal for age”. +Radiographic findings +artefact, collection, compression, dilation, effacement, her- +niation, hyperdensity, hypodensity, loss of differentia- +tion, malacic changes, mass effect, midline shift, oedema, +swelling. +Clinical impressions +abscess, +atrophy, +aneurysm, +calcification, +cavernoma, +cerebral small vessel disease, congenital abnormality, cyst, +evidence of surgery/intervention, fracture, gliosis, hemor- +rhage, hydrocephalus, ischemia, infection, tumor, vessel oc- +clusion, lesion, pneumocephalus. +Appendix D. Selecting and annotating the iCAIRD test set +We provide additional details below on the process by which +the iCAIRD test set was selected and annotated. +Scan selection: +For hemorrhage and ischemia cases, our +primary source was the Scottish Stroke Care Audit (SSCA) +records for which we had access for the stroke episodes in the +dataset; we searched these records for stroke episodes classed as +“hemorrhagic”, “ischemic”, or “hemorrhagic transformation”. +For cases of tumors and rarer hemorrhages (epidural and sub- +dural), we used a combination of ICD-10 code and free text +searches of the Scottish Morbidity Records (SMRs) and radi- +ology reports (e.g., “extradural”, “extra-dural”, “extra dural”, +“epidural”, “edh”, “subdural”, and “sdh”), respectively. +We +then excluded scans acquired prior to 2016 for image compres- +sion reasons. +Annotation and Review: We recruited 3 GG&C clinicians +(one Consultant Neuroradiologist and two senior Radiology +trainees) to perform pixel-level annotation, following the anno- +tation protocol prepared for this project. For the selected cases, +all hemorrhage, ischemia and tumor lesions present were anno- +tated, including any surrounding regions of edema for hemor- +rhagic lesions. The Consultant Neuroradiologist acted also as +reviewer; 40% of cases were randomly selected for review and +annotators also had the option of sending any of the remaining +60% for review when they required a second opinion. +Appendix E. Relationship between noise coarseness and +anomaly size +10 +20 +30 +40 +50 +Synthetic anomaly diameter (mm) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Performance (AUPRC) +α = 1 +α = 2 +α = 4 +α = 8 +α = 16 +α = 32 +α = 64 +α = 128 +Fig. E.11. DAE anomaly detection performance evaluated using synthetic +anomalies of different sizes with models trained with noise generated at +different resolutions α. Each point indicates a mean of three runs. The +best noise resolution (α = 16) generalizes the best across a wide variety of +synthetic anomaly sizes. + diff --git a/dNE_T4oBgHgl3EQf0hzM/content/tmp_files/load_file.txt b/dNE_T4oBgHgl3EQf0hzM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e892f020e867db8200bee747042e4351b039ddb9 --- /dev/null +++ b/dNE_T4oBgHgl3EQf0hzM/content/tmp_files/load_file.txt @@ -0,0 +1,1294 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf,len=1293 +page_content='Preprint (2023) The role of noise in denoising models for anomaly detection in medical images Antanas Kascenasa,b,, Pedro Sanchezc, Patrick Schrempfa, Chaoyang Wanga, William Clacketta, Shadia S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Mikhaela, Jeremy P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' United Kingdom dThe Alan Turing Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' United Kingdom A R T I C L E I N F O 2000 MSC: 68T99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 92C55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 68U10 Keywords: Anomaly detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Unsuper- vised learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Autoencoder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion A B S T R A C T Pathological brain lesions exhibit diverse appearance in brain images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' in terms of in- tensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' texture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' shape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Comprehensive sets of data and annotations are difficult to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Therefore, unsupervised anomaly detection approaches have been proposed using only normal data for training, with the aim of detecting outlier anomalous voxels at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising methods, for instance classical denoising autoencoders (DAEs) and more recently emerging diffusion models, are a promising approach, however naive application of pixelwise noise leads to poor anomaly detec- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We show that optimization of the spatial resolution and magnitude of the noise improves the performance of different model training regimes, with simi- lar noise parameter adjustments giving good performance for both DAEs and diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Visual inspection of the reconstructions suggests that the training noise influ- ences the trade-off between the extent of the detail that is reconstructed and the extent of erasure of anomalies, both of which contribute to better anomaly detection perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We validate our findings on two real-world datasets (tumor detection in brain MRI and hemorrhage/ischemia/tumor detection in brain CT), showing good detection on diverse anomaly appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Overall, we find that a DAE trained with coarse noise is a fast and simple method that gives state-of-the-art accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion models ap- plied to anomaly detection are as yet in their infancy and provide a promising avenue for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Code for our DAE model and coarse noise is provided at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com/A ntanasKascenas/DenoisingAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Introduction Anomaly detection is a fundamental task in medical image analysis, mimicking the initial review that a radiologist per- forms of imaging studies to identify abnormal regions which should be reviewed and characterized further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Supervised ma- chine learning methods have shown promising results, however comprehensive supervised pathology detection methods require extensive and heterogeneous training sets which are costly to annotate and difficult to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Conversely, unsupervised anomaly detection (UAD) methods require only identification e-mail: antanas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='kascenas@mre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='medical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='canon (Antanas Kascenas) of a healthy cohort of patients for training (therefore these meth- ods are sometimes regarded as semi-supervised), after which they may be applied to detect out-of-distribution anomalous re- gions in test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Autoencoder deep learning methods have been commonly used for UAD in brain scans (Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021), relying on the assumption that normal data as seen during training will be re- constructed better than unseen anomalous – potentially patho- logical – regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' A classical approach is denoising autoen- coders (DAEs) (Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2008) in which corrupting noise is added to the input and the network must learn to remove the noise in order to reconstruct the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This training task of removing noise can be regarded as a proxy for the test time task of removing anomalies in order to reconstruct an im- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='08330v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='IV] 19 Jan 2023 2 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) age of normal appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' It was shown in Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) that for detection of brain tumors in MRI data, train- ing with naive pixelwise noise gave poor anomaly detection performance, while training with coarse noise (see Algorithm 1) gave good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Following simple optimization of noise resolution and magnitude, a classical DAE outperformed more complex previous state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our contributions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We take the simple and effective DAE that was proposed by Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) for brain anomaly detection in medical 2D MRI images, and investigate its application to 3D CT images with a range of anomalies, showing that optimal noise resolution and magnitude parameters are largely transferable between modalities and anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We analyze noise type in the alternative denoising model paradigm of diffusion models, showing that similar adjust- ment of the type of noise gives accuracy gains also for this alternative denoising approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We additionally analyze an alternative noise type (Simplex noise) which has been recently advocated by Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022), showing our proposed coarse noise to be superior in most anomaly detection setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Finally, since we consider training an anomaly detection algorithm in a practical setting where a large uncurated dataset of scans is available, we demonstrate that NLP analysis of radiology reports can be effectively used to se- lect the training cohort of normal scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Related Work Anomaly detection is an open-ended task for which a variety of approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Autoencoder approaches to anomaly detection Many modifications to the standard autoencoder pipeline have been proposed to improve performance on the task of anomaly detection, which has the potentially conflicting twin goals of reconstructing normal regions of the original brain scan with high fidelity, while reconstructing any anomalous regions with poor fidelity (in order to distinguish them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Variational autoencoders (VAEs) (Zimmerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2019) constrain the latent bottleneck representation to follow a pa- rameterized multivariate Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Zimmerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) further add a context-encoding task and combine recon- struction error with density-based scoring to obtain the anomaly scores, while Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020) use an iterative gradient descent restoration process at test time in restoration-VAE, replacing the reconstruction error with a restoration error to estimate anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Architectural changes have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Atlason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2018) introduce convolutional autoen- coders and higher capacity spatial bottlenecks instead of fully- connected (dense) bottlenecks to achieve better reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Chen and Konukoglu (2018) use constrained autoencoders to improve latent representation consistency in anomalous images at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Bayesian skip-autoencoders Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020a) use skip connections with dropout to improve reconstruction and allow uncertainty to be measured via dropout stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020b) use scale-space autoencoders to compress and reconstruct different frequency bands of brain MRI using the Laplacian pyramid to achieve higher reconstruction fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The UAD autoencoder framework of encoder-decoder com- ponents and reconstruction error for anomaly scores has fea- tured in more complex approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Schlegl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2019) train a generative adversarial network called f-AnoGAN which reuses the generator and discriminator to train an autoencoder with both reconstruction and adversarial losses for the anomaly de- tection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022a) combine a vector quantized VAE (VQ-VAE) to encode an image with a transformer model to resample low-likelihood latent variables in order to produce reconstructions with fewer reproduced anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) have performed an evaluation of some common autoencoder methods for anomaly detection in brain MRI, finding restoration-VAE Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020) and f- AnoGAN Schlegl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2019) to be among the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' How- ever, more recently Meissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021a) showed that most autoencoder-based MRI UAD methods can be outperformed by a simple thresholding baseline, applied to the FLAIR sequence after histogram equalization preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This training-free approach detected hyperintense brain tumor and multiple scle- rosis lesions better than most UAD approaches that require healthy data to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our work relies on the same principle of using reconstruction error for anomaly detection as most autoencoding methods but we use noise instead of architectural constraints to make the autoencoding training task non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising methods The above evaluations of medical anomaly detection meth- ods largely omitted consideration of classical denoising autoen- coders (DAEs) (Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2008) and other methods exploit- ing noise, however a few approaches have shown promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Alex et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2017) applied DAEs as pretraining for brain lesion detec- tion with limited labels and for simple novelty detection using patch-based masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Collin and De Vleeschouwer (2021) use a DAE for anomaly detection in industrial vision with a stain noise model with randomized shape, color, size and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Bengs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) use 3D VAEs with spatial patches replaced with voxelwise noise to train an inpainting model for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Generative diffusion models (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020), in which noise is added and removed over many iterations, have been used in the context of anomaly detection (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) by assuming that models trained on only healthy data will fail during reconstruction of anomalous features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Recently, Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) showed that when noise coarseness and intensity are adjusted, a DAE can achieve com- petitive results for the detection of tumors in brain MRI images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Further, (Daras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) have shown recently that diffusion models can be trained with degradation functions other than Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In fact, Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022) showed that using Simplex noise in diffusion models can signif- icantly improve anomaly detection performance over traditional Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 3 Test time Input Reconstruction Reconstruction Residuals Anomaly scores Ground truth upsample & mask Normal image Noisy input post-process residuals Noise Denoising model Reconstruction procedure Noise generation diffusion models input additional parameter “t”, corresponding to noise magnitude Corruption process C Training time | | Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Workflow for denoising anomaly detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' During training (top), noise is added to the normal image, and the network is trained to reconstruct the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' At test time (bottom), different methods are applied to reconstruct and post-process the potentially anomalous input image to produce the anomaly score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For the simple denoising autoencoder (DAE) approach, the denoising model is applied once to the input and the anomaly score is simply the reconstruction residuals followed by median filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' However, the diffusion models apply more complex iterative noise addition followed by iterative denoising to obtain the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In this paper, we examine this theme of the role of noise in anomaly detection, investigating and comparing types of noise and denoising methods in a common setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Method In summary, we employ one of three types of noise (Gaus- sian, Simplex or coarse) to train neural network models to de- noise healthy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' At test time, anomalies are detected via reconstruction error (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Below we describe this pro- cess in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising Models for Anomaly Detection Denoising neural networks ϵθ receive corrupted data ˜x as input and are trained to recover original (uncorrupted) data ˆx = ϵθ (˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We consider the corruption process to have a con- ditional distribution C (˜x | x, n), degrading x into ˜x with the in- jection of some noise n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Training a denoising neural network ϵθ with parameters θ can then be written as: θ∗ = arg min θ Ex∼pdata, ˜x∼C(x) � ∥ϵθ (˜x) − x∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (1) The resulting network learns to reconstruct samples x that be- long to pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' However, we consider a distribution panomaly which is similar to pdata but contains features (anomalies) that are not present in pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' As shown by Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021), an anomalous sample x′ ∼ panomaly will not be reconstructed ap- propriately by the denoising network ϵθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The training and test pipelines are visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The anomalies are detected by taking the absolute difference between the input data and the resulting reconstruction |x′ − ˆx′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising Autoencoder (DAE) approach We implement a simple denoising deep autoencoder neural network, and use reconstruction error to detect and localize anomalies at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The network has a U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2015) style architecture with skip connections which en- ables significantly better image reconstructions compared to bottleneck architectures such as the VAE (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We note that any neural network architecture yielding dense predic- tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' segmentations) could be trained as a DAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Details of the network architecture and training procedure can be found in Section 6 and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' During training, we corrupt images according to: C (˜x | x) =⇒ ˜x = x + σn, (2) where σ is the standard deviation which controls the inten- sity magnitude and n is noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Classically, n is sampled from a Gaussian distribution N(0, I), but in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 we explore more efficient techniques in the context of anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' At inference time, the DAE is used to localize anomalies by calculating pixelwise/voxelwise anomaly scores A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' If we de- note the input image as x, the number of image channels as M (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=" for multiple imaging sequences or imaging modalities), n~XxX'A(x')4 Antanas Kascenas et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1T t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3T t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6T Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion model input at different timesteps t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise component is larger at further timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' a background mask of pixels with x intensities across channels equal to 0 as B, the median filtering operation as f, and the reconstruction as ˆx, then the anomaly score can be defined as: A(x) = f �������(1 − B) ⊙ M � m |xm − ˆxm| M ������� (3) No noise is used at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion model approach We next explore the diffusion model methods developed in Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022b) and Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Both methods fol- low a training strategy that was initially proposed by Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In contrast to the denoising model used in the DAE, dif- fusion denoising models are trained to predict the noise rather than to reconstruct the original image itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In particular, con- sider a model trained to find optimal parameters as θ∗ = arg min θ Ex∼pdata, t∼U(0,T), ˜x∼C(x,t) � λ(t) ∥ϵθ (˜x, t) − n∥2� , (4) where the timestep t is sampled from a uniform distribution be- tween 0 and T (T is a hyperparameter which we set to 1000) and λ(t) is a loss weighting term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Here the corruption process C (˜x | x, t) depends also on t which controls the strength of the corruption through αt as described in Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) , ac- cording to: C (˜x | x, t) =⇒ ˜x = √αtx + � 1 − αtn, (5) where the coefficient αt runs from α0 = 1 (original image) through to αT = 0 (noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Figure 2 shows examples of a corrupted image for different values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Training with multi- ple t values corresponds to training with multiple noise magni- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Training with C (˜x | x, t) has been extensively studied in the diffusion probabilistic modeling (DPM) literature (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For example, when using Gaussian noise, adding noise with high standard deviation causes the network to focus on coarse features while low standard deviation noise causes focus on texture and other high frequency detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Most importantly, training to denoise at multiple magnitudes enables image gen- eration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' we refer the reader to Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2020) for details on the image generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The ability of DPMs to denoise at different noise magnitudes as well as its generative power has inspired methods for anomaly detection using diffusion models (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Once the diffusion denoising model has been trained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' we in- vestigate two inference techniques to detect anomalies: Algorithm 1 Generation of noise with spatial resolution α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' and output shape a × b × c 1: procedure Noise(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' c) 2: n ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' I) ∈ Rα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='α 3: n ← upsample(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' c)) ▷ Bilinearly upsample 4: x ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' a) ▷ Uniformly sample in range (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' a) 5: y ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' b) 6: z ∼ U0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' c) 7: n ← translate(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' z)) ▷ Randomly translate 8: return n ▷ The generated noise is n 9: end procedure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Reconstruction (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) - In the AnoDDPM method, noise is injected at a selected magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' we use t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='25T because this was found to be the best in Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We then run the DDPM (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020) iterative generation from t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='25T → 0, using the noisy image as the starting point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 250 steps where T=1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We follow Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022) in averaging the reconstruc- tions across 5 runs of this generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' KL divergence + inpainting (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) - In this method, noise is injected with different magnitudes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' different t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4T, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6T]) and the difference is com- puted between the predicted output ϵθ (˜x, t) and the ex- pected output n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' A heatmap is obtained by averaging the difference images produced by different values of t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' since DPMs have a probabilistic interpretation, Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' term this the Kullback–Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The KL diver- gence heatmap is binarized at a threshold corresponding to the 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5 percentile value of the heatmap, to produce a mask for the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The masked region only is then reconstructed by the model using the DPM i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' “in- painted” (Lugmayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) by running iterative gen- eration from t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The final heatmap is the difference between the original and inpainted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Coarse noise generation It was shown in Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) that training with lower resolution noise leads to better anomaly detection than naive pixelwise Gaussian noise for a DAE detecting brain tu- mors in brain MRI data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' in this paper we are interested to fur- ther investigate the impact of the type of noise with different data and denoising models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We generate the lower resolution (“coarse”) noise by sampling random pixelwise Gaussian noise at a low resolution and bilinearly (trilinearly for 3D) upsam- pling it to the input resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We then randomly translate the generated noise to avoid consistent upsampling patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' See Figure 4 for examples of generated noise and Algorithm 1 for the pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' To examine the effect of noise, we take the DAE and vary the noise resolution α and the standard deviation σ used for gener- ating Gaussian noise before upsampling (see Figure 3) on the two datasets described later in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We find that a reason- ably coarse noise is critical, as DAE models trained using stan- dard pixel-level noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' generated at 128 × 128 resolution Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 Noise magnitude, σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='8 AUPRC MRI 1 2 4 8 16 32 64 128 Noise resolution, α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='75 AUPRC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='30 Noise magnitude, σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6 AUPRC CT 1 2 4 8 16 32 64 128 Noise resolution, α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6 AUPRC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise coarseness and magnitude ablation results on BraTS head MRI (left) and iCAIRD head CT (right) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Magnitude ablation uses noise resolution α = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Coarseness ablation uses σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Error bars show standard deviation across three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 1×1 2×2 4×4 8×8 16×16 32×32 64×64 128×128 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Samples of 2D noise generated at different resolutions, from 1×1 through to 128×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The background mask B is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' for 128 × 128 pixel 2D head MRI slices) or using the opposite extreme of image-level noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' generated at 1 × 1 resolu- tion) perform significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' DAEs appear to be less sen- sitive to the magnitude of the noise (σ of the generating Gaus- sian distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The noise parameters are robust and transfer well between modalities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' from MRI to CT) and patholo- gies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' tumor to hemorrhage) and 2D to 3D as long as the image field-of-view and resolution are comparable i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' we used 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='62mm2/pixel for 2D MRI and 2mm3/voxel for 3D CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For diffusion models, we similarly modify the corruption process C, adopting the noise generation process described in Algorithm 1 instead of pixel-wise Gaussian noise both during training and when applying each of the inference techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Datasets We evaluate anomaly detection in 2D brain MRI slices and 3D brain CT volumes using the two datasets described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' BraTS challenge dataset: Brain MRI We evaluate the anomaly detection performance on the sur- rogate task of brain tumor segmentation using data from the BraTS 2021 challenge (Menze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Bakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This data comprises native (T1), post-contrast T1- weighted (T1Gd), T2-weighted (T2), and T2 Fluid Attenuated Inversion Recovery (FLAIR) modality volumes for each patient from a variety of institutions and scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Selecting the training and test data We randomly split the dataset into 938 training, 62 valida- tion, and 251 test patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' During training, we use only slices that do not contain any tumor pixels, under the assumption that these non-tumor slices represent healthy tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' At test time, we consider the union of the tumor sub-region labels to be the anomalous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Preprocessing The data has already been co-registered, skull-stripped and interpolated to the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Labels are provided for tumor sub-regions: the GD-enhancing tumor, the peritumoral edema, and the necrotic and non-enhancing tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For the data input to the models, we stack all four modal- ities at the channel dimension for each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We normalize (rescale) the pixel intensity values in each modality of each scan by dividing by the 99th percentile foreground voxel intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Values are scaled to a range of [−1, 1] for diffusion methods 16 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Data filtering steps towards obtaining a healthy training set for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Filtering step Images Patients Initial Data cohort 16,559 7,122 After filtering on report labels from Schrempf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) 2,350 1,788 After filtering out follow-up scans 1,020 961 After rapid manual image review 996 939 Healthy training set 804 757 and [0, 1] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' All slices are downsampled to a resolution of 128×128 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='62mm/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' iCAIRD GG&C NHS dataset: Head CT We use head CT scans obtained through a collaboration with the Industrial Centre for Artificial Intelligence Research in Dig- ital Diagnostics (iCAIRD)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The data has been sourced from hospitals in the Greater Glasgow & Clyde (GG&C) area in Scotland and comprises all patients who were diagnosed with a stroke in the period 2013-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The data is pseudonymised and we obtain access onsite via the West of Scotland Safe Haven within NHS Greater Glasgow and Clyde via the Safe Haven Artificial Intelligence Platform (SHAIP) (Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We have obtained ethical approval to use this data2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The data was originally collected by identifying hospital ad- missions which were assigned International Classification of Diseases (ICD-10)3 codes relating to stroke diagnoses, and then selecting medical data from the stroke event hospital admission as well as the documentation from 18 months prior and all prior images held at the GG&C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In total, the dataset contains infor- mation about 15,882 stroke events from 10,143 patients and in- cludes CT images, radiology reports, clinical documents and structured clinical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use 16,559 head CT images avail- able from 7,122 patients for the purpose of this work and refer to this as the iCAIRD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Radiology report NLP for normal scan selection Identification of normal scans by manual examination of this large dataset would be time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Fortunately, corre- sponding free text radiology reports are available for most of the head CT images in the iCAIRD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The reports vary in depth and exposition reflecting the style and seniority of the reporting radiologists, but generally describe the radiographic findings and clinical impressions in the associated CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use this information to identify and exclude abnormal scans from our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' However, comprehensive manual exam- ination of radiology reports, while faster than examination of images, would still be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Therefore, we leverage a pre- viously developed automatic deep learning model (Schrempf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020, 2021) which was trained on 357 manually labeled 1https://icaird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com 2West of Scotland Safe Haven ethical approval number GSH19NE004 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='int/standards/classifications/classifi cation-of-diseases non-contrast head CT radiology reports and outputs labels for 14 radiographic findings and 19 clinical impressions (see Ap- pendix C for the list of labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Each label is assigned one of the 4 classes: positive, negative, uncertain or not mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Selecting the training data Defining Normal vs Abnormal: We aim to obtain a training set that is as healthy as possible in order to detect as many anomalies as possible at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' However, since the dataset is from an elderly stroke population (mean age of 72 years), re- ports without any positive findings (labels) are rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Therefore, there is a trade-off between how aggressively we filter versus the size of the final training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Hence, we include scans for which the associated reports contain only findings/impressions that are commonly found in an elderly population, specifically calcification, atrophy, cerebral small vessel disease and hypo- density (the latter is most commonly associated with atrophy and small vessel disease).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Applying this more generous defini- tion of “Normal” leaves a set of 2350 scans from 1788 patients (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Filtering out follow-up scans: Upon closer manual inspec- tion we find that many reports are non-exhaustive (note these are free text rather than structured reports), appearing not to list all of the findings present in the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This most com- monly occurs for follow-up scans where the associated report assumes knowledge of earlier scan reports, usually not explic- itly re-listing all findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' An example such report would be “No progression compared to previous scan from 10/22/2021.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Thus, absence of positive or uncertain labels does not necessar- ily equate to absence of pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Therefore we further filter down the remaining cases using keywords and pattern matching using spaCy (Honnibal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020), removing reports which contain references to previous imaging and comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This keyword filtering leaves 1020 scans from 961 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Rapid manual image review for obvious anomalies: Finally, we perform a rapid manual review by non-experts which elim- inates a further 24 scans mostly containing processing issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' bone reconstruction, significant artifact, significantly de- graded scan quality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use 804 scans from the remaining 996 cases as our healthy training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Selecting and annotating the test data In addition to the filtered healthy training data, we selected and annotated a separate set of scans with hemorrhages, is- chemia and tumors to quantitatively evaluate the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The annotation workflow consisted of several steps: curation, anno- tation, review and quality assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Further details are pro- vided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The resulting data was split into Test and Training sets as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Test set: The test set contains voxelwise annotations for 114 scans of which 104, 23 and 4 contain hemorrhage, ischemia and tumor ground truth respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use the union of the three pathologies for evaluating the anomaly detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Training data for supervised baselines: We further reserve 129 scans annotated with 116 hemorrhage, 30 ischemia and 6 tumor annotations for training the supervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Preprocessing We rigidly register the CT scans to a reference volume and crop to a fixed field-of-view which includes only the head re- gion of the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Volumes are then resampled to 2mm3 resolu- tion and windowed to Hounsfield Unit (HU) values from 0 to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' As for the MRI data, intensities are rescaled to a range of [−1, 1] for diffusion methods and [0, 1] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use ran- dom flipping and affine transformation data augmentation for training of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' qure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='ai CQ500: Head CT We use the CQ500 dataset from qure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='ai (Chilamkurthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2018) for qualitative evaluation (see Figure 7) of the head CT methods as the data contains similar pathologies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' hemor- rhages, ischemia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This dataset does not, however, contain any voxel-level ground truth and could not be used for quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Baselines We compare against a range of common reconstruction-error based methods as well as providing supervised segmentation model results trained using ground truth for context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2D brain MRI baselines We compare the denoising anomaly detection model perfor- mance against four methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Firstly, we implement a stan- dard VAE (Zimmerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2019) and f-AnoGAN (Schlegl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2019) models with pixelwise reconstruction error as the anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Secondly, we use the same VAE model but im- plement an iterative gradient-based restoration process (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020) to produce restoration images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Finally, we apply the simple thresholding approach from Meissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021a) modified to use median filtering as proposed by (Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use the hyperparameters from the original works for the deep learning methods but tune manually where necessary to improve training stability and anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3D brain CT baselines We compare the denoising anomaly detection model perfor- mance on 3D head CT data against two reconstruction-error based methods: VAE reconstruction (Zimmerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2019) and VAE restoration (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Implementation details 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise Coarse noise is generated by sampling random Gaussian pix- elwise noise at resolutions of 16×16 and 16×16×16 for 2D and 3D respectively, before bilinearly/trilinearly upsampling to the input resolution of 128×128 for 2D brain MRI and 80×112×88 for 3D head CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The generated noise is then randomly trans- lated to randomize the centers of the coarse noise peaks that may occur due to upsampling from very low resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise is generated independently for each image modality in the case of 2D MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We investigate the parameters of the noise, as re- ported in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Simplex noise is generated using the implementation pro- vided by Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For DAE experiments with Sim- plex noise, we scale the generated noise magnitude by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Denoising autoencoder For 2D MRI data, we use a U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2015) encoder-decoder architecture with three downsam- pling/upsampling stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Each encoder stage consists of two weight-standardized convolutions (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2019) with ker- nel sizes of 3 and 64, 128, 256 output channels for the three stages respectively followed by Swish activations (Ramachan- dran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2018) and group normalization (Wu and He, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Average 2 × 2 pooling is used for downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The decoder architecture mirrors the encoder in reverse, using transposed convolutional layers for upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Architecture visualization and further details can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For 3D head CT data, we use an analogous architecture in 3D with three downsampling/upsampling stages and 32, 64, 128 output channels for the three stages respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use mean L2 reconstruction loss in the foreground as the training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 2D DAE Models are trained for 67,200 iter- ations with a batch size of 16 slices using Adam (Reddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2018) with a cosine annealed (Loshchilov and Hutter, 2017) maximum learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='0001 with a period of 200 itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 3D DAE Models are trained for 25,600 iterations with a batch size of 3 volumes using Adam with OneCycleLR learning rate schedule (Smith and Topin, 2019) with a maximum learn- ing rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion model We implement a diffusion model with a U-Net-like architec- ture based on implementation provided by Dhariwal and Nichol (2021) which includes residual layers, global attention, dropout and a projection of the timestep embedding to each residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use T = 1000 diffusion steps with linear noise sched- ule training models to predict the noise and optimizing the mean squared loss between the noise which was used for sampling and the predicted noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For 2D MRI data we use the AdamW optimizer (Loshchilov and Hutter, 2018) with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='0001 and weight de- cay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='01 with a batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Model weights are averaged by taking the exponential moving average (EMA) with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='9999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The 2D U-Net architecture and diffusion training code can be found on github 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For 3D CT data we use the AdamW optimizer with a OneCy- cleLR learning rate schedule (Smith and Topin, 2019) with a maximum learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='0001 and batch size of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Model weights are averaged by taking the EMA with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com/Julian-Wyatt/AnoDDPM 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com/vios-s/Diff-SCM 8 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) Gaussian Simplex Coarse Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The three noise types tested to train diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' VAE reconstruction VAE models use a similar architecture to their DAE coun- terparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Skip connections are removed and a bottleneck with dimensionality of 128 is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For the training objective, we compute the sum of mean L2 reconstruction error and KL- divergence with a weight of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use the same training procedure and anomaly score formula as for their DAE counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' VAE restoration Using the VAE model described above, we implement a restoration method (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2020) to produce the anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We perform the restoration procedure using 100 iter- ations on individual slices/volumes basing our implementation on public source code6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Note that due to the iterative nature of the restoration procedure it takes significantly longer (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' ×100) to produce predictions compared to the single inference iteration DAE/VAE reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' f-AnoGAN We adapt the original public implementation7 for the brain MR data task as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use an additional generator, dis- criminator and encoder block to account for the higher resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Strided convolutions and transposed convolutions are used for downsampling and upsampling respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We use a batch size of 32 and learning rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='00001 for the generator, discriminator and encoder respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The encoder was trained using κ = 1 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Thresholding We follow Meissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021a) to obtain results for the thresholding baseline but omit the connected component filter- ing as we have found median filtering to be more effective and computationally efficient (Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' FLAIR se- quence volumes are used as the anomaly score volumes, fol- lowing processing by histogram equalization of the foreground (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' excluding surrounding air) and median filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com/yousuhang/Unsupervised-Lesion-Detec tion-via-Image-Restoration-with-a-Normative-Prior 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='com/tSchlegl/f-AnoGAN Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Relationship between DAE and different diffusion anomaly detec- tion inference methods and noise used for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For the method of (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) we include also the results of the intermediate KL step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Numbers show area under the precision-recall curve (AURPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Mean results reported across 3 runs ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise Inference method Gaussian Simplex Coarse 2D Head MRI Reconstruction (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='197 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='464 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='653 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='063 KL + inpainting (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='305 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='640 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='028 → KL step only (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='258 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='796 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='022 DAE (Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='325 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='723 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='833 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 3D Head CT Reconstruction (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='312 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='623 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='573 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='012 KL + inpainting (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='069 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='357 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='512 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 → KL step only (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='098 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='432 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='629 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='002 DAE (Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='233 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='611 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='693 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Supervised segmentation baselines We train supervised baselines to provide context on the ex- pected performance range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The supervised head MRI baseline was trained using a 2D U-Net model with the same architecture as the DAE using 938 annotated volumes with tumor ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The supervised head CT baseline was trained using the nnU-Net package (Isensee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021) using 129 annotated vol- umes with hemorrhage, ischemia and tumor ground truth as de- scribed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Postprocessing We use the same postprocessing in all tested methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We apply median filtering with a kernel size of 5 which effectively reduces the false positives in the anomaly score heatmaps as shown in Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021) by filtering out insignificant reconstruction noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Results We now examine the difference in performance between noise types (Gaussian, Simplex, Coarse), between models (VAE, DAE, Diffusion models), across different modalities (MRI and CT), and between noise resolutions applied to dif- ferent anomaly sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We finally inspect the model outputs to observe the difference in behavior qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Metrics We evaluate the anomaly detection performance of the meth- ods with two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Firstly, we measure the area under the precision-recall curve (AUPRC) at the pixel level computed for the whole test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' AUPRC evaluates anomaly scores directly without requiring to set an operating point for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Secondly, we calculate ⌈Dice⌉, a Dice score which measures the segmentation quality using the optimal threshold for bina- rization found by sweeping over possible values using the test ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' ⌈Dice⌉ represents the upper bound for the Dice scores that would be obtainable in a more practical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Noise comparison We reported earlier (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4) on experiments with the noise parameters for training a DAE, showing that the noise resolution makes a significant difference to the test-time perfor- mance of a DAE, observing similar effects across two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Remarkably, the same parameters were optimal in the brain MRI and in the head CT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We now compare between noise types, namely Gaussian noise, Simplex noise (advocated by Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022)), and our proposed coarse noise, using the optimal parameters identified in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We measure the impact on perfor- mance of the DAE and of two diffusion model inference meth- ods proposed by Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022b) and Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our proposed coarse noise achieves most accurate performance, significantly improving the results compared to models trained with standard Gaussian noise, and in most cases improving over models trained with Simplex noise (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Interestingly for the method of Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022b), when the model is trained with Sim- plex or coarse noise, the intermediate KL step (similar to ap- plying a DAE) gives better results than the subsequent diffusion inpainting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Model comparison We compare models trained with coarse noise on the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' To put the unsupervised anomaly detection perfor- mance results in context, we also provide supervised U-Net baselines, trained on a moderate number of labeled volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Pathology detection performance as evaluated on BraTS Head MRI Tumor test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Metrics are the test set wide pixel-level area under the precision-recall curve (AUPRC) and ideal Dice score (⌈Dice⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Mean results reported across 3 runs ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Method AUPRC ⌈Dice⌉ Thresholding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='749 f-AnoGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='365±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='449±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='014 VAE (reconstruction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='555±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='548±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='003 VAE (restoration) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='750±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='689±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 Diffusion (reconstruction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='653±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='610±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='060 Diffusion (KL + inpainting) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='689±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='675±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='015 → KL step only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='796±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='723±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='013 DAE (α = 16, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='833±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='773±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 U-Net (supervised) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='972±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='914±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='002 Quantitative results can be seen in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Overall, denoising methods appear to offer more accurate anomaly de- tection than other unsupervised methods, with the simple DAE giving overall best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion models perform bet- ter than unsupervised baselines except for the simple threshold- ing baseline (provided for MRI but not possible for the multi- intensity lesions in CT), but worse than the DAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' As seen in Table 2, the intermediate KL step of Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022b)’s method outperforms the results of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Relationship between noise resolution and anomaly size We further examine the relationship between the DAE noise used at training time and performance on anomalies during test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In particular, we investigate whether the coarseness of the noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' noise resolution α before upsampling) has a large impact on the size of test anomalies that the DAE successfully detects as this could imply the need to tune the noise parameters for specific anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In order to isolate the effect, we evaluate DAEs trained with different noise coarseness on synthetic anomalies in 3D Head CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We synthesize bright spherical anomalies inside the brain at random locations, sampling the diameter uniformly from the range of 5mm to 50mm and then multiplying the nor- mal tissue intensity within the sphere by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Full results are shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We observe no rela- tionship between the noise resolution and the performance on anomalies in specific size range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' That is, selecting a reason- ably coarse noise shows consistent best performance across a wide range of synthetic anomaly sizes with no affinity towards a specific anomaly size when compared to DAEs trained with different noise coarseness parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Visualization of model outputs Selected examples of image slices and the corresponding de- noising reconstructions and anomaly detection heatmaps for different methods are shown in Figure 6 for head MRI and Fig- ure 7 for head CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The DAE performs consistently well on eas- ier tumor cases in MRI and larger hemorrhages in CT, produc- ing strong signal predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Weaker signal predictions some- times correspond to subtler tumors in MRI and ischemia cases in CT, and sometimes correspond to false positive detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Pathology detection performance as evaluated on iCAIRD Head CT Hemorrhage/Ischaemia/Tumour test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Metrics are the test set wide pixel-level area under the precision-recall curve (AUPRC) and ideal Dice score (⌈Dice⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Mean results reported across 3 runs ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Method AUPRC ⌈Dice⌉ VAE (reconstruction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='382±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='432±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 VAE (restoration) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='542±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='537±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='011 Diffusion (reconstruction) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='573±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='600±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='013 Diffusion (KL + inpainting) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='512±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='547±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='008 → KL step only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='629±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='608±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='003 DAE (α = 16, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='693±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='674±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='003 nnU-Net (supervised) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='817±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='786±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='004 10 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) FLAIR T1 T1Gd T2 GT a) b) c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Sample anomaly score predictions on 2D head MRI (BraTS2021) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Columns a), b), c) show diffusion reconstruction (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022), KL divergence (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b) and DAE anomaly scores respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' When we examine more closely the impact of model choice and noise type on the reconstruction (see Figure 8), we observe the trade-off between reconstruction of the image detail vs era- sure of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The diffusion model tends to better erase the anomaly compared to the DAE, particularly when trained with Simplex or even Gaussian noise, but also yields poorer re- construction of the image, erasing also fine details of normal anatomy such as the folds of the gyri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The DAE achieves best erasure, or at least suppression, of the anomaly when coarse noise is used, as well as best retention of the normal anatomical detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Discussion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' What type of noise is best?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our results show that the noise used for training denoising models has a large impact on anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our proposed coarse noise significantly improves the perfor- mance of both DAEs and diffusion models, outperforming naive Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Our coarse noise largely also outperforms the more complex Simplex noise alternative (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022), which similarly introduces low frequencies to the noise pattern (alongside higher frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Visual inspection of the recon- structions suggests that the noise used for model training im- pacts the trade-off between the extent of the detail that is recon- structed and the extent of erasure of anomalies, both of which contribute to better anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Remarkably, we found the same noise parameters to be op- timal across the two datasets described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Further, noise parameters appeared not to be related to anomaly size, as reported in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Thus, it appears that noise coarseness parameters are independent of the size of test anomalies and can be set without knowledge of the nature of the target anomalies ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We note however that these datasets were pro- cessed in a similar way and are of similar anatomy (head/brain), therefore it would be desirable to do rigorous testing of many anatomies, modalities, and intensity/resolution pre-processing in order to come to a general conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' What type of model is best?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In general, we find that employing denoising as the learn- ing mechanism enables architectures with skip connections to be used, leading to higher fidelity reconstructions which are more effective for anomaly detection than those achieved by VAE methods with bottleneck architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Considering only the denoising approaches, we find that sim- ple denoising autoencoder (DAE) models currently outperform more advanced diffusion models across the two datasets that we evaluated, when trained with optimal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' However, diffusion models show an exciting ability to erase anomalies and generate convincing high definition reconstructions, with the caveat that Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 11 Input Supervised a) b) c) d) e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Sample anomaly score predictions on contrasting example slices from the 3D head CT (CQ500) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Columns a) and b) show the show coarse noise diffusion reconstructions and anomaly scores (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022), column c) shows the coarse noise diffusion KL divergence anomaly scores (Pinaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022b), and columns d) and e) show coarse noise DAE reconstructions and the associated anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Input Gaussian Simplex Coarse Diffusion DAE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Comparison of reconstruction between DAE and diffusion using re- construction procedure by Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022) across the three different noise types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' normal anatomical detail may also be erased, limiting their ef- fectiveness at discriminating normal from abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Therefore, while diffusion models are producing state-of-the-art results in image generation, further investigation is needed to find appro- priate methods to apply diffusion models to anomaly detection specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' In terms of practical use, we note that diffusion methods (similarly to VAE restoration methods) come at a cost since all inference methods evaluated in this paper take hundreds of it- erations to produce final predictions, resulting in much longer inference times than for the DAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Limitations of our anomaly evaluation A limitation of the evaluations presented in this paper is that we have focused on a subset of anomalies which are present in the datasets, albeit also the anomalies which are of most clinical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This is explicit for the iCAIRD GG&C NHS Head CT dataset, in which we annotated 3 pathologies (hemorrhage, ischemia, tumor) but extracted NLP labels for several more (see Appendix C), filtering on these to obtain a healthy training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Consequently, our metrics only approximate general anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We leave comprehensive annotation of anomalies as an important avenue for future work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' in fact, the performance on rarer pathologies for which expert annotations are not available is potentially more important than on common pathologies since this is where training traditional supervised approaches might be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Alternatives to residual error for anomaly detection Finally, DAEs and the diffusion inference methods we have applied rely on reconstruction error in order to detect anoma- lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Reconstruction error might be suitable for prominent anomalies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' large hemorrhages) but struggle with anoma- lies subtler in intensity contrast (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' ischemia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Discriminative 12 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2022)) that infer the anomaly score directly have been achieving success, notably in the Medical Out-of-Distribution (MOOD) Analysis MICCAI Challenge (Zimmerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' They might be more suitable for subtle anomalies, since they do not use the residual error (which will be small for subtle inten- sity changes) as the anomaly signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' see Meissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021b) for more in-depth analysis of the pitfalls associated with us- ing residual error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Differently to reconstruction-based methods, discriminative methods are typically trained by synthesizing ab- normal data to discriminate from the healthy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This has pros and cons, allowing easier application of domain knowl- edge about the nature of anomalies and explicit control over the definition of “abnormal”, at the risk of losing generality and overfitting to the selected synthetic anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Conclusion In this paper we have demonstrated the effectiveness of a sim- ple coarse noise model in both simple classical DAEs and more complex recently proposed diffusion models for anomaly de- tection across two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We find that the parametrization of the noise model has a wide tolerance, giving robust transfer across datasets and denoising methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' As part of this work, we implemented an anomaly detection pipeline in a real-world scenario involving the collation of a healthy training set by run- ning NLP methods on radiology reports, thereby showing that a largely automated pipeline is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Overall the classical DAE outperforms other methods, in terms of implementation simplicity, accuracy, and inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' While detection is successful for more obvious instances of anomalies such as tumors, hemorrhages and ischemia, fur- ther accuracy improvements are required to achieve reliable detection of subtle anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Diffusion models applied to anomaly detection are as yet in their infancy and provide a promising avenue for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare that they have no known competing fi- nancial interests or personal relationships that could have influ- enced the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Acknowledgements This work is part of the Industrial Centre for AI Research in Digital Diagnostics (iCAIRD) which is funded by Innovate UK on behalf of UK Research and Innovation (UKRI) project num- ber 104690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We thank the West of Scotland Safe Haven at NHS Greater Glasgow and Clyde for their assistance in creating this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We would also like to acknowledge assistance of Canon Medical Research Europe Limited in providing the Canon Safe Haven Artificial Intelligence Platform (SHAIP) tool, assisting with the deidentification of data and the provision of a secure machine learning workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We acknowledge Engineering and Physical Sciences Re- search Council (EPSRC) for funding part of this work through the EPSRC Centre for Doctoral Training in Applied Photonics (CDTAP) managed by Heriot-Watt University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' This work was supported by the University of Edinburgh, the Royal Academy of Engineering and Canon Medical Re- search Europe via PhD studentships of Pedro Sanchez (grant RCSRF1819\\8\\25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Tsaftaris acknowledges the support of Canon Medi- cal and the Royal Academy of Engineering and the Research Chairs and Senior Research Fellowships scheme (grant RC- SRF1819\\8\\25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Many thanks to Sin Yee Foo, Harris Hameed, and Paul Don- nelly from GG&C NHS for creating the pathology annotations which we used for our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Many thanks to Paul Thomson and Ewan Hemingway for their help with developing the imaging pre-processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' References Alex, 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289–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Zimmerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', Kohl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', Petersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', Isensee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', Maier-Hein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Context-encoding variational autoencoder for unsupervised anomaly detec- tion, in: Medical Imaging with Deep Learning (MIDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 14 Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) Supplementary Material Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' DAE vs VAE reconstruction comparison FLAIR T1 T1Gd T2 Input VAE DAE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Sample healthy brain reconstructions from VAE and DAE mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The DAE gives more precise reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The VAE reconstruction quality could be improved by increasing bottleneck dimensionality, how- ever this would negatively impact anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Neural network architectures GN in|out Denoising autoencoder in|out 64|4 4|64 64|128 125|256 256|512 512|256 128|64 256|128 64|4 4|64 64|128 128|256 64|256 128|64 256|128 256|256 μ128 σ128 ~ Swish GN k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' = 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' = 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' = 16 out_channels = 256 stride = 1 Variational autoencoder ConvBlock module Swish GN out|out Swish GN in|out Weight standardized convolution layer: kernel size (k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=') = 3, stride=1 unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Numbers denote channel numbers in|out Convolution layer without weight standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Swish Swish activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Group normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' ConvBlock module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Numbers denote channel numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Average pooling, kernel size=2, stride=2 Transposed convolution layer, kernel size=2, stride=2 unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' U-Net skip connection Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Architectures of 2D DAE and VAE models used in brain MRI experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Antanas Kascenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' / Preprint (2023) 15 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Radiology report labels Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' List of report labels extracted from radiology reports using the method of Schrempf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We do not exclude scans with associated positive/uncertain labels which are underlined from our healthy training set, since we decide that scans with only these labels (and no others) are “normal for age”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Radiographic findings artefact, collection, compression, dilation, effacement, her- niation, hyperdensity, hypodensity, loss of differentia- tion, malacic changes, mass effect, midline shift, oedema, swelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Clinical impressions abscess, atrophy, aneurysm, calcification, cavernoma, cerebral small vessel disease, congenital abnormality, cyst, evidence of surgery/intervention, fracture, gliosis, hemor- rhage, hydrocephalus, ischemia, infection, tumor, vessel oc- clusion, lesion, pneumocephalus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Selecting and annotating the iCAIRD test set We provide additional details below on the process by which the iCAIRD test set was selected and annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Scan selection: For hemorrhage and ischemia cases, our primary source was the Scottish Stroke Care Audit (SSCA) records for which we had access for the stroke episodes in the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' we searched these records for stroke episodes classed as “hemorrhagic”, “ischemic”, or “hemorrhagic transformation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For cases of tumors and rarer hemorrhages (epidural and sub- dural), we used a combination of ICD-10 code and free text searches of the Scottish Morbidity Records (SMRs) and radi- ology reports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=', “extradural”, “extra-dural”, “extra dural”, “epidural”, “edh”, “subdural”, and “sdh”), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' We then excluded scans acquired prior to 2016 for image compres- sion reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Annotation and Review: We recruited 3 GG&C clinicians (one Consultant Neuroradiologist and two senior Radiology trainees) to perform pixel-level annotation, following the anno- tation protocol prepared for this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' For the selected cases, all hemorrhage, ischemia and tumor lesions present were anno- tated, including any surrounding regions of edema for hemor- rhagic lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The Consultant Neuroradiologist acted also as reviewer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' 40% of cases were randomly selected for review and annotators also had the option of sending any of the remaining 60% for review when they required a second opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Relationship between noise coarseness and anomaly size 10 20 30 40 50 Synthetic anomaly diameter (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='8 Performance (AUPRC) α = 1 α = 2 α = 4 α = 8 α = 16 α = 32 α = 64 α = 128 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' DAE anomaly detection performance evaluated using synthetic anomalies of different sizes with models trained with noise generated at different resolutions α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' Each point indicates a mean of three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} +page_content=' The best noise resolution (α = 16) generalizes the best across a wide variety of synthetic anomaly sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE_T4oBgHgl3EQf0hzM/content/2301.08330v1.pdf'} diff --git a/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf b/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf new file mode 100644 index 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b/edE1T4oBgHgl3EQfLgP4/content/tmp_files/2301.02979v1.pdf.txt @@ -0,0 +1,1057 @@ +CameraPose: Weakly-Supervised Monocular 3D Human Pose Estimation +by Leveraging In-the-wild 2D Annotations +Cheng-Yen Yang1∗, Jiajia Luo2, Lu Xia2, Yuyin Sun2, Nan Qiao2, Ke Zhang2, +Zhongyu Jiang1, Jenq-Neng Hwang1, Cheng-Hao Kuo2 +1 Department of Electrical and Computer Engineering, University of Washington, WA, USA +2 Amazon Lab126, USA +{cycyang,zyjiang,hwang}@uw.edu, +{lujiajia,luxial,yuyinsun,kezha,qiaonan,chkuo}@amazon.com +Abstract +To improve the generalization of 3D human pose estima- +tors, many existing deep learning based models focus on +adding different augmentations to training poses. However, +data augmentation techniques are limited to the ”seen” +pose combinations and hard to infer poses with rare ”un- +seen” joint positions. To address this problem, we present +CameraPose, a weakly-supervised framework for 3D hu- +man pose estimation from a single image, which can not +only be applied on 2D-3D pose pairs but also on 2D alone +annotations. By adding a camera parameter branch, any +in-the-wild 2D annotations can be fed into our pipeline +to boost the training diversity and the 3D poses can be +implicitly learned by reprojecting back to 2D. Moreover, +CameraPose introduces a refinement network module with +confidence-guided loss to further improve the quality of +noisy 2D keypoints extracted by 2D pose estimators. Ex- +perimental results demonstrate that the CameraPose brings +in clear improvements on cross-scenario datasets. Notably, +it outperforms the baseline method by 3mm on the most +challenging dataset 3DPW. In addition, by combining our +proposed refinement network module with existing 3D pose +estimators, their performance can be improved in cross- +scenario evaluation. +1. Introduction +Human pose estimation (HPE) is a task to predict the +configuration of a particular set of human body parts from +some visual input such as images or videos. Depending on +the output format, it can be further divided into 2D and 3D +HPE, respectively. Different from the 2D HPE that pre- +dicts the human keypoints with x, y coordinates, the 3D +* This work was mostly done when Cheng-Yen Yang was an intern at +Amazon Lab126. +Figure 1. Training data expansion overview. Data augmentation +on existing 2D poses can improve the diversity of training to some +extend. By taking advantage of in-the-wild 2D annotations, more +rare but challenging poses can be utilized to further improve the +model generalization. +HPE regresses x, y, z which can be more helpful to solve +difficult tasks, such as action and motion prediction[3, 7], +posture and gesture recognition [14, 22], augmented real- +ity and virtual reality [10, 12], healthcare [6, 19]. Although +deep learning based methods have boosted the performance +of 3D HPE [23, 24, 27, 28, 39], the error will typically in- +crease to around two times from Human3.6M [15] to 3DHP +[24] for cross-dataset scenario due to the poor model gener- +alization [11]. +arXiv:2301.02979v1 [cs.CV] 8 Jan 2023 + +PREVIOUSWORKS +EvoSkeleton (Lietal.) +PoseAug (Gong et al.) +Bone Angle +Bone Length +Rotation +Translation +OURMETHOD +CameraPose +Camera +Branch +Reprojection +3DPose +Branch +2DReprojectionLoss +2DAnnotation +OnlyDatasetsTable 1. MPJPE on Human3.6M using different source of 2D key- +points source: HRNet and ground-truth. +3D Pose Estimator +Human3.6M +(MPJPE) +2D Keypoints Source +HRNet +Ground-truth +Zhao et al. [38] +57.5 +44.4 +Martinez et al. [23] +53.0 +43.3 +Pavllo et al. [28] +52.2 +41.8 +Recent works argue that poor model generalization +can be mitigated by increasing the variance in training +data. +Therefore, many augmentation-related algorithms +have been proposed to improve the 3D HPE accuracy. How- +ever, no matter it is image-based augmentation [25, 31], +synthetic-based augmentation [5, 35], predefined transfor- +mation [20] or GAN-based augmentation [11], the vari- +ances added to the training data is still limited to the orig- +inal 2D-3D pair. Figure 1 shows examples of augmented +2D-3D pairs with different algorithms. We can observe that +the generated new pair 2D-3D cannot provide pose changes +(lying to sitting etc.). Due to the limitation in the training +data, the scenes or scenario are still relatively simple to the +in-the-wild environment, which hinder the real-world appli- +cation of these algorithms. +Different from the existing methods that rely on data +augmentation for training data expansion, we proposed a +novel weakly-supervised framework, CameraPose, to im- +prove model generalization on 3D HPE by taking advantage +of plentiful 2D annotations. Compared to the expensive 3D +annotations, 2D annotations are less expensive, and many +challenging 2D datasets [1, 17, 21] containing rich actions, +poses, and scenes are available in the literature. The pro- +posed CameraPose network can combine any existing 2D +or 3D datasets in a single framework by adding a camera +parameter estimation branch. Our approach also integrated +the GAN-base pose augmentation framework to improve +the training data diversity and ensure the camera branch’s +generalization. +Existing 3D HPE networks usually directly use 2D key- +points from some pre-trained detectors as input to train 3D +joints. However, inferred 2D keypoints will lead to the sit- +uation illustrated in Fig.2. The errors from the 2D joints +estimation step will generate 3D prediction errors on some +keypoints. In addition, augmentation on inaccurate 2D key- +points will further enlarge the errors in 3D joints. As shown +in Table 1, the ground-truth inputs significantly boosted +the accuracy in all testing cases with different pose estima- +tors. Therefore, it is necessary to improve the 2D keypoints +before feeding them into our 3D estimator network. To mit- +igate the error in 2D input, we propose to incorporate a re- +finement network that aims to infer better 2D joints based +on the positions and confidence scores of detected 2D joints. +Our contributions are three-fold: 1) We propose a camera +Figure 2. Example of feeding different source of 2D joints predic- +tion into the same 3D lifting network. Due to the inaccurate right +elbow prediction from the HRNet[32], the errors from the same +keypoint will be enlarged in the 3D poses. +parameter branch that will generate per-instance camera pa- +rameter inference so that any existing 2D keypoints datasets +(without 3D labeling) can be utilized in model training. 2) +We propose a Refinement Network to improve the accuracy +of 2D joints, which can be helpful in the GAN-based aug- +mentation stage, as well as the final 3D joints predictions. +3) We introduce the reprojection loss, confidence-guided re- +finement loss, together with the camera loss in the loss de- +sign to make the network differentiable. +2. Related Works +Fully-Supervised 3D HPE. There are a lot of papers and +research that use the 2D-3D annotation pairs for a fully- +supervised training manner. +Tekin et al. +[33] directly +regress the 3D human pose from a spatio-temporal volume +of bounding boxes, and Martinez et al. [23] regress the 3D +human pose from a naive MLP using 2D keypoints as input +and 3D keypoints as output. +On similar datasets, these end-to-end methods often per- +form very well. Their capacity to generalize to different +settings, on the other hand, is restricted. +Many studies +use cross dataset training or data augmentation to address +this issue [31, 25, 5, 35]. +Most recently, Li et al. +[20] +directly augment 2D-3D pose pairs by randomly apply- +ing partial skeleton recombination and joint angle pertur- +bation on source datasets. Then Gong et al. [11] used a +generative-based model to manipulate the transformation of +3D ground-truth and then do the reprojection back to image +space to get the corresponding 2D keypoints. This can be +trained along with the 3D lifting network and some discrim- +inators to ensure the augmented poses are realistic and in- + +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +X +Input: HRNet 2D Keypoints +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +-0.50 +0.75 +1.00 +X +Input: Ground-truth 2D KeypointsFigure 3. Overall framework of our proposed CameraPose. +It consisted of three main parts: +(1) RefineNet, (2) Pose Genera- +tor/Discriminator, and (3) Weakly-Supervised Reprojection Camera Branch. When trained with 2D-3D annotated datasets, all of the +loss will be used while with 2D only datasets, only the 2D projection loss will be considered to update the weights. +crease the diversity of the training dataset. While effective, +the major downside of all supervised approaches is that they +do not generalize well to unseen poses. Therefore, their ap- +plication to in-the-wild scenes is limited. +Some even use a portion amount of dataset to do the +training for human pose estimation through methods like +transfer learning [24, 8, 34]. As they all try to mixed 2D +pose from in-the-wild images and 3D poses from laboratory +settings to learn the deep features through shared represen- +tation. These methods generalize better to unseen poses be- +cause they learn distributions of realistic 3D postures and +their characteristics. They can recreate out-of-distribution +positions to a degree, but they have trouble with entirely +undetected poses. +Weakly-Supervised 3D HPE. Some approaches use un- +paired 2D-3D annotations to get some 3D priors or basis to +do the 3D human pose estimation from a monocular camera. +Drover et al. [9] proposed a projection layer that randomly +projects the predicted 3D poses back into 2D poses and then +feeds into a discriminator. Chen et al. [4] introduced cycle +consistency loss into [9] extending the training with a step +of lifting the projected 2D pose once again into the 3D pose. +Habibie et al. [13] designed an architecture that comprises +an encoding of explicit 2D and 3D features, and uses su- +pervision by a separately learned projection model from the +predicted 3D pose. Wandt et al. [36] proposed RepNet to +tackle the problem with reprojection constraints by using +an adversarial-based method with a sub-network that can +estimate the camera. However, we argue the gap between +supervised algorithms and unsupervised algorithms can be +large on some challenging datasets. +As for multi-view settings, Rochette et al. [30] using +multi-view consistency by moving the stereo reconstruction +problem into the loss. Kocabas et al. [18] proposed an- +other multi-view approach by applying epipolar geometry +to predicted 2D pose under different views to construct the +pseudo-ground-truth. Iqbal et al. [16] proposed a end-to- +end learning framework adopting a 2.5D pose representa- +tion without any 3D annotations. Wandt et al. [37] then +proposed a self-supervised method that requires no prior +knowledge about the scene, 3D skeleton, or camera calibra- +tion and also introduced the 2D joint confidences into the +3D lifting pipeline. However, these algorithms are hard to +be applied to single-view or in-the-wild predictions due to +their multi-view pipeline design. +HPE with Data Augmentation. Data augmentation can +help the model generalization ability by enlarging the train- +ing data [31, 25, 5, 35]. +Most recently, Li et al. +[20] +directly augment 2D-3D pose pairs by randomly apply- +ing partial skeleton recombination and joint angle pertur- +bation on source datasets. +Then Gong et al. +[11] used +a generative-based model to manipulate the transformation +of 3D ground-truth then do the reprojection back to image +space to get the corresponding 2D keypoints. This can be +trained along with the 3D lifting network and some discrim- +inators to ensure the augmented poses are realistic and in- +crease the diversity of the training dataset. +3. Proposed Method +The CameraPose network consisted of three main parts: +(1) Refinement Network, (2) Pose Generator/Discriminator, +and (3) Weakly-Supervised Camera Parameter Branch. Fig- +ure 3 summarizes our CameraPose architecture design. +Let x ∈ R2×NJ denotes the 2D keypoints and X ∈ +R3×NJ denotes the corresponding 3D joint position in the +camera coordinate system with NJ represents the number +of joints in the framework. Our proposed network will train +on two different cases of datasets: (1) 2D-3D annotated +dataset ϕ = (x, X) ,and (2) 2D annotations only dataset + +3DPoseEstimation Loss +小 +3D Pose +Branch +2DRefinement +Loss +GT 2D Pose +Refinement +Pose +3DLifting +Network +Generator +Network +GT Intrinsic Params +Intrinsic Params +2D Pose + confidence scores +Refined 2D Pose +Camera Param +(fa,fy,Cr,Cg) +Perspective +(clean joints) +(fr,fy.Cr,Cy) +(noisy joints) +Branch +Reproject +Pose +GT Offset t +Offset t +Discriminator +Reprojected 2D Pose +CameraParameter Loss +2DReprojection Lossϕ′ = (x’, −) by optimizing the following equation: +min +θ3D,θref Lϕ +� +Pθ3D +� +Rθref (x) +� +, ϕ +� ++Lϕ′ +� +Pθ3D +� +Rθref (x′) +� +, ϕ′� +(1) +where θ3D and θref represent the weights of our 3D lift- +ing model and refinement network. Furthermore we extend +the design of pose augmentorA to enlarge the 2D-3D anno- +tated dataset with the augmented dataset A(ϕ) = (x∗, X∗). +Therefore our end-to-end optimization procedure will be- +come: +min +θ3D,θrefmax +θA Lϕ +� +ϕ ∪ A(ϕ) +� ++ Lϕ′� +ϕ′� +. +(2) +Table 2. Mathematical notations used in the equations. +Notation +Description +NJ +number of joints used +NS +number of samples in the batch +ϕ +datasets with 2D-3D annotations +ϕ′ +datasets with 2D annotations only +ϕ∗ +datasets generated by the pose generator +(x, X) +ground-truth 2D-3D annotations from ϕ +(x′, −) +ground-truth 2D annotations from ϕ′ +(x∗, X∗) +augmented 2D-3D annotations from ϕ∗ +ˆX +predicted 3D poses from 3D lifting network +3.1. Refinement Network +Instead of refining on the original noisy 2D keypoints, +we utilize the confidence score combined with the 2D (x, y) +coordinates as input to the refinement network. We first nor- +malize the coordinates of keypoints to (−1, 1) with respect +to the input image height and width. We also normalized +the confidence scores to a comparable scale by Eq. 3: +c′ +ij = +cij +||Ci||1 +(3) +where || · ||1 denotes for L1 norm and Ci stands for the all +the heatmaps in the i-th training sample while cij stands for +the maximum value (confidence score) on the j-th heatmap. +The normalized confidence score will be used as the weight +to compute the joint-wise mean-square error in Eq. 4. +The neural network architecture of our Refinement Net- +work is a standard residual block consisting of fully con- +nected layers with a hidden dimension of 512. The refine- +ment loss Lref is formulated as: +Lref = +1 +NS · NJ +NS +� +i +NJ +� +j +c′ +ij(xij − ˆxij)2 +(4) +where we compute the mean-square-error over the number +of training samples NS of the predicted poses ˆx and nor- +malized ground-truth poses x with joint-wise normalized +confidence-weight c′. +Figure 4. An example of heatmap visualization. Image in the up- +per left corner is the original image overlaid with the keypoints +extracted by HRNet. +All the rest images showed the overlaid +heatmaps from different keypoints. The maximum scores of each +keypoints are different and lower scores indicate lower confidence +level. +3.2. Camera Parameter Branch +In this paper, the 2D-3D pose pairs are calculated in the +camera coordinate system, so the camera parameters can be +simplified to be the intrinsic matrix Mint in Eq. 5 and a +3D offset t3D. For intrinsic matrix Mint we are essentially +predicting a 4-dimensional vector, namely fx, fy, cx, cy, the +focal lengths fx, fy, and principal center offsets cx, cy along +the x and y direction respectively. +Mint = + + +fx +0 +cx +0 +fy +cy +0 +0 +1 + + +(5) +and for the 3D offset t3D we are predicting a 3-dimensional +vector: +t3D = + + +tx +ty +tz + + . +(6) +The camera parameter branch consists of 2 residual +blocks with a hidden dimension of 512, which can be +plugged in to any standard 3D pose estimators. There are +three losses that can be involved depending on the annota- +tions. The 2D reprojection loss L2D as shown in Eq. 7 cal- +culates the Euclidean distance between the reprojected 2D +poses and ground truth. The mean-square error (MSE) is +used in loss calculation for both the camera parameter loss +and 3D inference loss as shown in Eqs. 8 and 9 respectively. +L2D,ϕ′ = 1 +N +N +� +i +Nj +� +j +( +ˆ +M inti · (ˆXij + ˆt3D,i) − xij)2, (7) +Lcam = ||M int − +ˆ +M int||2 +2 + ||t3D − ˆt3D||2 +2, +(8) + +lp0.7668 +Hp0.846 +RFoot 0.8664 +HP0.775.L3D = +1 +NS · NJ +NS +� +i +Nj +� +j +(Xij − ˆXij)2 +(9) +where ˆX stands for the predicted 3D pose from our 3D lift- +ing network. +Since CameraPose can work on 2D-3D pose pairs as well +as 2D alone pose estimations, the loss design can be differ- +ent according to the availability of labels. In the case of +all annotations are available during the training stage, the +camera loss can be calculated as: +Lϕ = λcamLcam + λ2D,ϕL2D,ϕ + λ3DL3D +(10) +In the case of 2D annotation alone training step, the loss +calculation will be from 2D reprojection error: +Lϕ′ = λ2D,ϕ′L2D,ϕ′ +(11) +3.3. Pose Generator and Discriminator +Similar to the framework in [11], we utilized both gen- +erator and discriminator to further improve the diversity +in training poses. As shown in Figure 5, the generator is +plugged in to the 2D pose generation stage, and the dis- +criminator is applied on both the 2D and 3D pose inference. +The generator is actually formed by 3 simple multi-layer +perceptions that generated different parameters for 3 differ- +ent augmentation operations respectively: (1) changing the +bone angle Xba, (2) changing the bone length Xbl and (3) +changing the camera view and position of the input 3D pose +R · Xbl + t. +The discriminator part of the framework can be divided +into 2 portions, the D2D and D3D as we want to make +sure that both the augmented X∗ and x∗ formed plausible +human poses in both image coordinate and camera coor- +dinate. But in our work we not only want to ensure the +goodness of the augmented poses from the generator, we +also want to utilized the discriminator to regulated our re- +projected 2D poses for those 2D annotations only dataset +cases. The discriminators also adapt the part-aware Kine- +matic Chain Space (KCS) proposed in [11], they are fully +connected networks with a structure similar to the pose re- +gression network using the KCS representation [36] of 2D +or 3D poses as input. Here we use the LS-GAN loss: +L2d +dis = 1 +2Ex[(D2D(KCS(x)) − 1)2] +(12) ++ 1 +2Ex[(D2D(KCS({x∗, x′ +2D})) − 1)2] +(13) +L3d +dis = 1 +2Ex[(D3D(KCS(X)) − 1)2] +(14) ++ 1 +2Ex[(D3D(KCS(X∗)) − 1)2] +(15) +Figure 5. Visualization of the pose generator and discriminator. +As we augmented from the original 2D-3D annotated dataset ϕ = +(x, X) using the 3D pose as the input to the generator which give +3 different sets of parameters γba, γbl and (R, t) to sequentially +modified the 3D pose into our augmented dataset ϕ∗ = (x∗, X∗). +as the pose discrimination loss to train the generator and +discriminator. +3.4. Overall Loss +The overall framework is made differentiable and can be +trained in the end-to-end fashion. We update different mod- +ules alternatively by minimizing loss in Eq. 4, Eq. 10, Eq. +11 as well as generators and discriminators with some pre- +assigned hyper-parameters λ. +Then we interactively train the entire model and update +the weights of 3D lifting network using the losses: +Lϕ = λref,ϕLref + λcamLcam + λ2D,ϕL2D,ϕ + λ3DL3D +(16) +and +Lϕ′ = λref,ϕ′Lref + λ2D,ϕ′L2D,ϕ′. +(17) +depending on the different datasets ϕ or ϕ′ we are using +for the batch. We will introduce more training details and +hyper-parameter settings in the Sec. 4.3. +4. Experiments +4.1. Datasets +For the 2D-3D paired annotations, we utilize the most +popular datasets 3D HPE dataset Human3.6M [15], 3DHP +[24] and 3DPW [34]. Both Human3.6M and 3DHP were +collected indoor in some laboratory environment through + +1.0 +Pose +0.0 +Generator +0.5 +0.0 +0.5 +3.5 +0.5 +0.5 +0.0 + 0.0 +Tba +19 +0.5 +(R,t) +0.0 +0.5 +0.5 +00g001 +5.0 +5.5 +1.0 +4.5 +0.5 0.0 +5.0 +1.5 +. +Perspective +3DPose +Reprojection +Discriminator +T +2DPose +DiscriminatorTable 3. Different human pose estimation datasets used in our work. The datasets in bold font are used for the training while other dataset +in italic are used for cross-dataset evaluation. The rest of the datasets will be used to visualize and serve as qualitative analysis targets. +Dataset +# of Sample +2D Annotations +3D Annotations +Camera Parameters +Human3.6M [15] +3.6M +v +v +v +MPI-INF-3DHP [24] +1.3M +v +v +v +3DPW [34] +51k +v +v +Ski-Pose PTZ [29] +20k +v +v +v +MPII [1] +25k +v +MS-COCO [21] +250k +v +Figure 6. Qualitative comparison for the Human3.6M [15] (left), 3DHP [24] (right) and 3DPW [34] (bottom) generalization ability analysis +using the pretrained baseline [11] and our purposed method. Both the baseline and our model were trained only with Human3.6M so 3DHP +and 3DPW are considered as cross-dataset in this case. The green arrows highlight locations where the models predict differently. +the MoCap (motion capture) system [26] with multiple cal- +ibrated cameras. The 3DPW is a more challenging dataset +collected in outdoor environment using IMU (inertial mea- +surement unit) sensors with mobile phone lens. +For 2D annotations only datasets, we used MPII [1] +which contains a variety of in-the-wild everyday human ac- +tivities. Another popular 2D dataset MS-COCO [21] is +also used for qualitative analysis purposes. Although the +2D annotation dataset such as MPII is much less than Hu- +man3.6M or 3DHP in terms of sample size, these 2D anno- +tation datasets contain more challenging human poses with +different activities. Note that both Human3.6M and 3DHP +are video based datasets, so that the total number of images +is much larger than MPII and MSCOCO. We summarized +the datasets utilized in our experiments in Table 3. +4.2. Preprocessing +Different datasets have distinctive annotations on joints, +which make the model training difficult. In this paper, we +used the Human3.6M format as standard one, and inter- +preted missing joints by labeling nearby joints for other +datasets. All the joints that are not included in Human3.6M +format will be discarded. +Many existing 3D HPE algorithms use the groundtruth +as model input for evaluation. However, groundtruth is not +available in real use cases. To evaluate the model perfor- +mance on the real-world applications, we also used existing +2D detector HRNet to extract the 2D keypoints as model +input and rerun results on different datasets. +Due to the various labeling schemes or joint formats dif- +ference, we preprocess other schemes into the Human3.6M +format by simple interpolation of some related joints and re- +moval of the unused joints. For example, there is no pelvis; + +Image +Image +GT +Baseline +GT +Baseline +CameraPose +CameraPoseTable 4. Result on Human3.6M, 3DHP and 3DPW using the 2D ground-truth keypoints as the input in terms of MPJPE, note that we use +the same model for evaluation on all datasets to mimic cross-dataset evaluation. Best results are shown in bold font. +Method +Human3.6M (MPJPE) +3DHP (MPJPE) +3DPW (PA-MPJPE) +Wnadt et al. [37] +74.3 +104.0 +- +Rhodin et al. [29] +80.1 +121.8 +- +Zhao et al. [38] +44.4 +97.4 +- +Martinez et al. [23] +43.3 +85.3 +- +Cai et al. [2] +41.7 +87.8 +- +Pavllo et al. [28] +41.80 +92.64 +76.38 +Gong et al. [11] +39.02 +76.13 +66.27 +Ours (CameraPose) +38.87 +78.85 +63.26 +we simply create such joint by computing the mid-point of +the left and right hip of any given label. Even though such +interpolations are not always perfect due to the nature of +each dataset, this preprocessing procedure allows us to have +a better idea and comparison on cross-dataset scenarios. +4.3. Training +CameraPose network is trained on 2 datasets: +Hu- +man3.6M (2D + 3D) and MPII (2D). For the former, we +followed most 3D human pose estimation training protocols +using the subjects S1, S5, S6, S7, S8 from Human3.6M as +our 2D-3D training data, and subjects S9, S11 for evalua- +tion purposes. For the latter, we filtered and selected around +10k training samples by checking the joints annotations. +For evaluation, MPI-INF-3DHP and 3DPW were used to +get quantitative results in terms of MPJPE (mean-per-joint- +position-error) and PA-MPJPE (aligned with ground-truths +by rigid transformation). +The model training can be divided into 3 steps. +The +refinement network was trained as the first step for 100 +epochs with learning rate being 0.0001 and weight decay +at epochs 30, 60 and 90, respectively. Next step, the 3D +lifting network along with the pose generator and discrimi- +nator was trained using Human3.6M dataset for 10 epochs +with a learning rate of 0.0001. This step is for warm-up and +GAN tuning which can make the following model training +more stable. Finally, the model was trained in an end-to- +end fashion using both 2D-3D pairs annotations as well as +2D alone annotations. In each iteration, we first updated the +weights of generator and discriminator to make the genera- +tors more stable. Then the 3D lifting network was updated +based on the augmented poses plus the 2D-3D annotated +dataset. After that, 2D only annotations were utilized to +tune the camera parameter branch. The model was trained +for 75 epochs with a learning rate of 0.0005 and weight de- +cay at 30, 60, respectively. And the weighting for loss we +choose λcam = 0.01, λ2D,ϕ = 0.5, λ2D,ϕ′ = 0.2, and +λ3D = 1.0. +Figure 7. Visualization for qualitative analysis of 3D human pose +estimation on MPII [1] (Testing), MS-COCO [21], and SKiPose- +PTZ [29]. Our model can still generate reliable 3D poses even +when the target poses are in general rare or never seen from the +training. +4.4. Quantitative Results +CameraPose Network Accuracy. We compared Camera- +Pose with other state-of-the-art methods [2, 28, 38, 11, 23] +trained on Human3.6M. For the temporal-based methods +[2, 28], we implemented the single frame version for a +fair comparison. Table 4 summarized the experimental re- +sults of different methods. For each column, the MPJPE +or PA-MPJPE are calculated for evaluation, obtained from +the same model trained and selected based on the evaluation +dataset of Human3.6M. Some existing algorithms selected +distinctive best models on different testing datasets, which +may not reflect the generalization of models well. Instead, +we selected a single model based on the accuracy of the +validation of Human3.6M to make it more realistic for real- +world application. +As shown in Table 4, our method outperforms the SOTA +on the most challenging dataset 3DPW by a noticeable mar- +gin (3mm and 13mm). It also has significantly higher accu- + +TTable 5. Experimental results of the effect of refinement network as we examine the effectiveness of our refinement module using the +HRNet detections on training and evaluation purposes. +Method +Training Source +(2D Estimator) +Human3.6M +(MPJPE) +3DHP +(MPJPE) +Pavllo et al. [28] +Human3.6M (HRNet) +57.90 +103.86 +Gong et al. [11] +Human3.6M (HRNet) +55.18 +99.50 +Gong et al. [11] w/ Refinement Network +Human3.6M (HRNet) +54.32 +97.45 +CameraPose w/ Refinement Network +Human3.6M (HRNet) +54.20 +97.35 +CameraPose w/o Refinement Network +Human3.6M (HRNet) +54.38 +98.12 +racy than other weakly-supervised methods like [29] and +[37]. Our model also achieves the highest accuracy on the +Human3.6M dataset. Experimental results clearly show the +strong generalization capability of our proposed method. +Adding the camera parameter branch can help the model to +learn from in-the-wild datasets with 2D annotations, which +is very effective for hard examples. +The results on the 3DHP are slightly lower than SOTA +methods, and we claim it is due to the fact that the 2D an- +notations we added from MPII are more helpful for chal- +lenging cases such as the 3DPW dataset. The best accuracy +on the 3DHP dataset can be 75.54 MPJPE using our model, +which outperforms the current SOTA if we select a specific +model for the 3DHP dataset. +Refinement Network Accuracy. To show the effectiveness +of the refinement network, we trained different models with +different settings as shown in the Table 5. +We used HRNet as 2D detectors to extract the 2D key- +points on all the training and evaluation datasets. We added +the refinement network to both the SOTA method [11] and +our proposed model. By adding the refinement network, +both PoseAug and our model have improved accuracy on +both Human3.6M and 3DHP. In addition, our model outper- +forms the SOTA on both testing datasets. Therefore, both +our proposed camera parameter network and the refinement +network are useful for 3D HPE. +4.5. Qualitative Visualization +3D Pose Estimation. We choose 3 datasets (Human3.6M, +3DHP and 3DPW) to qualitative compare our proposed +method and baseline [11]. As shown in Figure 6, our model +has more accurate predictions on challenging datasets such +as 3DPW. Note that we utilize cross-scenario training to +make sure there is no overlap between training and test- +ing datasets. We also visualize our results on datasets with- +out 3D annotations such as MPII, MSCOCO, and SkiPose- +PTZ [29] in Figure 7. The visualization results are very +plausible, which indicates the capability of our model for +in-the-wild prediction. +2D Reprojection. To validate the camera parameter branch, +we visualize the results of our model at a different stage. +Figure 8 shows the original image, input 2D keypoints from +HRNet, inferred 3D poses, and reprojected 2D poses from +Figure 8. 3D-2D reprojection visualization on MPII [1]. Column +from the left: original images, 2D keypoints from HRNet, inferred +3D keypoints, reprojected 2D keypoints. The camera parameters +predicted by CameraPose can successfully reprojected the 3D pose +back into the image coordinate. +left to right columns. It clearly shows that our CameraPose +can predict well on unseen poses and the reprojected 2D +poses are meaningful too. +5. Conclusions +We propose CameraPose, a weakly-supervised frame- +work for 3D human pose estimation from a single image +that can aggregate 2D annotations by designing a camera +parameter branch. Given any noisy 2D keypoints from pre- +trained 2D pose estimator, CameraPose is able to refine the +keypoints with a confidence-guided loss and feed them into +the 3D lifting network. Since our approach uses the camera +parameters learned from the camera branch to do the repro- +jection back to 2D, it can solve the problem of the lacking of +the 2D-3D datasets with rare poses or outdoor scenes. We +evaluate our proposed method on some benchmark datasets; +the results show that our model can achieve higher accuracy +on challenging datasets and be able to predict meaningful +3D poses given in-the-wild images or 2D keypoints. + +Original Image +Input2DKeypoints +3DPrediction +Reprojected 2D +(MPII Test) +(HRNet) +(CameraPose) +(CameraPose)References +[1] Mykhaylo Andriluka, Leonid Pishchulin, Peter Gehler, and +Bernt Schiele. 2d human pose estimation: New benchmark +and state of the art analysis. 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In The IEEE International +Conference on Computer Vision (ICCV), Oct 2017. + diff --git a/edE1T4oBgHgl3EQfLgP4/content/tmp_files/load_file.txt b/edE1T4oBgHgl3EQfLgP4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ef6ad9f5b4c531d6a73cb234d3e42721e03fd87 --- /dev/null +++ b/edE1T4oBgHgl3EQfLgP4/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf,len=564 +page_content='CameraPose: Weakly-Supervised Monocular 3D Human Pose Estimation by Leveraging In-the-wild 2D Annotations Cheng-Yen Yang1∗, Jiajia Luo2, Lu Xia2, Yuyin Sun2, Nan Qiao2, Ke Zhang2, Zhongyu Jiang1, Jenq-Neng Hwang1, Cheng-Hao Kuo2 1 Department of Electrical and Computer Engineering, University of Washington, WA, USA 2 Amazon Lab126, USA {cycyang,zyjiang,hwang}@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='edu, {lujiajia,luxial,yuyinsun,kezha,qiaonan,chkuo}@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='com Abstract To improve the generalization of 3D human pose estima- tors, many existing deep learning based models focus on adding different augmentations to training poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' However, data augmentation techniques are limited to the ”seen” pose combinations and hard to infer poses with rare ”un- seen” joint positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' To address this problem, we present CameraPose, a weakly-supervised framework for 3D hu- man pose estimation from a single image, which can not only be applied on 2D-3D pose pairs but also on 2D alone annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' By adding a camera parameter branch, any in-the-wild 2D annotations can be fed into our pipeline to boost the training diversity and the 3D poses can be implicitly learned by reprojecting back to 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Moreover, CameraPose introduces a refinement network module with confidence-guided loss to further improve the quality of noisy 2D keypoints extracted by 2D pose estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Ex- perimental results demonstrate that the CameraPose brings in clear improvements on cross-scenario datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Notably, it outperforms the baseline method by 3mm on the most challenging dataset 3DPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In addition, by combining our proposed refinement network module with existing 3D pose estimators, their performance can be improved in cross- scenario evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Introduction Human pose estimation (HPE) is a task to predict the configuration of a particular set of human body parts from some visual input such as images or videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Depending on the output format, it can be further divided into 2D and 3D HPE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Different from the 2D HPE that pre- dicts the human keypoints with x, y coordinates, the 3D This work was mostly done when Cheng-Yen Yang was an intern at Amazon Lab126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Training data expansion overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Data augmentation on existing 2D poses can improve the diversity of training to some extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' By taking advantage of in-the-wild 2D annotations, more rare but challenging poses can be utilized to further improve the model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' HPE regresses x, y, z which can be more helpful to solve difficult tasks, such as action and motion prediction[3, 7], posture and gesture recognition [14, 22], augmented real- ity and virtual reality [10, 12], healthcare [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Although deep learning based methods have boosted the performance of 3D HPE [23, 24, 27, 28, 39], the error will typically in- crease to around two times from Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M [15] to 3DHP [24] for cross-dataset scenario due to the poor model gener- alization [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='02979v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='CV] 8 Jan 2023 PREVIOUSWORKS EvoSkeleton (Lietal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=') PoseAug (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=') Bone Angle Bone Length Rotation Translation OURMETHOD CameraPose Camera Branch Reprojection 3DPose Branch 2DReprojectionLoss 2DAnnotation OnlyDatasetsTable 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' MPJPE on Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M using different source of 2D key- points source: HRNet and ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3D Pose Estimator Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (MPJPE) 2D Keypoints Source HRNet Ground-truth Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [38] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='4 Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [23] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3 Pavllo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [28] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='8 Recent works argue that poor model generalization can be mitigated by increasing the variance in training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Therefore, many augmentation-related algorithms have been proposed to improve the 3D HPE accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' How- ever, no matter it is image-based augmentation [25, 31], synthetic-based augmentation [5, 35], predefined transfor- mation [20] or GAN-based augmentation [11], the vari- ances added to the training data is still limited to the orig- inal 2D-3D pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Figure 1 shows examples of augmented 2D-3D pairs with different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We can observe that the generated new pair 2D-3D cannot provide pose changes (lying to sitting etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Due to the limitation in the training data, the scenes or scenario are still relatively simple to the in-the-wild environment, which hinder the real-world appli- cation of these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Different from the existing methods that rely on data augmentation for training data expansion, we proposed a novel weakly-supervised framework, CameraPose, to im- prove model generalization on 3D HPE by taking advantage of plentiful 2D annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Compared to the expensive 3D annotations, 2D annotations are less expensive, and many challenging 2D datasets [1, 17, 21] containing rich actions, poses, and scenes are available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The pro- posed CameraPose network can combine any existing 2D or 3D datasets in a single framework by adding a camera parameter estimation branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Our approach also integrated the GAN-base pose augmentation framework to improve the training data diversity and ensure the camera branch’s generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Existing 3D HPE networks usually directly use 2D key- points from some pre-trained detectors as input to train 3D joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' However, inferred 2D keypoints will lead to the sit- uation illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The errors from the 2D joints estimation step will generate 3D prediction errors on some keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In addition, augmentation on inaccurate 2D key- points will further enlarge the errors in 3D joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As shown in Table 1, the ground-truth inputs significantly boosted the accuracy in all testing cases with different pose estima- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Therefore, it is necessary to improve the 2D keypoints before feeding them into our 3D estimator network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' To mit- igate the error in 2D input, we propose to incorporate a re- finement network that aims to infer better 2D joints based on the positions and confidence scores of detected 2D joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Our contributions are three-fold: 1) We propose a camera Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Example of feeding different source of 2D joints predic- tion into the same 3D lifting network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Due to the inaccurate right elbow prediction from the HRNet[32], the errors from the same keypoint will be enlarged in the 3D poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' parameter branch that will generate per-instance camera pa- rameter inference so that any existing 2D keypoints datasets (without 3D labeling) can be utilized in model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 2) We propose a Refinement Network to improve the accuracy of 2D joints, which can be helpful in the GAN-based aug- mentation stage, as well as the final 3D joints predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3) We introduce the reprojection loss, confidence-guided re- finement loss, together with the camera loss in the loss de- sign to make the network differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Related Works Fully-Supervised 3D HPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' There are a lot of papers and research that use the 2D-3D annotation pairs for a fully- supervised training manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Tekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [33] directly regress the 3D human pose from a spatio-temporal volume of bounding boxes, and Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [23] regress the 3D human pose from a naive MLP using 2D keypoints as input and 3D keypoints as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' On similar datasets, these end-to-end methods often per- form very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Their capacity to generalize to different settings, on the other hand, is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Many studies use cross dataset training or data augmentation to address this issue [31, 25, 5, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Most recently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [20] directly augment 2D-3D pose pairs by randomly apply- ing partial skeleton recombination and joint angle pertur- bation on source datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Then Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [11] used a generative-based model to manipulate the transformation of 3D ground-truth and then do the reprojection back to image space to get the corresponding 2D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' This can be trained along with the 3D lifting network and some discrim- inators to ensure the augmented poses are realistic and in- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 X Input: HRNet 2D Keypoints 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='00 X Input: Ground-truth 2D KeypointsFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Overall framework of our proposed CameraPose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' It consisted of three main parts: (1) RefineNet, (2) Pose Genera- tor/Discriminator, and (3) Weakly-Supervised Reprojection Camera Branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' When trained with 2D-3D annotated datasets, all of the loss will be used while with 2D only datasets, only the 2D projection loss will be considered to update the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' crease the diversity of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' While effective, the major downside of all supervised approaches is that they do not generalize well to unseen poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Therefore, their ap- plication to in-the-wild scenes is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Some even use a portion amount of dataset to do the training for human pose estimation through methods like transfer learning [24, 8, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As they all try to mixed 2D pose from in-the-wild images and 3D poses from laboratory settings to learn the deep features through shared represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' These methods generalize better to unseen poses be- cause they learn distributions of realistic 3D postures and their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' They can recreate out-of-distribution positions to a degree, but they have trouble with entirely undetected poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Weakly-Supervised 3D HPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Some approaches use un- paired 2D-3D annotations to get some 3D priors or basis to do the 3D human pose estimation from a monocular camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Drover et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [9] proposed a projection layer that randomly projects the predicted 3D poses back into 2D poses and then feeds into a discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [4] introduced cycle consistency loss into [9] extending the training with a step of lifting the projected 2D pose once again into the 3D pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Habibie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [13] designed an architecture that comprises an encoding of explicit 2D and 3D features, and uses su- pervision by a separately learned projection model from the predicted 3D pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Wandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [36] proposed RepNet to tackle the problem with reprojection constraints by using an adversarial-based method with a sub-network that can estimate the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' However, we argue the gap between supervised algorithms and unsupervised algorithms can be large on some challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As for multi-view settings, Rochette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [30] using multi-view consistency by moving the stereo reconstruction problem into the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Kocabas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [18] proposed an- other multi-view approach by applying epipolar geometry to predicted 2D pose under different views to construct the pseudo-ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Iqbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [16] proposed a end-to- end learning framework adopting a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5D pose representa- tion without any 3D annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Wandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [37] then proposed a self-supervised method that requires no prior knowledge about the scene, 3D skeleton, or camera calibra- tion and also introduced the 2D joint confidences into the 3D lifting pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' However, these algorithms are hard to be applied to single-view or in-the-wild predictions due to their multi-view pipeline design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' HPE with Data Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Data augmentation can help the model generalization ability by enlarging the train- ing data [31, 25, 5, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Most recently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [20] directly augment 2D-3D pose pairs by randomly apply- ing partial skeleton recombination and joint angle pertur- bation on source datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Then Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [11] used a generative-based model to manipulate the transformation of 3D ground-truth then do the reprojection back to image space to get the corresponding 2D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' This can be trained along with the 3D lifting network and some discrim- inators to ensure the augmented poses are realistic and in- crease the diversity of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Proposed Method The CameraPose network consisted of three main parts: (1) Refinement Network, (2) Pose Generator/Discriminator, and (3) Weakly-Supervised Camera Parameter Branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Fig- ure 3 summarizes our CameraPose architecture design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Let x ∈ R2×NJ denotes the 2D keypoints and X ∈ R3×NJ denotes the corresponding 3D joint position in the camera coordinate system with NJ represents the number of joints in the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Our proposed network will train on two different cases of datasets: (1) 2D-3D annotated dataset ϕ = (x, X) ,and (2) 2D annotations only dataset 3DPoseEstimation Loss 小 3D Pose Branch 2DRefinement Loss GT 2D Pose Refinement Pose 3DLifting Network Generator Network GT Intrinsic Params Intrinsic Params 2D Pose + confidence scores Refined 2D Pose Camera Param (fa,fy,Cr,Cg) Perspective (clean joints) (fr,fy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='Cr,Cy) (noisy joints) Branch Reproject Pose GT Offset t Offset t Discriminator Reprojected 2D Pose CameraParameter Loss 2DReprojection Lossϕ′ = (x’, −) by optimizing the following equation: min θ3D,θref Lϕ � Pθ3D � Rθref (x) � , ϕ � +Lϕ′ � Pθ3D � Rθref (x′) � , ϕ′� (1) where θ3D and θref represent the weights of our 3D lift- ing model and refinement network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Furthermore we extend the design of pose augmentorA to enlarge the 2D-3D anno- tated dataset with the augmented dataset A(ϕ) = (x∗, X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Therefore our end-to-end optimization procedure will be- come: min θ3D,θrefmax θA Lϕ � ϕ ∪ A(ϕ) � + Lϕ′� ϕ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' (2) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Mathematical notations used in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Notation Description NJ number of joints used NS number of samples in the batch ϕ datasets with 2D-3D annotations ϕ′ datasets with 2D annotations only ϕ∗ datasets generated by the pose generator (x, X) ground-truth 2D-3D annotations from ϕ (x′, −) ground-truth 2D annotations from ϕ′ (x∗, X∗) augmented 2D-3D annotations from ϕ∗ ˆX predicted 3D poses from 3D lifting network 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Refinement Network Instead of refining on the original noisy 2D keypoints, we utilize the confidence score combined with the 2D (x, y) coordinates as input to the refinement network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We first nor- malize the coordinates of keypoints to (−1, 1) with respect to the input image height and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We also normalized the confidence scores to a comparable scale by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3: c′ ij = cij ||Ci||1 (3) where || · ||1 denotes for L1 norm and Ci stands for the all the heatmaps in the i-th training sample while cij stands for the maximum value (confidence score) on the j-th heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The normalized confidence score will be used as the weight to compute the joint-wise mean-square error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The neural network architecture of our Refinement Net- work is a standard residual block consisting of fully con- nected layers with a hidden dimension of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The refine- ment loss Lref is formulated as: Lref = 1 NS · NJ NS � i NJ � j c′ ij(xij − ˆxij)2 (4) where we compute the mean-square-error over the number of training samples NS of the predicted poses ˆx and nor- malized ground-truth poses x with joint-wise normalized confidence-weight c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' An example of heatmap visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Image in the up- per left corner is the original image overlaid with the keypoints extracted by HRNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' All the rest images showed the overlaid heatmaps from different keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The maximum scores of each keypoints are different and lower scores indicate lower confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Camera Parameter Branch In this paper, the 2D-3D pose pairs are calculated in the camera coordinate system, so the camera parameters can be simplified to be the intrinsic matrix Mint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 5 and a 3D offset t3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For intrinsic matrix Mint we are essentially predicting a 4-dimensional vector, namely fx, fy, cx, cy, the focal lengths fx, fy, and principal center offsets cx, cy along the x and y direction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Mint = \uf8ee \uf8f0 fx 0 cx 0 fy cy 0 0 1 \uf8f9 \uf8fb (5) and for the 3D offset t3D we are predicting a 3-dimensional vector: t3D = \uf8ee \uf8f0 tx ty tz \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' (6) The camera parameter branch consists of 2 residual blocks with a hidden dimension of 512, which can be plugged in to any standard 3D pose estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' There are three losses that can be involved depending on the annota- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The 2D reprojection loss L2D as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 7 cal- culates the Euclidean distance between the reprojected 2D poses and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The mean-square error (MSE) is used in loss calculation for both the camera parameter loss and 3D inference loss as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 8 and 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' L2D,ϕ′ = 1 N N � i Nj � j ( ˆ M inti · (ˆXij + ˆt3D,i) − xij)2, (7) Lcam = ||M int − ˆ M int||2 2 + ||t3D − ˆt3D||2 2, (8) lp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='7668 Hp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='846 RFoot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='8664 HP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='L3D = 1 NS · NJ NS � i Nj � j (Xij − ˆXij)2 (9) where ˆX stands for the predicted 3D pose from our 3D lift- ing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Since CameraPose can work on 2D-3D pose pairs as well as 2D alone pose estimations, the loss design can be differ- ent according to the availability of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In the case of all annotations are available during the training stage, the camera loss can be calculated as: Lϕ = λcamLcam + λ2D,ϕL2D,ϕ + λ3DL3D (10) In the case of 2D annotation alone training step, the loss calculation will be from 2D reprojection error: Lϕ′ = λ2D,ϕ′L2D,ϕ′ (11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Pose Generator and Discriminator Similar to the framework in [11], we utilized both gen- erator and discriminator to further improve the diversity in training poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As shown in Figure 5, the generator is plugged in to the 2D pose generation stage, and the dis- criminator is applied on both the 2D and 3D pose inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The generator is actually formed by 3 simple multi-layer perceptions that generated different parameters for 3 differ- ent augmentation operations respectively: (1) changing the bone angle Xba, (2) changing the bone length Xbl and (3) changing the camera view and position of the input 3D pose R · Xbl + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The discriminator part of the framework can be divided into 2 portions, the D2D and D3D as we want to make sure that both the augmented X∗ and x∗ formed plausible human poses in both image coordinate and camera coor- dinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' But in our work we not only want to ensure the goodness of the augmented poses from the generator, we also want to utilized the discriminator to regulated our re- projected 2D poses for those 2D annotations only dataset cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The discriminators also adapt the part-aware Kine- matic Chain Space (KCS) proposed in [11], they are fully connected networks with a structure similar to the pose re- gression network using the KCS representation [36] of 2D or 3D poses as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Here we use the LS-GAN loss: L2d dis = 1 2Ex[(D2D(KCS(x)) − 1)2] (12) + 1 2Ex[(D2D(KCS({x∗, x′ 2D})) − 1)2] (13) L3d dis = 1 2Ex[(D3D(KCS(X)) − 1)2] (14) + 1 2Ex[(D3D(KCS(X∗)) − 1)2] (15) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Visualization of the pose generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As we augmented from the original 2D-3D annotated dataset ϕ = (x, X) using the 3D pose as the input to the generator which give 3 different sets of parameters γba, γbl and (R, t) to sequentially modified the 3D pose into our augmented dataset ϕ∗ = (x∗, X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' as the pose discrimination loss to train the generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Overall Loss The overall framework is made differentiable and can be trained in the end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We update different mod- ules alternatively by minimizing loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 10, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 11 as well as generators and discriminators with some pre- assigned hyper-parameters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Then we interactively train the entire model and update the weights of 3D lifting network using the losses: Lϕ = λref,ϕLref + λcamLcam + λ2D,ϕL2D,ϕ + λ3DL3D (16) and Lϕ′ = λref,ϕ′Lref + λ2D,ϕ′L2D,ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' (17) depending on the different datasets ϕ or ϕ′ we are using for the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We will introduce more training details and hyper-parameter settings in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Datasets For the 2D-3D paired annotations, we utilize the most popular datasets 3D HPE dataset Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M [15], 3DHP [24] and 3DPW [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Both Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M and 3DHP were collected indoor in some laboratory environment through 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 Pose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 Generator 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 Tba 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 (R,t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 00g001 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Perspective 3DPose Reprojection Discriminator T 2DPose DiscriminatorTable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Different human pose estimation datasets used in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The datasets in bold font are used for the training while other dataset in italic are used for cross-dataset evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The rest of the datasets will be used to visualize and serve as qualitative analysis targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Dataset # of Sample 2D Annotations 3D Annotations Camera Parameters Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M [15] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M v v v MPI-INF-3DHP [24] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3M v v v 3DPW [34] 51k v v Ski-Pose PTZ [29] 20k v v v MPII [1] 25k v MS-COCO [21] 250k v Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Qualitative comparison for the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M [15] (left), 3DHP [24] (right) and 3DPW [34] (bottom) generalization ability analysis using the pretrained baseline [11] and our purposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Both the baseline and our model were trained only with Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M so 3DHP and 3DPW are considered as cross-dataset in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The green arrows highlight locations where the models predict differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' the MoCap (motion capture) system [26] with multiple cal- ibrated cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The 3DPW is a more challenging dataset collected in outdoor environment using IMU (inertial mea- surement unit) sensors with mobile phone lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For 2D annotations only datasets, we used MPII [1] which contains a variety of in-the-wild everyday human ac- tivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Another popular 2D dataset MS-COCO [21] is also used for qualitative analysis purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Although the 2D annotation dataset such as MPII is much less than Hu- man3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M or 3DHP in terms of sample size, these 2D anno- tation datasets contain more challenging human poses with different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Note that both Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M and 3DHP are video based datasets, so that the total number of images is much larger than MPII and MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We summarized the datasets utilized in our experiments in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Preprocessing Different datasets have distinctive annotations on joints, which make the model training difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In this paper, we used the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M format as standard one, and inter- preted missing joints by labeling nearby joints for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' All the joints that are not included in Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M format will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Many existing 3D HPE algorithms use the groundtruth as model input for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' However, groundtruth is not available in real use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' To evaluate the model perfor- mance on the real-world applications, we also used existing 2D detector HRNet to extract the 2D keypoints as model input and rerun results on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Due to the various labeling schemes or joint formats dif- ference, we preprocess other schemes into the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M format by simple interpolation of some related joints and re- moval of the unused joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For example, there is no pelvis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Image Image GT Baseline GT Baseline CameraPose CameraPoseTable 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Result on Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M, 3DHP and 3DPW using the 2D ground-truth keypoints as the input in terms of MPJPE, note that we use the same model for evaluation on all datasets to mimic cross-dataset evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Best results are shown in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Method Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (MPJPE) 3DHP (MPJPE) 3DPW (PA-MPJPE) Wnadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [37] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0 Rhodin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [29] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='1 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='8 Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [38] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='4 Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [23] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3 Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [2] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='8 Pavllo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [28] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='80 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='64 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='38 Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [11] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='02 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='13 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='27 Ours (CameraPose) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='87 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='85 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='26 we simply create such joint by computing the mid-point of the left and right hip of any given label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Even though such interpolations are not always perfect due to the nature of each dataset, this preprocessing procedure allows us to have a better idea and comparison on cross-dataset scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Training CameraPose network is trained on 2 datasets: Hu- man3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (2D + 3D) and MPII (2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For the former, we followed most 3D human pose estimation training protocols using the subjects S1, S5, S6, S7, S8 from Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M as our 2D-3D training data, and subjects S9, S11 for evalua- tion purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For the latter, we filtered and selected around 10k training samples by checking the joints annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For evaluation, MPI-INF-3DHP and 3DPW were used to get quantitative results in terms of MPJPE (mean-per-joint- position-error) and PA-MPJPE (aligned with ground-truths by rigid transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The model training can be divided into 3 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The refinement network was trained as the first step for 100 epochs with learning rate being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0001 and weight decay at epochs 30, 60 and 90, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Next step, the 3D lifting network along with the pose generator and discrimi- nator was trained using Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M dataset for 10 epochs with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' This step is for warm-up and GAN tuning which can make the following model training more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Finally, the model was trained in an end-to- end fashion using both 2D-3D pairs annotations as well as 2D alone annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In each iteration, we first updated the weights of generator and discriminator to make the genera- tors more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Then the 3D lifting network was updated based on the augmented poses plus the 2D-3D annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' After that, 2D only annotations were utilized to tune the camera parameter branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The model was trained for 75 epochs with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0005 and weight de- cay at 30, 60, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' And the weighting for loss we choose λcam = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='01, λ2D,ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5, λ2D,ϕ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='2, and λ3D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Visualization for qualitative analysis of 3D human pose estimation on MPII [1] (Testing), MS-COCO [21], and SKiPose- PTZ [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Our model can still generate reliable 3D poses even when the target poses are in general rare or never seen from the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Quantitative Results CameraPose Network Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We compared Camera- Pose with other state-of-the-art methods [2, 28, 38, 11, 23] trained on Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For the temporal-based methods [2, 28], we implemented the single frame version for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Table 4 summarized the experimental re- sults of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' For each column, the MPJPE or PA-MPJPE are calculated for evaluation, obtained from the same model trained and selected based on the evaluation dataset of Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Some existing algorithms selected distinctive best models on different testing datasets, which may not reflect the generalization of models well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Instead, we selected a single model based on the accuracy of the validation of Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M to make it more realistic for real- world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As shown in Table 4, our method outperforms the SOTA on the most challenging dataset 3DPW by a noticeable mar- gin (3mm and 13mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' It also has significantly higher accu- TTable 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Experimental results of the effect of refinement network as we examine the effectiveness of our refinement module using the HRNet detections on training and evaluation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Method Training Source (2D Estimator) Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (MPJPE) 3DHP (MPJPE) Pavllo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [28] Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (HRNet) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='90 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='86 Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [11] Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (HRNet) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='18 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='50 Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' [11] w/ Refinement Network Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (HRNet) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='32 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='45 CameraPose w/ Refinement Network Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (HRNet) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='20 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='35 CameraPose w/o Refinement Network Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M (HRNet) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='38 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='12 racy than other weakly-supervised methods like [29] and [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Our model also achieves the highest accuracy on the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Experimental results clearly show the strong generalization capability of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Adding the camera parameter branch can help the model to learn from in-the-wild datasets with 2D annotations, which is very effective for hard examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The results on the 3DHP are slightly lower than SOTA methods, and we claim it is due to the fact that the 2D an- notations we added from MPII are more helpful for chal- lenging cases such as the 3DPW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The best accuracy on the 3DHP dataset can be 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='54 MPJPE using our model, which outperforms the current SOTA if we select a specific model for the 3DHP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Refinement Network Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' To show the effectiveness of the refinement network, we trained different models with different settings as shown in the Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We used HRNet as 2D detectors to extract the 2D key- points on all the training and evaluation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We added the refinement network to both the SOTA method [11] and our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' By adding the refinement network, both PoseAug and our model have improved accuracy on both Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M and 3DHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In addition, our model outper- forms the SOTA on both testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Therefore, both our proposed camera parameter network and the refinement network are useful for 3D HPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Qualitative Visualization 3D Pose Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We choose 3 datasets (Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content='6M, 3DHP and 3DPW) to qualitative compare our proposed method and baseline [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' As shown in Figure 6, our model has more accurate predictions on challenging datasets such as 3DPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Note that we utilize cross-scenario training to make sure there is no overlap between training and test- ing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We also visualize our results on datasets with- out 3D annotations such as MPII, MSCOCO, and SkiPose- PTZ [29] in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The visualization results are very plausible, which indicates the capability of our model for in-the-wild prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 2D Reprojection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' To validate the camera parameter branch, we visualize the results of our model at a different stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Figure 8 shows the original image, input 2D keypoints from HRNet, inferred 3D poses, and reprojected 2D poses from Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 3D-2D reprojection visualization on MPII [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Column from the left: original images, 2D keypoints from HRNet, inferred 3D keypoints, reprojected 2D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' The camera parameters predicted by CameraPose can successfully reprojected the 3D pose back into the image coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' left to right columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' It clearly shows that our CameraPose can predict well on unseen poses and the reprojected 2D poses are meaningful too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Conclusions We propose CameraPose, a weakly-supervised frame- work for 3D human pose estimation from a single image that can aggregate 2D annotations by designing a camera parameter branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Given any noisy 2D keypoints from pre- trained 2D pose estimator, CameraPose is able to refine the keypoints with a confidence-guided loss and feed them into the 3D lifting network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Since our approach uses the camera parameters learned from the camera branch to do the repro- jection back to 2D, it can solve the problem of the lacking of the 2D-3D datasets with rare poses or outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' We evaluate our proposed method on some benchmark datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' the results show that our model can achieve higher accuracy on challenging datasets and be able to predict meaningful 3D poses given in-the-wild images or 2D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' Original Image Input2DKeypoints 3DPrediction Reprojected 2D (MPII Test) (HRNet) (CameraPose) (CameraPose)References [1] Mykhaylo Andriluka, Leonid Pishchulin, Peter Gehler, and Bernt Schiele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' 2d human pose estimation: New benchmark and state of the art analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfLgP4/content/2301.02979v1.pdf'} +page_content=' In IEEE Conference on Com- puter Vision and Pattern 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Jan 2023 +1 +STARS-ISAC: How Many Sensors Do We +Need? +Zheng Zhang, Student Member, IEEE, Yuanwei Liu, Senior Member, IEEE, +Zhaolin Wang, Graduate Student Member, IEEE, Jian Chen, Member, IEEE, +Abstract +A simultaneously transmitting and reflecting surface (STARS) enabled integrated sensing and com- +munications (ISAC) framework is proposed, where a novel bi-directional sensing-STARS architecture +is devised to facilitate the full-space communication and sensing. Based on the proposed framework, +a joint optimization problem is formulated, where the Cram´er-Rao bound (CRB) for estimating the 2- +dimension direction-of-arrival of the sensing target is minimized. Two cases are considered for sensing +performance enhancement. 1) For the two-user case, an alternating optimization algorithm is proposed. +In particular, the maximum number of deployable sensors is obtained in the closed-form expressions. +2) For the multi-user case, an extended CRB (ECRB) metric is proposed to characterize the impact +of the number of sensors on the sensing performance. Based on the proposed metric, a novel penalty- +based double-loop (PDL) algorithm is proposed to solve the ECRB minimization problem. To tackle the +coupling of the ECRB, a general decoupling approach is proposed to convert it to a tractable weighted +linear summation form. Simulation results reveal that 1) the proposed PDL algorithm can achieve a near- +optimal performance with consideration of sensor deployment; 2) without violating the communication +under the quality of service requirements, reducing the receive antennas at the BS does not deteriorate +the sensing performance; and 3) it is preferable to deploy more passive elements than sensors in terms +of achieving optimal sensing performance. +Index Terms +Beamforming design, integrated sensing and communications (ISAC), simultaneously transmitting +and reflecting surface (STARS), sensor deployment. +Zheng Zhang and Jian Chen are with the School of Telecommunications Engineering, Xidian University, Xi’an 710071, +China (e-mail: zzhang 688@stu.xidian.edu.cn; jianchen@mail.xidian.edu.cn). Yuanwei Liu and Zhaolin Wang are with the +School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail: +yuanwei.liu@qmul.ac.uk; zhaolin.wang@qmul.ac.uk;). + +2 +I. INTRODUCTION +With the commercialization of the fifth generation (5G) wireless networks, the 2030-oriented +sixth generation (6G) wireless communication systems drew growing attention in both academia +[1] and industry [2], [3]. 6G seeks a fundamental paradigm shift in wireless network architecture, +which breaks the physical boundaries of sensing and communications to support more emerging +applications, such as extended reality (XR), auto-driving, and Metaverse. To realize this vision, +a key enabling technique, integrated sensing and communication (ISAC), has been proposed to +unify the two functions via the same hardware platform and signal processing module [4], [5]. +To elaborate, through the dedicated co-designed framework, ISAC is capable of significantly +enhancing the utilization efficiency of the network resources, thereby resulting in low imple- +mentation overheads. Furthermore, through deep integration, ISAC is also envisioned to realize +mutual assistance and win-win benefit between the two functions [6]. +To provide ubiquitous wireless connection with low energy consumption, reconfigurable in- +telligent surface (RIS) has emerged as another promising and cost-effective technique for future +wireless networks [7], [8]. Technically, RIS can be regarded as a metasurface-based planar array, +which is composed of lots of passive tunable elements. The electromagnetic response at each +element can be proactively adjusted via an external smart controller, which aims to reconfigure +the amplitude and phase shifts of the incident signal and thus realize a smart radio environment. +However, since the conventional RIS can merely reflect the incident signals and provide half- +space coverage, the design flexibility is stringently limited by its geographical location and panel +orientation. As a remedy, a new RIS paradigm, namely simultaneously transmitting and reflecting +surface (STARS), has been proposed [9], [10]. Compared to the conventional reflecting-only RIS, +STARS can provide full-space electromagnetic environment reconfiguration [11]. +A. Prior Works +There have been lots of efforts devoted to the ISAC networks [12]–[16]. More specifically, +the authors of [12] devised a pair of antenna setups for multi-antenna ISAC systems, where +a high-accuracy beampattern strategy is proposed to improve the sensing performance while +guaranteeing the communication requirements. As a further advance, the authors of [13] proposed +the optimal sensing waveform strategies, where the performance tradeoff between communication +and sensing was investigated. To introduce more degrees-of-freedom (DoFs) for target sensing, +a sophisticated ISAC framework was proposed in [14], where the independent radar waveforms + +3 +and communication symbols were exploited to form the multiple beams for high-quality sensing. +However, the aforementioned works only focused on the waveform design at the transmitter +while neglecting the sensing performance imposed by the received echo at the receiver. To +provide a comprehensive evaluation of the sensing performance, the authors of [15] introduced +the Cram´er-Rao bound (CRB) as the sensing performance metric of an unbiased estimation +at the receiver. Furthermore, the authors of [16] developed a fairness-oriented unified resource +allocation framework, where the BS was employed to carry out the device-free sensing services +while satisfying communication QoS demands. +More recently, it is claimed that the proper exploitation of RISs in ISAC systems can further +boost the sensing performance [17]–[21]. In [17], the authors proposed a RIS-assisted ISAC +framework, in which a RIS is employed to establish reliable line-of-sight (LoS) links for distance +and velocity estimation. To support scenarios with multiple point-like targets, the authors of [18] +developed a majorization-minimization algorithm for target tracking by collaboratively designing +the transmit beampattern and RIS coefficients. In [19], the authors conceived a joint optimization +scheme regarding the sensing waveform and the RIS coefficients from the perspective of sensing +mutual information. Considering the practical restriction of discrete phase shifts, a constant- +modulus sensing waveform was designed in [20], in which the multi-user interference was +minimized under the sensing CRB constraint. In [21], the authors considered utilizing the RIS to +provide sensing services to the blind-zone target, where the CRB minimization problems were +investigated in the cases of point target and extended target, respectively. +However, the direct combination of RIS and ISAC in the aforementioned works inevitably +increased the number of reflections experienced by the echo signals, which restricted the sensing +performance. To address this issue, the authors of [22] pioneered a RIS-self-sensing architecture, +where the dedicated sensors are deployed at the RIS to carrying out the sensing functionality. +Shortly thereafter, the authors of [23] proposed a two-phase semi-passive RIS-assisted ISAC +scheme, where the RIS supported the uplink communications in the first phase while carrying +out the multi-user location sensing in the second one. Exploiting the same semi-passive sensing- +at-RIS architecture, the authors of [24] studied the effect of sensing functionality on the commu- +nications, where the RIS was employed to sense the user location to facilitate the communication +beamforming design. Most recently, there was a preliminary exploration of STARS-enabled ISAC +networks in [25], where a sensing-at-STARS structure was proposed to achieve the 2-dimension +direction-of-arrivals (DOAs) estimation. + +4 +B. Motivations and Contributions +Based on the aforementioned RIS-enabled ISAC works [17]–[25], we can obtain following +two observations. +• Although there have been a few works focusing on the RIS/STARS-enabled ISAC systems, +the communication users and/or targets are only considered to be located on one side of +the RIS/STARS1. Whereas in the practical networks, the users and targets are probably in +different geographical positions at different times, even on both sides of the RIS/STARS. +Apparently, such a problem cannot be coped with by the existing schemes, which thus calls +for a more general strategy for the RIS/STARS-enabled ISAC systems. +• For the sensor-at-RIS/STARS architecture [22]–[25], a critical endogenous problem has +not been answered yet, i.e., should we deploy more sensors or passive elements (PEs) at +the RIS/STARS? Given the fixed number of total elements of the RIS/STARS, deploying +more sensors can increase the echo sampling resolution. Whereas deploying more PEs can +introduce more spatial DoFs to favor both communication and sensing performance. Hence, +there may exist a tradeoff between the number of sensors and PEs, which requires further +investigation. +Motivated by the above observations, we propose a STARS-enabled ISAC framework, where +the communication users and the targets are located on both sides of the STARS, with a particular +focus on the tradeoff between the number of sensors and PEs. Our main contributions are +summarized below. +• We propose a novel bi-directional sensing-STARS architecture, where the micro-sized sen- +sors with encapsulated antennas are integrated into the transparent substrate of STARS to +provide full-space communication and sensing service. To tackle the energy/signal leakage +issue in uplink STARS transmissions, a time switching (TS) protocol based two-phase +scheme is proposed, where the STARS periodically switches between the reflection and +transmission modes to support different users/targets. With this transmission framework, the +closed-form CRB expression is derived as the sensing performance metric for estimating +the 2-dimension DOAs. +1In the work [25], STARS is employed to divide the whole space into the communication region and sensing region, where +the communication users and target are only situated on the corresponding half-space region. + +5 +• We first consider a two-user network. A CRB minimization problem is formulated subject +to the communication quality of service (QoS) constraint of the ergodic achievable rate. +To facilitate the optimization, the approximated ergodic rate is derived in the closed-form +expression. Then, we propose an alternating optimization (AO) algorithm, where the optimal +sensing waveform, transmit power, and reflection/transmission coefficients are alternately +obtained by utilizing the standard semidefinite program (SDP) method. To provide further +insights, the maximum number of sensors that can be deployed is derived in the closed-form +expression, which unveils that the maximum number of sensors is only relevant to the QoS +requirements of communications. +• We further consider a multi-user network, where a new metric of extended CRB (ECRB) +is proposed to transform the impact of the number of sensors on the sensing accuracy +into an explicit form. We aim to minimize the proposed ECRB under the communication +QoS requirements. To efficiently solve the formulated mixed integer non-linear program +(MINLP), a generic decomposing method is devised to transform the non-convex objective +function of the ECRB into a weighted linear summation form of the constant ECRB matrices. +Based on the transformation, a penalty-based double-loop (PDL) algorithm is proposed to +solve the resultant non-convex optimization problem. +• Numerical results demonstrate the effectiveness and convergence of the proposed algorithms. +It is also verified that the proposed PDL can enable a near-optimal allocation of the number +of sensors. Besides, two insights are observed: 1) with the proposed bi-directional sensing- +STARS architecture, reducing the number of receive antennas at the BS does not deteriorate +the sensing performance provided that QoS requirements are satisfied; and 2) given a fixed +total number of STARS elements, deploying more PEs at the STARS is more attractive than +sensors in terms of sensing performance enhancement. +The organization of this paper is as follows. In Section II, we present the system model and +performance metrics. In Section III, we conceive an AO algorithm to minimize CRB under the +given number of sensors. Section IV develops a PDL algorithm for the joint optimization of +beamforming and the number of sensors. The numerical results are illustrated in Section V. +Finally, the conclusion is presented in Section VI +Notations: we use the boldface capital X and lower-case letter x to represent matrix and +vector, respectively. For any N × M-dimensional matrix X ∈ CN×M, XT and XH denote the +transpose and Hermitian conjugate operations. Similarly, rank(X), Tr(X), ∥X∥, ∥X∥F represent + +6 +BS +T1 +UK +U1 +T2 +Sensing element +Passive element +Sensing element +Passive element +Communication signal: +Probing signal or sensing echo: +Radar sensor with +encapsulated antennas +Tunable element +1 +UK +... +1 1 +UK + +Fig. 1. The bi-directional sensing-STARS enabled uplink ISAC network. +the rank value, trace value, spectral norm operation and Frobenius norm operation. X ⪰ 0 +denotes that X is a positive semidefinite matrix, while x ∼ CN (0, X) denotes that x is a +circularly symmetric complex Gaussian (CSCG) vector with zero mean and covariance matrix +X. For a matrix X, vec(X) and X−1 denote the vectorizing and inverse matrix operation. For +any vector x, diag(x) denotes a diagonal matrix whose main diagonal elements equal to the +elements of x. |x| and ∥x∥ denote the modulus of x and the Euclidean norm of the vector x, +respectively. E(·) is the statistical expectation operation. IN is a N-dimensional identity matrix, +and 0N is a N-dimensional zero matrix. ℜ(·) and ℑ(·) denote the real component and imaginary +component of the complex value. For any complex scalar z, ˜z denotes the conjugate of z. For +any real scalar x,⌊x⌋ and ⌈x⌉ denote the round-down and round-up operations. ⊙ denotes the +Hadamard product. +II. SYSTEM MODEL +A. Network Description +As shown in Fig. 1, we consider a STARS-enabled uplink ISAC network, where a STARS is +deployed to establish reliable LoS links for K blind-zone users {U1, · · · , UK} to communicate +with a BS while relaying the probing signal intended for the targets {T1, T2}. The whole space +is divided by the STARS into two separate region, each of which contains a target requiring +estimation. It is assumed that the direct links of BS-users, users-targets, and BS-targets channels +are blocked due to the obstacles. To mitigate the full-duplex self-interference at the BS, the +BS is assumed to be equipped with an Mt-antenna transmit uniform linear array (ULA) and +an Mr-antenna receive ULA [15], and all the users are single-antenna nodes. The STARS is + +7 +composed of a uniform planar array (UPA) with N = NvNh sub-wavelength elements, where +Nv and Nh denote the number of elements located vertically and horizontally in the x-o-z plane, +respectively. +To support the full-space communication and sensing, we propose a bi-directional sensing- +STARS architecture, which divides the N STARS elements into two parts. The former N1 PEs +are employed to support the uplink communication, and the latter N2 = N − N1 elements +(referred to as sensing elements) are equipped with the sensors for targets estimation. More +specifically, the PEs operate in the reflection or transmission mode for information transmission +[10]. For each sensing element, a micro-sized low-cost sensor with antennas in packages is +integrated inside the transparent substrate of STARS, where the adjacent tunable element op- +erates in the full transmission mode, with the unit amplitude coefficient and zero phase-shift +manipulation for the dual-sided incident signals [26]. Thus, the sensor is cable of receiving +full-space echo waves without suffering penetration attenuation caused by STARS element. All +the inter-antenna/element distances are assumed to be sub-wavelength, so the adjacent channel +reflected/transmitted by the STARS element can be deemed to be independent channels [27]. +The considered network is assumed to be a narrow-band system, where the BS and users +transmit probing signal and communication signal in one coherence block of T consecutive +sample frames [15]. All the channels are assumed to be static at one coherence block, but vary +over different coherence blocks [23]. The channel coefficients from PEs to the BS and Uk are +respectively denoted as Gr ∈ CN1×Mr, Gt ∈ CN1×Mt and hk,S ∈ CN1×1. All the channels are +assumed to follow Rician fading model since STARS can be flexibly deployed to favor the LoS +links. Hence, the communication channels Hc ∈ {Gr, Gt, hk,S} are modeled as +Hc = Lc +�� +κ +1 + κ +ˆHc + +� +1 +1 + κ +˜Hc +� +, +(1) +where κ denotes the Rician factor, ˆHc represents the LoS component, and ˜Hc denotes the non- +LoS component. Lc = +� +L0d−αc +c +∈{Lr, Lt, Lk,S} denotes the corresponding path loss, where dc is +the communication distance, αc represents the path-loss exponent, and L0 denotes the path loss +at the reference distance of 1 meter (m). For the sensing process, the probing signal and echo +wave undergoes the PEs→target→sensors channels, which is modeled as +H[ı],s = α[ı]b(ϕ[ı], φ[ı])aT(ϕ[ı], φ[ı]), +1 ≤ ı ≤ 2, +(2) + +8 +where α[ı] ∈ C is the reflection coefficient containing the radar cross section (RCS) of Tı and the +round-trip path-loss, ϕ[ı] denotes the azimuth angle of arrival/departure from the STARS to the +target Tı, and φ[ı] denote the elevation angle of arrival/departure from the STARS to the target +Tı. Note that a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) denote the steering vectors of PEs and sensors, where +the n-th elements of a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) are given by [23] +a(ϕ[ı], φ[ı])[n] = e[¯nπ cos φ[ı] sin ϕ[ı]+(n−1−Nh¯n)π sin φ[ı]], +1 ≤ n ≤ N1, +(3) +b(ϕ[ı], φ[ı])[n] = e[¯nπ cos φ[ı] sin ϕ[ı]+(n−1−Nh¯n)π sin φ[ı]], +N1 + 1 ≤ n ≤ N, +(4) +where ¯n = ⌊n−1 +Nh ⌋. To investigate the fundamental performance limit of the considered network, +the perfect channel information state (CSI) is assumed for all the channels. +B. Signal Model +Without loss of generality, we focus on the transmission at the t-th time frame and propose +a TS-based framework, which equally divides the each time frame into following two phases. +1) Phase I: The PEs operates in the reflection mode and users K1 ≜ {U1, · · · , UK1} transmit +the communication signal c[1](t) = � +k∈K1 +√Pkck(t) with E{|ck(t)|2} = 1 to the PEs. Meanwhile, +the BS exploits the multiple beams to send the dedicated probing signal s[1](t) = �I[1] +j=1 ˇsj(t) ∈ +CM×1 with a general-rank covariance matrix R[1],s = E{s[1](t)sH +[1](t)}. On receiving the omnidi- +rectional signal, the PEs reflects the communication signal to the BS while combining the c[1](t) +and s[1](t) as a new probing signal to perform estimation. +2) Phase II: The PEs operates in the transmission mode. The users K2 ≜ {UK1+1, · · · , UK} +send the communication signal c[2](t) = � +k∈K2 +√Pkck(t) to the PEs. Meanwhile, the BS +transmits probing signal s[2](t) = �I[2] +j=1 ˇsj(t) ∈ CM×1 (R[2],s = E{s[2](t)sH +[2](t)}) to the PEs. +At the STARS, the PEs is exploited to transmit the c[2](t) to the BS while reconfiguring the +probing signal to detect T2. +Accordingly, the received signal at the BS in the ı-th phase is given by +y[ı](t) = hH +k,SΘr/tGr +� +k∈Kı +� +Pkck(t) + n[ı](t), +(5) +where n[ı](t) ∼ CN(0, σ2IMr) denotes the ı-th phase additive white Gaussian noise (AWGN) +at the BS, and Θr/t = Θr in case of ı = 1 while denoting Θt in case of ı = 2. There- +into, the reflection/transmission coefficient matrix is defined as Θr/t = diag(ur/t) with ur/t = + +9 +[�βr/t,1eθr/t,1, . . . , �βr/t,Neθr/t,N]T. To recover ck(t), we assume that a unit-norm linear com- +bination vector v[ı],k ∈ CMr×1 is employed at the BS. The extracted signal for Uk is given +by +y[ı](t)v[ı],k = hH +k,SΘr/tGrv[ı],k +� +Pkck(t) +� +�� +� +desired signal ++ hH +j,SΘr/tGrv[ı],k +� +j̸=k,j∈Kı +� +Pjcj(t) +� +�� +� +interference ++n[ı](t)v[ı],k. +(6) +While for the sensing, the equivalent probing signal from PEs at the t-th time frame of the ı-th +phase is given by +x[ı](t) = + + + + + +HU,S +√¯Pc(t) + Gts[1](t), +if ı = 1, +Gts[2](t), +if ı = 2, +(7) +where HU,S = [h1,S, · · · , hK1,S], ¯P = diag[P1, · · · , PK1], and c(t) = [c1, · · · , cK1]T. Accordingly, +the covariance matrix of x[ı](t) is given by +R[ı],x = E[x[ı](t)x[ı](t)H] = + + + + + +HU,S¯PHH +U,S + GtR[1],sGH +t , +if ı = 1, +GtR[2],sGH +t , +if ı = 2, +(8) +Thus, the received echo wave at the sensors over T consecutive time frames of the ı-th phase +is given by +Y[ı],s = H[ı],sΘr/tX[ı] + N[ı], +(9) +where X[ı] = [x[ı](1), · · · , x[ı](T)] and N[ı] = [n[ı](1), · · · , n[ı](T)]. +C. Performance Metric +As stated above, the achievable rate at the BS in the ı-th phase to decode ck(t) is expressed +as +R[ı],k = 1 +2 log2 + + +1 + +Pk|hH +k,SΘr/tGrv[ı],k|2 +� +j̸=k,j∈Kı +Pj|hH +j,SΘr/tGrv[ı],k|2 + σ2 + + + , +1 ≤ ı ≤ 2, +(10) +For the sensing, we focus on the CRB performance with unknown parameters ς[ı] = [ϕ[ı], φ[ı], +ℜ(α[ı]), ℑ(α[ı])] ∈ R4×1. To facilitate deriving CRB expression, we vectorize equation (9), which +can be rewritten as +y[ı],s = vec(Y[ı],s) = vec +� +H[ı],sΘr/tX[ı] +� ++ n[ı], +(11) + +10 +where n[ı] = vec(N[ı]) ∼ CN (0, σ2IMT). Let q[ı] = vec +� +H[ı],sΘr/tX[ı] +� +, the Fisher information +matrix (FIM) F[ı] ∈ R4×4 for estimating ς[ı] can be expressed as a Jacobian matrix with the h-th +row and v-th column element being given by [28] +F[ı][h, v] = 2ℜ +� +∂qH +[ı] +∂ς[ı],h +R−1 +n[ı] +∂q[ı] +∂ς[ı],v +� ++ Tr +� +R−1 +n[ı] +∂Rn[ı] +∂ς[ı],h +R−1 +n[ı] +∂Rn[ı] +∂ς[ı],v +� += 2 +σ2ℜ +� +∂qH +[ı] +∂ς[ı],h +∂q[ı] +∂ς[ı],v +� +, +1 ≤ h, v ≤ 4, +(12) +where Rn[ı] = σ2IMT. Accordingly, we can repartition F[ı] as +F[ı] = + +JΨ[ı]Ψ[ı] +JΨ[ı]α[ı] +JT +Ψ[ı]α[ı] +Jα[ı]α[ı] + + , +(13) +where Ψ[ı] = [ϕ[ı], φ[ı]], α[ı] = [ℜ(αı), ℑ(αı)], while the specific expressions of JΨ[ı]Ψ[ı], JΨ[ı]α[ı] +and Jα[ı]α[ı] are given in Appendix A. Then, with the inverse formula of the second order matrix, +we can derive the CRB expression of the ı-th phase with regard to Ψ[ı] [29] as +CRB(Ψ[ı]) = +� +JΨ[ı]Ψ[ı] − JΨ[ı]α[ı]J−1 +α[ı]α[ı]JT +Ψ[ı]α[ı] +�−1 +. +(14) +III. BEAMFORMING OPTIMIZATION UNDER FIXED SENSOR NUMBER +In this section, we concentrate on joint sensing waveform and communication beamforming +optimization with the fixed number of sensors. To draw instructive insights for practical system +design, we consider a special network setup of two users, and utilize the ergodic rate as the +average communication performance metric. The the closed-form approximation expression of +ergodic achievable rate is derived. Accordingly, an AO algorithm to efficiently solve the non- +convex problem. +A. Problem Formulation +Since inter-user interference is non-existent in the two-user case, the achievable rate at the BS +in the ı-th phase to decode ck(t) is rewritten to +R[ı],k = 1 +2 log2 +� +1 + Pk|hH +k,SΘr/tGrv[ı],k|2 +σ2 +� +. +(15) +To provide the generalised insights to the CRB optimization, we employ the ergodic achievable +rate as the average performance metric for the communication. Accordingly, we target at minimiz- +ing the CRB performance for DOA Ψ[ı] at each phase. Subject to the average QoS requirements of + +11 +users, the joint optimization of the transmit power at the users, reflection/transmission coefficients +of the PEs, the receive beamforming and the sensing waveform at the BS is considered. The +optimization problem in the ı-th phase is formulated as follows. +min +P[ı],Θr/t,R[ı],s,v[ı],k +Tr(CRB(Ψ[ı])) +(16a) +s.t. +Tr(R[ı],s) ≤ PBS,max, +(16b) +Pk ≤ PU,max, +1 ≤ k ≤ K, +(16c) +E{Rk} ≥ Rer,t, +(16d) +∥v[ı],k∥2 = 1, +1 ≤ k ≤ K, +(16e) +θr,n, θt,n ∈ [0, 2π], 1 ≤ n ≤ N, +(16f) +βr,n, βt,n ∈ [0, 1], 1 ≤ n ≤ N, +(16g) +where P[1] = [P1, · · · , PK1], P[2] = [PK1+1, · · · , PK], Rer,t denotes the ergodic QoS rate of +users, PU,max denotes the maximal transmit power at the users, and PBS,max denotes the transmit +power budget at the BS. (16b) and (16c) denote the transmit power constraint at the BS and +users’ sides, respectively; (16d) represents the ergodic QoS constraint of users; (16e) denotes +the normalization constraint of the receive beamforming; (16f) and (16g) are the phase-shift +and amplitude constraints of the PEs. Notably, the intractable expression of CRB and the non- +convex constraints (16d) and (16e) make problem (16) non-convex and challenging to solve. In +the following, we consider optimizing the sensing waveform and communication beamforming +in an alternating manner. +B. Problem Reformulation +Before handling the challenging problem, we can observe that v[ı],k only exists in the constraint +(16d) and has no direct influence on the CRB performance, which indicates that only the +feasible v[ı],k are required. For maximum compliance with QoS constraint, the optimal receive +beamforming is given by v[ı],k = +(hH +k,SΘr/tGr)H +∥hH +k,SΘr/tGr∥ , to obtain the best communication performance. +Hence, (15) can be transformed into R[ı],k = +1 +2 log2 +� +1 + +Pk∥hH +k,SΘr/tGr∥2 +σ2 +� +. With this in mind, +we further rewrite ∥ˆhH +k,SΘr/t ˆGr∥2 as Tr( ˆHk,S ˆHH +k,SUr/t), with definition of Ur/t = ur/tuH +r/t and +ˆHk,S = diag(ˆhH +k,S) ˆGr. Then, we resort Lemma 1 to obtain the approximation expression of +E{Rk}. + +12 +Lemma 1: The ergodic achievable rate in (16d) can be approximated as +E{R[ı],k} ≈ 1 +2 log2 +� +1+ PkL2 +k,SL2 +r +σ2(1+κ)2 +� +κ2Tr( ˆHk,S ˆHH +k,SUr/t)+κTr( ˆGr ˆGH +r Ur/t)+ +κMrTr +� +diag(ˆhH +k,S)Ur/tdiag(ˆhk,S) +� ++ MrTr(Ur/t) +�� +. +(17) +Proof: See Appendix B. +The approximation accuracy can be verified in Fig. 3. To proceed, we further introduce Lemma +2 to convexify the non-convex objective function (16a). +Lemma 2: According to [30], we have that for any X ⪰ 0 and Y ⪰ 0, if X ⪰ Y is guaranteed, +the inequality of Tr(X−1) ≤ Tr(Y−1) holds. +As such, with the fact that FIM matrix +� +JΨ[ı]Ψ[ı] − JΨ[ı]α[ı]J−1 +α[ı]α[ı]JT +Ψ[ı]α[ı] +� +is positive semidefi- +nite, we can introduce an auxiliary variable E[ı] ⪰ 0 to equivalently transform (16a) into Tr(E−1 +[ı] ), +with satisfying following linear matrix inequality (LMI) constraint. + +JΨ[ı]Ψ[ı] − E[ı] +JΨ[ı]α[ı] +JT +Ψ[ı]α[ı] +Jα[ı]α[ı] + + ⪰ 0. +(18) +With the transformations above, problem (16) can be reformulated as follows. +min +E[ı],Ur/t,R[ı],s,P[ı] +Tr(E−1 +[ı] ) +(19a) +s.t. +(16b), (16c), (18), +(19b) +PkL2 +k,SL2 +r +(1 + κ)2 +� +κ2Tr( ˆHk,S ˆHH +k,SUr/t)+κMrTr +� +diag(ˆhH +k,S)Ur/tdiag(ˆhk,S) +� ++ κTr( ˆGr ˆGH +r Ur/t) + MrTr(Ur/t) +� +≥ σ2(22Rer,t − 1), +(19c) +E[ı] ⪰ 0, +Ur/t ⪰ 0, +(19d) +Ur/t[n, n] ≤ 1, +1 ≤ n ≤ N, +(19e) +rank(Ur/t) = 1. +(19f) +C. Joint Beamforming Optimization +1) Transmit power and sensing waveform optimization: With the fixed Ur/t, the problem (19) +is reduced to the following subproblem. +min +R[ı],s,E[ı],P[ı] +Tr(E−1 +[ı] ) +(20a) + +13 +s.t. +(16b), (16c), (18), (19c), +(20b) +E[ı] ⪰ 0, +(20c) +which is a SDP and can be optimally solved by the convex toolbox, e.g., CVX. +2) Reflection/Transmission coefficients optimization: With fixed {R[ı],s, P[ı]}, problem (19) is +rewritten as +min +E[ı],Ur/t +Tr(E−1 +[ı] ) +(21a) +s.t. +(18), (19c) − (19f). +(21b) +To handle the non-convex LMI constraint (18), we adopt the singular value decomposition (SVD) +to equivalently convert the quadratic terms {JΨ[ı]Ψ[ı], JΨ[ı]α[ı], Jα[ı]α[ı]} to the tractable forms. +Specifically, by decomposing Rx[ı] into � +q sqdq, we have +Θr/tRx[ı]ΘH +r/t = +� +q +diag(sq)ur/tuH +r/tdiag(dq) += +� +q +Squr/tuH +r/tDq = +� +q +SqUr/tDq. +(22) +Then, we substitute (22) into FIM matrix, the constraint (18) becomes convex with respect to +Ur/t. While for the non-convex constraint (19f), we employ the penalty-based rank-one relaxation +approach [31], which exploits the successive convex approximation (SCA) technique to relax +the equivalent rank-one constraint Tr(Ur/t) − ∥Ur/t∥2 = 0 as a convex penalty term in objective +function (21a). Accordingly, the problem (21) is reformulated as +min +E[ı],Ur/t +Tr(E−1 +[ı] ) − 1 +2ρ1 +� +Tr(Ur/t) − ∥U[n−1] +r/t +∥2 − Tr(¯u[n−1] +r/t +(¯u[n−1] +r/t +)H(Ur/t − U[n−1] +r/t +)) +� +, (23a) +s.t. +(18), (19c) − (19e), +(23b) +where U[n−1] +r/t +denotes the optimized result in the (n − 1)-th iteration, ¯u[n−1] +r/t +denotes the leading +eigenvector of U[n−1] +r/t +, and ρ1 represents the penalty factor. The resultant problem (23) is a convex +program, where the rank-one solution can be optimally obtained when p is sufficiently small +[31, Proposition 2]. The specific SCA algorithm to optimize Ur/t is summarized in Algorithm +1. +Remark 1: (MLE Validation) In this paper, we consider employing the maximum likelihood +estimation (MLE) approach in [21, Appendix E] to obtain the estimated DoA Ψes +[ı] under the + +14 +Algorithm 1 SCA algorithm for rank-one solution. +1: Initialize initial U[n−1] +r/t +and p1 with n = 1. Set a convergence accuracy ǫ1 and calculate the +leading eigenvector ¯u[n−1] +r/t +. +2: repeat +3: +update U[n] +r/t by solving problem (23). +4: +update the leading eigenvector ¯u[n] +r/t. +5: +set n = n + 1 and ρ1 = ρ1 +c1 (c1 > 1). +6: until the penalty term in objective function (23a) drops below ǫ1. +Algorithm 2 AO algorithm. +1: Initialize initial U[l−1] +r/t +and Tr(CRB(Ψ[ı]))[l−1] with l = 1. Set a convergence accuracy ǫ2. +2: repeat +3: +update the optimal receive beamforming vopt +[ı],k = +(hH +k,SΘr/tGr)H +∥hH +k,SΘr/tGr∥ . +4: +update optimal {R[l] +[ı],s, P[l] +[ı]} by solving problem (20). +5: +update optimal U[l] +r/t by carrying out Algorithm 1. +6: +set l = l + 1 and calculate Tr(CRB(Ψ[ı]))[l]. +7: until |Tr(CRB(Ψ[ı]))[l] − Tr(CRB(Ψ[ı]))[l−1]| ≤ ǫ2. +optimized waveform, where the the correctness of the proposed CRB optimization framework +is demonstrated in Fig. 5. +Corollary 1: (Maximal Number of Sensor Deployment) For the special case of single receive +antenna, i.e., Mr = 1, the optimal reflection/transmission coefficients and the maximal number +of sensors to be deployed can be derived as following closed-form expressions. + + + + + +θopt +r,n = ∠ˆh1,S[n] − ∠ˆgr[n], +βopt +r,n = 1 +θopt +t,n = ∠ˆh2,S[n] − ∠ˆgr[n], +βopt +t,n = 1, +(24) +Nmax +2 += N − +� +1 +2κ2 +�� +(2κ + 1)2 + 4κ2σ2(22Rer,t − 1)(1 + κ)2 +PU,maxL2 +k,SL2 +r +− (2κ + 1) +�� +, +(25) +where ˆgr is the degenerated channel of ˆGr. +Proof: See Appendix C. +D. Overall Algorithm + +15 +The overall algorithm is summarized in Algorithm 2, which optimizes the sensing waveform +and reflection/transmission coefficients alternatively. By denoting the CRB value at l-th iteration +as a function of R[ı],s and Ur/t, i.e., g(R[ı],s, P[ı], Ur/t), the following inequality always holds +g(R[l−1] +[ı],s , P[l−1] +[ı] +, U[l−1] +r/t ) +(a) +≥ g(R[l] +[ı],s, P[l] +[ı], U[l−1] +r/t ) +(b) +≥ g(R[l] +[ı],s, P[l] +[ı], U[l] +r/t), +(26) +where inequality signs (a) and (b) hold because the optimal sensing waveform and optimal +reflection/transmission coefficients are both guaranteed in the step 4 and step 5 at the same +AO iteration. Meanwhile, since g(R[ı],s, P[ı], Ur/t) is continuous over the compact feasible set +of problem, there exists a finite positive number that serves as a lower bound on the objective +value. This proves that our proposed AO algorithm remains non-increasing over the iterations. +On the other hand, the computational complexity of AO algorithm mainly relies on solving SDP +problems (20) and (23). The overall complexity based on the interior-point method is given by +O +� +log( 1 +ǫ2) +� +(M2 +t + 5)3.5 + log( 1 +ǫ1)(N2 +1 + 4)3.5�� +[32]. +IV. HOW MANY SENSORS DO WE NEED? +In this section, we consider the general multi-user system with the joint optimization of +beamforming design and the number of sensors. By modifying the traditional CRB expression, +a new metric of ECRB is proposed, which can evaluate the sensing performance while taking +the sensors’ deployment into the consideration. Based on the proposed ECRB, a PDL algorithm +is devised to jointly optimize the ISAC waveform, reflection/transmission coefficients and the +number of PEs/sensors. +A. Extended CRB Derivation +To facilitate the optimization of sensor number, we define two N-dimensional matrices A = +diag[IN1, 0N2] and B = diag[0N1, IN2], where A and B are the element selection matrices for +PEs and sensors, respectively. With the definition above, we can rewrite the steering vector +a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) as the extended form. +a(ϕ[ı], φ[ı]) = Aε(ϕ[ı], φ[ı]) +b(ϕ[ı], φ[ı]) = Bε(ϕ[ı], φ[ı]). +(27) +Here, ε(ϕ[ı], φ[ı]) ∈ CN×1 denotes the steering vector of the STARS, n-th element of which is +defined in (3) with 1 ≤ n ≤ N. For convenience of denotation, we abbreviate ε(ϕ[ı], φ[ı]) as ε[ı] +in the following. Then, we introduce Proposition 1 to derive the expression of the extended FIM +matrix. + +16 +Proposition 1: The h-th row and v-th column element of extended FIM matrix is given by +F[ı][h, v] = +2TC(h,v) +[ı] +σ2 +Tr +� +B¯Cς[ı][v]AΘr/tRX[ı]ΘH +r/tA¯CH +ς[ı][h]B +� +, 1 ≤ h, v ≤ 4, +(28) +where +C(i,j) +[ı] += + + + + + + + + + + + + + + + + + + + +|α[ı]|2, if 1 ≤ i, j ≤ 2, +˜α[ı], +if ι = 3, ι ≤ 2, +˜α[ı], +if ι = 4, ι ≤ 2, +1, +if 3 ≤ i, j ≤ 4, +¯Cς[ı][i] = + + + + + +∂ε[ı] +∂Ψ[ı][i]εT +[ı] + ε[ı] +∂εT +[ı] +∂Ψ[ı][i] if 1 ≤ i ≤ 2, +ε[ı]εT +[ı], +if 3 ≤ i ≤ 4, +(29) +where ι = max{i, j} and ι = min{i, j}, and Θr/t ∈ CN×N is the reflection/transmission +coefficient matrix of STARS. +Proof: See Appendix D. +Therefore, the ECRB expression can be derived by substituting the extended FIM matrix expres- +sion into (14). Similarly, the extended achievable rate expression can be expressed as R[ı],k = +1 +2 log2 +� +1 + +Pk|hH +k,SΘr/tAGrv[ı],k|2 +� +j̸=k,j∈Kı Pj|hH +k,SΘr/tAGrv[ı],k|2+σ2 +� +with hk,S ∈ CN×1 and Gr ∈ CN×Mr. +B. Problem Formulation +With the derivations above, we aim to minimize the ECRB value for estimating Ψ[ı] of each +phase under the QoS constraints, by jointly optimizing the transmit power at the users, the +reflection/transmission coefficients of the PEs, the number of PEs/sensors at the STARS, the +receive beamforming and sensing waveform at the BS. Based on the definitions of Ur/t = ur/tuH +r/t +and Hk,S = diag(hH +k,S)Gr, the problem formulation in the ı-th phase is given by +min +E[ı],P[ı],Ur/t, +V[ı],k,A,B,R[ı],s +Tr(E−1 +[ı] ) +(30a) +s.t. +(16b), (16c), (18), (19d) − (19f), +(30b) +PkTr(AHk,SV[ı],kHH +k,SAUr/t)≥γt +� � +j̸=k,j∈Kı +PjTr(AHj,SV[ı],kHH +j,SAUr/t)+σ2� +, (30c) +A[n, n] ∈ {0, 1} , B[n, n] ∈ {0, 1}, +1 ≤ n ≤ N, +(30d) +A + B = IN, +(30e) +Tr(V[ı],k) = 1, +V[ı],k ⪰ 0, +(30f) + +17 +where γt = (22Rt − 1) with Rt representing the QoS requirements of users, and V[ı],k = +v[ı],k(v[ı],k)H. (30d) and (30e) denote the integer variable constraints for selection matrices. +Note the problem (30) is a MINLP that cannot be optimally solved by conventional convex +optimization methods, except for exhaustive search. To strike a balance between optimality and +complexity, we propose a PDL algorithm obtain the near-optimal solution of problem (30), which +optimizes the constructed augmented Lagrangian (AL) problem in the inner loop while updating +the penalty factor in the outer loop. +C. Augmented Lagrangian Problem Construction +To convert problem (30) to a tractable form, we introduce the auxiliary variables [p1, · · · , pN−1], +which satisfies pn ∈ {0, 1} and �N−1 +n=1 pn = 1. Then, we can equivalently rewrite (28) as +F[ı][h, v] = +2TC(h,v) +[ı] +σ2 +N−1 +� +n=1 +pnTr +� +¯Fn,vΘr/tRX[ı]ΘH +r/t¯FH +n,h +� +, +(31) +where the constant matrix ¯Fn,v = ¯Cς[ı][v]An − An ¯Cς[ı][v]An with An = [In, 0N−n]. Also, the +QoS constraint (30c) can be transformed into +PkHk,k +[ı],n ≥ γt +� +� +j̸=k,j∈Kı +PjHk,j +[ı],n + σ2 +� +, +(32) +where Hk,j +[ı],n = �N−1 +n=1 pnTr(AnHj,SV[ı],kHH +j,SAnUr/t). Note that when pn = 1 and pm = 0 +(m ̸= n) hold, the selection matrix A can be exactly determined, i.e., A = An. Thus, the +problem (30) can be converted to the following AL form without selection matrices. +min +E[ı],P[ı],Ur/t,V[ı],k,R[ı],s,pn,ρ2 +Tr(E−1 +[ı] ) + 1 +2ρ2 +N−1 +� +n=1 +(pn − p2 +n) +(33a) +s.t. +(16b), (16c), (18), (19d) − (19f), (30f), (32), +(33b) +N−1 +� +n=1 +pn = 1. +(33c) +Thereinto, when ρ2 → ∞, the penalty term pn − p2 +n approaches 0, which would be equivalent +to integer constraint pn ∈ {0, 1}. In the inner loop, we adopt the AO framework to optimize the +transmit power P[ı], the ISAC waveform R[ı],s, the reflection/transmission coefficients Ur/t and +the weight coefficient pn. + +18 +D. Joint Beamforming and Elements Optimization +1) Receive beamforming optimization: With fixed {P[ı], Ur/t, R[ı],s, pn}, problem (33) is equiv- +alent to the following feasible detection problem with respect to {V[ı],k}. +find +V[ı],k +(34a) +s.t. +rank(V[ı],k) = 1, +(34b) +(30f), (32), +(34c) +which can be efficiently handled by using penalty-based rank-one relaxation method [31]. The +converted problem is given by +min +V[ı],k +1 +2ρ1 +� +Tr(V[ı],k) − ∥V[n−1] +[ı],k ∥2 − Tr(¯v[n−1] +[ı],k (¯v[n−1] +[ı],k )H(V[ı],kV[n−1] +[ı],k )) +� +(35a) +s.t. +(30f), (32), +(35b) +where V[n−1] +[ı],k +is the given point determined by (n − 1)-th iteration and ¯v[n−1] +[ı],k +is the leading +eigenvector of V[n−1] +[ı],k . Note that the optimal v[ı],k can be obtained by carrying out SCA iterations +with the accuracy ǫv. +2) Transmit power and sensing waveform optimization: With fixed {Ur/t, v[ı],k, pn}, problem +(33) is reduced to +min +R[ı],s,E[ı],P[ı] +Tr(E−1 +[ı] ) +(36a) +s.t. +(16b), (18), (20c), (32), +(36b) +The optimal solutions {R[ı],s, P[ı]} can be easily obtained by solving the SDP problem (36). +3) Reflection/transmission coefficient optimization: With fixed {R[ı],s, P[ı], v[ı],k, pn} and the +transformation of (22), problem (33) can be re-expressed as +min +E[ı],Ur/t +Tr(E−1 +[ı] ) +(37a) +s.t. +(18), (19d) − (19f), (32), +(37b) +which can be optimally solved by Algorithm 1. + +19 +Algorithm 3 SCA algorithm for weight coefficient optimization. +1: Initialize p[m−1] +n +and Tr(CRB(Ψ[ı]))[m−1] with m = 1. Set a convergence accuracy ǫ3. +2: repeat +3: +update p[m] +n +by solving problem (38). +4: +calculate the Tr(CRB(Ψ[ı]))[m] and set m = m + 1. +5: until |Tr(CRB(Ψ[ı]))[m] − Tr(CRB(Ψ[ı]))[m−1]| ≤ ǫ3. +4) Weight coefficient optimization: With the fixed {R[ı],s, P[ı], v[ı],k, Ur/t}, the following AL +problem is obtained. +min +E[ı],pn +Tr(E−1 +[ı] ) + 1 +2ρ2 +N−1 +� +n=1 +(pn − p2 +n) +(38a) +s.t. +(18), (20c), (32), (33c). +(38b) +To handle the non-convex penalty term pn − p2 +n, we adopt the first-order Taylor expansion to +construct the linear upper bound approximation expression at m-th iteration, i.e., +pn − p2 +n ≤ pn − (p[m−1] +n +)2 − 2p[m−1] +n +(pn − p[m−1] +n +), +(39) +where p[m−1] +n +denotes the optimized value of pn at (m − 1)-th iteration. By substituting the +right-hand side of (39) into the penalty term of the objective function (38a), the problem (38) +becomes convex with respect to pn and can be solved over the SCA iterations. The details of +the SCA algorithm to optimize weight coefficient is summarized in Algorithm 3. +In the outer loop, we consider initializing ρ2 as a large value to make the integer constraint +trivial at the beginning, so that we can find a good starting point for original problem. Then, we +gradually decrease ρ2 over the outer loop iterations to obtain the feasible solution. +E. Overall Algorithm +The overall algorithm is summarized in Algorithm 4. Similarly, since the optimal trans- +mit power, sensing waveform, reflection/transmission coefficients and weight coefficients are +guaranteed at each step, the proposed PDL algorithm theoretically converges to the stationary +point solution over the non-increasing iterations. For the computational complexity, the entire +complexity of Algorithm 4 relies on the complexity to solve the SDP problems (35), (36) +(37) and (38). By exploiting the interior-point method, the computational complexity of PDL +algorithm at the ı-th phase can be expressed as O +� +lout log( 1 +ǫ2) +� +log( 1 +ǫv)(M2 +r )3.5 + (M2 +t + 4 + + +20 +Algorithm 4 Penalty-based double-loop algorithm. +1: Initialize {U[m−1] +r/t +, p[m−1] +n +, Tr(CRB(Ψ[ı]))[m−1]} with m = 1 and ρ[l−1] +2 +with l = 1. Set the +convergence accuracy {ǫ2, ρth}. +2: repeat +3: +repeat +4: +update v[m] +[ı],k by solving problem (35). +5: +update {R[m] +[ı],s, P[m] +[ı] } by solving problem (36). +6: +update U[m] +r/t by carrying out Algorithm 1 to solve problem (37). +7: +update p[m] +n +by carrying out Algorithm 3 to solve problem (38). +8: +calculate the CRB value Tr(CRB(Ψ[ı]))[m]and set m = m + 1. +9: +until |Tr(CRB(Ψ[ı]))[m] − Tr(CRB(Ψ[ı]))[m−1]| ≤ ρth. +10: +ρ[l] +2 = ρ[l] +2 +c2 (c2 > 1) and set l = l + 1. +11: until The penalty term �N−1 +n=1 (pn − p2 +n) drops below ǫ2. +BS(0,0,0) +x +y +o +342 +10m +T1 +T2 +U1 +UK +20m +STAR-RIS +(25 ,25 ,0) +2 +STAR-RIS +(25 ,25 ,0) +2 +2 +o +18 +Fig. 2. Top view of the simulation setup. +Kı)3.5 + log( 1 +ǫ1)(N2 + 4)3.5 + log( 1 +ǫ3)(N + 3)3.5�� +[32], where Kı = K1 in case of ı = 1, +Kı = K − K1 in case of ı = 2, and lout denotes the iteration number required in the outer loop. +V. NUMERICAL RESULTS +In this section, the numerical results are provided to demonstrate the effectiveness of the +proposed strategy. The top view of a three-dimensional coordinate network simulation setup is +illustrated in Fig. 2, where the BS is located at (0, 0, 0) m, and the STARS is located at a distance +of 50 m from the BS, i.e., (25 +√ +2, 25 +√ +2, 0) m. Uk is randomly distributed on a circle with a +radius of dk m centred on STARS, while T1 and T1 are located at a distance of 10 m from + +21 +0 +10 +20 +30 +40 +50 +0 +1 +2 +3 +4 +5 +6 +7 +Ergodic rate (bps/Hz) +Fig. 3. Accuracy of the approximated ergodic rate versus +κ with Mr = 8, Mt = 16, d1 = 20 m, K = 2, d2 = 10 +m, and PU,max = 15 dBm. +0 +5 +10 +15 +20 +25 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Maximum number of sensors +Fig. 4. The maximum number of sensor deployment ver- +sus the transmit power budget at the users with N = 20, +Mr = 1, Mt = 4, K = 2, dk = 20 m, and PBS,max = 15 +dBm. +the STARS with the directions of (342◦, 30◦) and (18◦, 30◦). The path loss at the unit reference +distance is set as L0 = −30 dB, the path-loss exponents for communication links are set as 2.2, +the path-loss exponents of sensing links are set as 2.5, the noise power is set as −115 dBm, +T = 10, Nv = 5, and Nh = +N +Nv. The other simulation parameters are listed in the caption of +each figure. Moreover, each figure is the average result over the 100 independent Monte-Carlo +experiments. +A. Verification of Lemma 1 and Corollary 1 +The accuracy of the approximated ergodic rate expression derived in Lemma 1 is verified +in Fig. 3, where the identity reflection/transmission coefficients Ur/t = IN1 are adopted and the +numerical ergoric rate is calculated based on the 10000 independent channels. It is shown that the +approximated ergodic rate is accurate to the actual ergodic rate for different N1, and U2 achieves +higher ergodic rate. These observations are expected since 1) the average non-LoS components +follow the complex Gaussian distribution with unit variance due to the law of large numbers; +and 2) U1 suffers the severer large-scale path loss than U2. It is also observed that the achievable +ergodic rate firstly increases with the rise of κ and gradually turns into stable. This is since that +1) when the κ is relatively small, increasing κ can make the LoS components prodominate and +provide the stronger transmission links; and 2) when κ becomes large, the Rician channels tend +to be the constant pure LoS channels. + +22 +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Angle (deg) +0 +0.5 +1 +Azimuth +Elevation +-80 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Angle (deg) +0 +0.5 +1 +Normalized amplitude of MLE spectrum +Azimuth +Elevation +Fig. 5. +The estimated azimuth and elevation angles via +the MLE method with N = 20, N1 = 5, Mr = Mt = 8, +K = 2, dk = 20 m, Rer,t = 2.5 bps/Hz, and PU,max = +PBS,max = 35 dBm. +5 +10 +15 +20 +25 +30 +35 +40 +45 +Number of iterations +0.030202 +0.030203 +0.030204 +0.030205 +0.030206 +Root CRB (deg) +0.044566 +0.044568 +0.04457 +Root CRB (deg) +Convergence of AO algorithm +R phase +T phase +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Number of iterations +10-2 +Root CRB (deg) +10-2 +10-1 +Root CRB (deg) +Convergence of PDL algorithm +R phase +T phase +Fig. 6. Convergence of the proposed algorithms with the +common parameters N = 20, Mr = 8, and dk = 20 m. +In Fig. 4, we evaluate the the maximum-number sensor deployment policy derived in Corollary +1 for both the transmission (T) and reflection (R) phases, where the actual maximum number of +sensors are obtained by solving the feasible detection problem with different number of sensors +for 100 independent channel realizations. Firstly, we can observe that the analytical maximum +number of sensors is exactly equal to the numerical results. Also can be seen, the maximum +number of sensors for T phase and R phase are equal. This is intuitive because the maximum- +number sensor deployment is only restricted by the corresponding QoS target rates, which are +the same for different phases in the considered network. Moreover, when PU,max increases and +Rer,t decreases, less PEs would be required to satisfy the QoS constraint, so a significant upward +trend in the number of sensors can be observed. +B. Effectiveness of Proposed Algorithms +To demonstrate the correctness of the proposed algorithms from the perspective of sensing +accuracy, the MLE spectrum lines under the two-user and fixed number of sensors are illustrated +in Fig. 5, where the estimated DOA angle can be determined by the exhaustive search of the +highest value point of the MLE spectrum [21, Appendix E]. As such, we can observe that the +MLE estimates angles are (−17.9904◦, 29.9999◦) and (17.9904◦, 29.9999◦), respectively, which +perfectly align with the presupposed DOA angles (342◦, 30◦) and (18◦, 30◦) and validate the +effectiveness of the optimized sensing waveform and reflection/transmission coefficients. Fig. 6 +plots the convergence performance of the proposed algorithms, where Mt = 8, N1 = 15, K = 2, + +23 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +10-2 +10-1 +100 +Root CRB (deg) +R phase: STAR-RIS-AO +T phase: STAR-RIS-AO +R phase: C-RIS-AO +T phase: C-RIS-AO +R phase: STAR-RIS-COC +T phase: STAR-RIS-COC +Fig. 7. +The root CRB versus the QoS target rate with +N = 20, N1 = 10, Mr = Mt = 8, K = 2, dk = 20 m, +PU,max = 15 dBm, and PBS,max = 30 dBm . +35 +37.5 +40 +42.5 +45 +47.5 +50 +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Root CRB (deg) +R phase: STAR-RIS-PDL +T phase: STAR-RIS-PDL +R phase: C-RIS-PDL +T phase: C-RIS-PDL +R phase: STAR-RIS-exhaustive-search +T phase: STAR-RIS-exhaustive-search +Fig. 8. The root CRB versus the transmit power budget +at the BS with with N = 20, Mr = 8, Mr = 12, K = 4, +dk = 20 m, Rer,t = 1 bps/Hz, and PU,max = 45 dBm. +Rer,t = 2.5 bps/Hz, PU,max = 25 dBm, PBS,max = 35 dBm are adopted for AO algorithm, while +Mt = 12, K = 4, Rer,t = 0.5 bps/Hz, PU,max = PBS,max = 45 dBm are set for PDL algorithm. +Note that for coinciding with the practical DOA angles, we transform the radian-based root CRB +into the degree unit in Fig. 6. As can be observed, both algorithms are capable of converging to +the stationary point solutions within the finite iterations. This is because the optimal solutions at +each step for solving the subproblem can be guaranteed in both two algorithms, which ensures +a non-increasing trend over the iterations. Meanwhile, we can observe that R phase achieves +smaller CRB compared to the T phase. It can be explained by the fact that in the R phase, the +STARS reconfigure the communication signal from the users and the sensing signal from the BS +as a new probing signal for estimation, which introduces more DoFs and enhance the received +echo energy, thus improving the sensing performance. +C. Performance Comparison +To verify the performance of the proposed scheme, we consider following benchmark schemes +for comparison: +• C-RIS-AO: In the “conventional-RIS-AO” (C-RIS-AO) scheme, we consider replacing the +STARS by a reflecting-only RIS and a transmitting-only RIS, where the proposed AO +algorithm is modified to jointly optimize the coefficients of PEs at conventional RIS, transmit +power at users and the receive beamforming at the BS under the two-user and fixed sensor + +24 +number setup. For fairness comparison, each conventional RIS is assumed to possess N1 +2 +PEs and N2 +2 sensors. +• STARS-COC: In the “STARS-communication-oriented coefficients (COC)” scheme, the re- +flection/transmission coefficients of PEs is determined by maximizing the minimum ergodic +rate of the user, i.e., ΘCOC +r/t += max +Θr/t +min +k +E{R[ı],k}. +• C-RIS-PDL: In the “C-RIS-PDL” scheme, we deploy an N +2 -element reflecting-only RIS and +an N +2 -element transmitting-only RIS at the same location of STARS, where the proposed +PDL algorithm is modified to jointly optimize the number of sensors, the coefficients of +PEs, transmit power at users and the receive beamforming at the BS. +• STARS-exhaustive-search: In the “STARS-exhaustive-search” scheme, we independently +optimize N −1 subproblems with respect to the reflection/transmission coefficients, transmit +power and the beamforming at the BS, where pn = 1 (�N−1 +n +pn = 1) is set for the n-th +subproblem. Then, we choose the smallest CRB as the final solution of this scheme. +As depicted in Fig. 7 and Fig. 8, the proposed scheme can always achieve better sens- +ing performance compared to the conventional RIS. This can be expected since the STARS +enables the double number of elements, which introduces more spatial DoFs for supporting +the communication and sensing. In Fig. 7, it is also observed that the sensing performance of +the considered network deteriorates with the increasing QoS target rate, and the STARS-COC +scheme achieves the worst performance. An intuitive explanation for the phenomenon is: 1) both +the sensing and communication relies on the coefficients setup of PEs, so when the QoS rate +increases, STARS has to prioritise support for communications to satisfy the QoS constraints, +which leads to a reduction in sensing performance; and 2) since the STARS-COC scheme only +focuses on the communication performance improvement, which would inevitably sacrifice the +accuracy of sensing. Moreover, we can observe that the CRB achieved by the STARS-COC +scheme does not vary with the QoS changes from Fig. 7. This is because the impact of QoS +target rate on the sensing performance can only be realized by affecting the coefficients of +PEs, which indicates that the sensing performance is independent of the QoS constraint under +any fixed reflection/transmission coefficients setup. For the Fig. 8, we can observe that the +proposed PDL algorithm can achieves the comparable performance of the optimal solution via the +exhaustive search, which significantly validates the equivalence of the derived ECRB expression +in Proposition 1. + +25 +12 +14 +16 +18 +20 +22 +24 +26 +28 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0.04 +0.045 +0.05 +0.055 +Root CRB (deg) +Fig. 9. +The root CRB versus the number of STARS +elements with K = 4, dk = 20 m, Rer,t = 0.5 bps/Hz, +and PU,max = PBS,max = 45 dBm. +4 +6 +8 +10 +12 +14 +16 +Number of sensors +2 +4 +6 +8 +10 +12 +14 +Root CRB (deg) +10-3 +R phase: K = 4 +T phase: K = 4 +R phase: K = 8 +T phase: K = 8 +Fig. 10. +Benchmarks under 2-user setup with N = 20, +Mr = 8, Mt = 12, dk = 20 m, Rer,t = 0.5 bps/Hz, +PU,max = PBS,max = 45 dBm, and Nv = 10. +D. Impact of STARS Elements +Fig. 9 investigates the influence of STARS element number on the sensing performance. Firstly, +It can be observed that the root CRB monotonically decreases as N increases. This is because:1)a +larger N can provide a higher array gains; and 2) under the near-optimal allocation of number +of sensors, increasing N also increases the number of sensors, which improves the sensing +reception ability at the sensing array. We can also observe an interesting result that increasing +number of transmit antennas at the BS significantly reduces the root CRB, but increasing number +of receive antennas at the BS has almost no effect on the CRB. This is resulted from the bi- +directional sensing-STARS architecture. Specifically, by exploiting this architecture, the receive +antenna at the BS is only responsible for receiving the communication signal, which implies that +we only need the number of receive antennas that meets the QoS requirements in the considered +network. However, increasing transmit antenna can expand the DoFs of the sensing waveform, +which provides a more flexible design of the sensing waveform for supporting the sensing. +In Fig. 10, we illustrate the impact of the number of sensors under the fixed total number of +STARS elements. It can be seen that when the number of sensors is less than 7, deploying more +sensors than PEs are preferable, and when the number of sensors is larger than 7, increasing the +number of sensors can hardly bring the constructive impact to the network anymore. It reveals that +there exists a trade-off between the the number of sensors and PEs. In detail, with a small number +of sensors, it is required to deploy more sensors to obtain sufficient echo sampling resolution to +extract target information. Whereas, in the case of large number of sensors, the information loss + +26 +of the sensing targets caused by insufficient echo sampling resolution is relatively small. Thus, +deploying more PEs to provide more DoFs for both communication and sensing becomes the +dominant factor in the sensing performance. Furthermore, we observe that increasing number +of users degrades the sensing performance, which is expected since the reflection/transmission +coefficients of PEs needs to be more oriented towards enhancing communications for satisfying +the QoS requirements of users. +VI. CONCLUSION +A new STARS-empowered ISAC scheme was proposed, where a bi-directional sensing-STARS +architecture was devised to support the full-space communication and sensing tasks. Two effi- +cient algorithms were developed to obtain the near-optimal solutions of the CRB minimization +problems. The correctness and effectiveness of proposed scheme was demonstrated by the +experiment results. It was also unveiled that: 1) STARS was capable of providing superior +performance compared to the conventional RIS; 2) under the unique design of bi-directional +sensing-STARS architecture, the number of receive antennas at the BS has little impact on the +sensing performance; 3) increasing number of PEs is more appealing than sensors for sensing +performance improvement. This work validated the potential of STARS in supporting full-space +dual-functional transmissions, and revealed an endogenous tradeoff that how do we determine +the number of sensors to be deployed. Both of them provided useful guidance for practical +system design. +APPENDIX A: DERIVATION OF FIM MATRIX +According (11), the element matrix in FIM matrix F[i] can be expressed as +JΨ[ı]Ψ[ı] = 2 +σ2 + + +ℜ +� +∂qH +[ı] +∂ϕ[ı] +∂q[ı] +∂ϕ[ı] +� +ℜ +� +∂qH +[ı] +∂ϕ[ı] +∂q[ı] +∂φ[ı] +� +ℜ +� +∂qH +[ı] +∂φ[ı] +∂q[ı] +∂ϕ[ı] +� +ℜ +� +∂qH +[ı] +∂φ[ı] +∂q[ı] +∂φ[ı] +� + + , +(A-1) +where +∂q[ı] +∂Ψ[ı][h] = vec +� +α[ı] +� ∂b(ϕ[ı],φ[ı]) +∂Ψ[ı][h] aT(ϕ[ı], φ[ı])+b(ϕ[ı], φ[ı]) +∂aT (ϕ[ı],φ[ı]) +∂Ψ[ı][h] +� +Θr/tX[ı] +� +(1 ≤ h ≤ 2). +With the derivative chain rule, we have +∂ǫ(ϕ[ı], φ[ı]) +∂Ψ[ı][h] += ǫ(ϕ[ı], φ[ı]) ⊙ ˙ǫΨ[ı][h], +ǫ(ϕ[ı], φ[ı]) ∈ {a(ϕ[ı], φ[ı]), b(ϕ[ı], φ[ı])}, +(A-2) + +27 +where the n-th elements of ˙ǫΨ[ı][h] are given by +˙ǫΨ[ı][h][n] = + + + + + +¯nπ cos φ[ı] cos ϕ[ı], +if h = 1, +(−¯nπ sin φ[ı] sin ϕ[ı] + (n − 1 − Nh¯n)π cos φ[ı]), +if h = 2, +(A-3) +Let ˙QΨ[ı][h] = +∂b(ϕ[ı],φ[ı]) +∂Ψ[ı][h] aT (ϕ[ı], φ[ı])+b(ϕ[ı], φ[ı]) +∂aT (ϕ[ı],φ[ı]) +∂Ψ[ı][h] +, we can rewrite JΨ[ı]Ψ[ı] as +JΨ[ı]Ψ[ı] = 2T|α[ı]|2 +σ2 +ℜ + + + +Tr( ˙Qϕ[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) +Tr( ˙Qϕ[ı]Θr/tRx[ı]ΘH +r/t ˙QH +φ[ı]) +Tr( ˙Qφ[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) +Tr( ˙Qφ[ı]Θr/tRx[ı]ΘH +r/t ˙QH +φ[ı]) + + + + , +(A-4) +where RX[ı] ≈ 1 +T X[ı]XH +[ı]. Similarly, let Q[ı] = b(ϕ[ı], φ[ı])aT(ϕ[ı], φ[ı]), we can obtain +JΨ[ı]α[ı] = 2T +σ2 ℜ + + + +˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) +˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) +˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) +˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH +r/t ˙QH +ϕ[ı]) + + + + , +(A-5) +Jα[ı]α[ı] = 2T +σ2 Tr(Q[ı]Θr/tRx[ı]ΘH +r/tQH +[ı])I2. +(A-6) +This completes the derivation. +APPENDIX B: DERIVATION OF ERGODIC RATE +With the results in [33, Lemma 1], the ergodic rate is approximated as +E{R[ı],k} ≈ 1 +2 log2 +� +1 + PkE{∥hH +k,SΘr/tGr∥2} +σ2 +� +. +(B-1) +Substituting (1) into (B-1), we can obatin +E{∥hH +k,SΘr/tGr∥2}=E + + + +������ +�� +κL2 +k,S +1 + κ +ˆhH +k,S+ +� +L2 +k,S +1 + κ +˜hH +k,S +� +Θr/t +�� +κL2 +r +1 + κ +ˆGr+ +� +L2 +r +1 + κ +˜Gr +������� +2 + + , +(a) += +L2 +k,SL2 +r +(1 + κ)2 +� +κ2E{∥ˆhH +k,SΘr/t ˆGr∥2} + κE{∥ˆhH +k,SΘr/t ˜Gr∥2}+ +κE{∥˜hH +k,SΘr/t ˆGr∥2} + E{∥˜hH +k,SΘr/t ˜Gr∥2} +� +, +(b)= +� +κ2∥ˆhH +k,SΘr/t ˆGr∥2+κMr∥ˆhH +k,SΘr/t∥2+κ∥Θr/t ˆGr∥2 +F+Mr∥Θr/t∥2 +F +� +(1 + κ)2/L2 +k,SL2 +r +, (B-2) +where equality (a) holds because the ˜hk,S and ˜Gr are independent of each other, while equality +(b) holds since all the elements in ˜hk,S and ˜Gr follow the complex Gaussian distribution with +zero mean and unit variance. With the result of (B-2), we can easily obtain the approximated +ergodic rate expression in Lemma 1. This completes the proof. + +28 +APPENDIX C: PROOF OF COROLLARY 1 +For the considered problem (16), we can readily know that achieving the maximum-number +sensor deployment is tantamount to search for the minimum-dimensional reflection/transmission +coefficients and the feasible transmit power that satisfy the QoS constraint, i.e., +find +{Θmin +r/t , P} +(C-1a) +s.t. +(16d), (16f), (16g), +(C-2b) +where Θmin +r/t denotes the minimum-dimensional coefficient matrix. By substituting Mr = 1 to the +ergodic rate expression in Lemma 1, it is readily to show that when the reflection/transmission co- +efficients of PEs are aligned to the cascaded channels ˆhH +k,SΘr/tˆgr, i.e., the reflection/transmission +coefficients derived in (24), and Pk = PU,max holds, it achieves the best communication perfor- +mance with the least N1. As such, constraint (19c) can be transformed into +κ2 +(1 + κ)2N2 +1 + 2κ + 1 +(1 + κ)2N1 − (22Rer,t − 1)σ2 +PU,maxL2 +k,SL2 +r +≥ 0. +(C-3) +Resorting the standard quadratic-root formula and the non-negativity of N1, the minimum Nmin +1 +can be determined. Thus, the maximum Nmax +2 +can be derived as shown in (25) based on Nmin +1 ++ +Nmax +2 += N. This completes the proof. +APPENDIX D: DERIVATION OF EXTENDED FIM MATRIX +Firstly, we can rewrite (11) as +y[ı],s = vec +� +α[ı]Bε[ı]εT +[ı]AΘr/tX[ı] +� +�� +� +q[ı] +� ++ n[ı], +(D-1) +where X[ı] ∈ CN×T is the equivalent signal matrix with regarding the whole STARS as PEs. +Similar to Appendix A, we have +∂q[ı] +∂Ψ[ı][h] = vec +� +α[ı]B +� +∂ε[ı] +∂Ψ[ı][h]εT +[ı] + ε[ı] +∂εT +[ı] +∂Ψ[ı][h] +� +AΘr/tX[ı] +� +, +1 ≤ h ≤ 2, +(D-2) +where +∂ε[ı] +∂Ψ[ı][h] is given in (A-2) and (A-3) with 1 ≤ n ≤ N. Hence, the h-th row and v-th column +element of JΨ[ı]Ψ[ı] is given by +∂qH +[ı] +∂Ψ[ı][h] +∂q[ı] +∂Ψ[ı][v] = vec +� +α[ı]B ˙CΨ[ı][h]AΘr/tX[ı] +�Hvec +� +α[ı]B ˙CΨ[ı][v]AΘr/tX[ı] +� +, += T|α[ı]|2Tr +� +B ˙CΨ[ı][v]AΘr/tRX[ı]ΘH +r/tA ˙CH +Ψ[ı][h]B +� +, 1 ≤ h, v ≤ 2, +(D-3) + +29 +where ˙CΨ[ı][h] = +∂ε[ı] +∂Ψ[ı][h]εT +[ı] + ε[ı] +∂εT +[ı] +∂Ψ[ı][h]. In the same way, the h-th row and v-th column element +of JΨ[ı]α[ı] and the h-th element on the diagonal of Jα[ı]α[ı] can be rewritten as +∂qH +[ı] +∂Ψ[ı][h] +∂q[ı] +∂α[ı][v] = T()v−1˜α[ı]Tr +� +BC[ı]AΘr/tRX[ı]ΘH +r/tA ˙CH +Ψ[ı][h]B +� +, 1 ≤ h, v ≤ 2, +(D-4) +∂qH +[ı] +∂α[ı][h] +∂q[ı] +∂α[ı][h] = TTr +� +BC[ı]AΘr/tRX[ı]ΘH +r/tACH +[ı]B +� +, 1 ≤ h ≤ 2, +(D-5) +where C[ı] = ε[ı]εT +[ı]. Substituting (D-3)–(D-5) into (A-4)–(A-6), we can obtain the extended +FIM matrix in Proposition 1. Then, we prove the equivalence between the extended FIM matrix +F[ı] and the original FIM matrix Fo +[ı] under any given A and B. To elaborate, let bp ∈ CN2×1 +and ap ∈ CN1×1 denote the practical steering vectors at the sensors and PEs, the received echo +signal can be expressed as Ep +[ı] = α[ı]bp(ap)TΘr/tX[ı]. With this in hand, it is easy to obtain the +following identity. +���� +∂q[ı] +ς[ı][i] +���� = +������ +∂vec +�� +0, 0; Ep +[ı], 0 +�� +ς[ı][i] +������ += +����� +∂qp +[ı] +ς[ı][i] +����� , +1 ≤ i ≤ 4, +(D-6) +where qp +[ı] = vec(Ep +[ı]). Therefore, the h-th row and v-th column element of the extended FIM +matrix is exactly equivalent to that of the original FIM matrix, i.e., +F[ı][h, v] = +∂qH +[ı] +ς[ı][h] +∂q[ı] +∂ς[ı][v] = +∂(qp +[ı])H +ς[ı][h] +∂qp +[ı] +∂ς[ı][v] = Fo +[ı][h, v], +1 ≤ h, v ≤ 4. +(D-7) +This completes the proof. +REFERENCES +[1] K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y. -J. A. Zhang, “The roadmap to 6G: AI empowered wireless networks,” +IEEE Commun. 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Signal Process., vol. 8, no. 5, pp. 966–981, Oct. 2014. + diff --git a/ftE1T4oBgHgl3EQfywUm/content/tmp_files/load_file.txt b/ftE1T4oBgHgl3EQfywUm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ad7157c9ef0f351fb920f2d1a0074e28b487888 --- /dev/null +++ b/ftE1T4oBgHgl3EQfywUm/content/tmp_files/load_file.txt @@ -0,0 +1,972 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf,len=971 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='03436v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='IT] 9 Jan 2023 1 STARS-ISAC: How Many Sensors Do We Need?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Zheng Zhang, Student Member, IEEE, Yuanwei Liu, Senior Member, IEEE, Zhaolin Wang, Graduate Student Member, IEEE, Jian Chen, Member, IEEE, Abstract A simultaneously transmitting and reflecting surface (STARS) enabled integrated sensing and com- munications (ISAC) framework is proposed, where a novel bi-directional sensing-STARS architecture is devised to facilitate the full-space communication and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Based on the proposed framework, a joint optimization problem is formulated, where the Cram´er-Rao bound (CRB) for estimating the 2- dimension direction-of-arrival of the sensing target is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Two cases are considered for sensing performance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1) For the two-user case, an alternating optimization algorithm is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In particular, the maximum number of deployable sensors is obtained in the closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) For the multi-user case, an extended CRB (ECRB) metric is proposed to characterize the impact of the number of sensors on the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Based on the proposed metric, a novel penalty- based double-loop (PDL) algorithm is proposed to solve the ECRB minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To tackle the coupling of the ECRB, a general decoupling approach is proposed to convert it to a tractable weighted linear summation form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Simulation results reveal that 1) the proposed PDL algorithm can achieve a near- optimal performance with consideration of sensor deployment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) without violating the communication under the quality of service requirements, reducing the receive antennas at the BS does not deteriorate the sensing performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 3) it is preferable to deploy more passive elements than sensors in terms of achieving optimal sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Index Terms Beamforming design, integrated sensing and communications (ISAC), simultaneously transmitting and reflecting surface (STARS), sensor deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Zheng Zhang and Jian Chen are with the School of Telecommunications Engineering, Xidian University, Xi’an 710071, China (e-mail: zzhang 688@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' jianchen@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Yuanwei Liu and Zhaolin Wang are with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (e-mail: yuanwei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='liu@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' zhaolin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='wang@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' INTRODUCTION With the commercialization of the fifth generation (5G) wireless networks, the 2030-oriented sixth generation (6G) wireless communication systems drew growing attention in both academia [1] and industry [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6G seeks a fundamental paradigm shift in wireless network architecture, which breaks the physical boundaries of sensing and communications to support more emerging applications, such as extended reality (XR), auto-driving, and Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To realize this vision, a key enabling technique, integrated sensing and communication (ISAC), has been proposed to unify the two functions via the same hardware platform and signal processing module [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To elaborate, through the dedicated co-designed framework, ISAC is capable of significantly enhancing the utilization efficiency of the network resources, thereby resulting in low imple- mentation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Furthermore, through deep integration, ISAC is also envisioned to realize mutual assistance and win-win benefit between the two functions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To provide ubiquitous wireless connection with low energy consumption, reconfigurable in- telligent surface (RIS) has emerged as another promising and cost-effective technique for future wireless networks [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Technically, RIS can be regarded as a metasurface-based planar array, which is composed of lots of passive tunable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The electromagnetic response at each element can be proactively adjusted via an external smart controller, which aims to reconfigure the amplitude and phase shifts of the incident signal and thus realize a smart radio environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' However, since the conventional RIS can merely reflect the incident signals and provide half- space coverage, the design flexibility is stringently limited by its geographical location and panel orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As a remedy, a new RIS paradigm, namely simultaneously transmitting and reflecting surface (STARS), has been proposed [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Compared to the conventional reflecting-only RIS, STARS can provide full-space electromagnetic environment reconfiguration [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Prior Works There have been lots of efforts devoted to the ISAC networks [12]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' More specifically, the authors of [12] devised a pair of antenna setups for multi-antenna ISAC systems, where a high-accuracy beampattern strategy is proposed to improve the sensing performance while guaranteeing the communication requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As a further advance, the authors of [13] proposed the optimal sensing waveform strategies, where the performance tradeoff between communication and sensing was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To introduce more degrees-of-freedom (DoFs) for target sensing, a sophisticated ISAC framework was proposed in [14], where the independent radar waveforms 3 and communication symbols were exploited to form the multiple beams for high-quality sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' However, the aforementioned works only focused on the waveform design at the transmitter while neglecting the sensing performance imposed by the received echo at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To provide a comprehensive evaluation of the sensing performance, the authors of [15] introduced the Cram´er-Rao bound (CRB) as the sensing performance metric of an unbiased estimation at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Furthermore, the authors of [16] developed a fairness-oriented unified resource allocation framework, where the BS was employed to carry out the device-free sensing services while satisfying communication QoS demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' More recently, it is claimed that the proper exploitation of RISs in ISAC systems can further boost the sensing performance [17]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In [17], the authors proposed a RIS-assisted ISAC framework, in which a RIS is employed to establish reliable line-of-sight (LoS) links for distance and velocity estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To support scenarios with multiple point-like targets, the authors of [18] developed a majorization-minimization algorithm for target tracking by collaboratively designing the transmit beampattern and RIS coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In [19], the authors conceived a joint optimization scheme regarding the sensing waveform and the RIS coefficients from the perspective of sensing mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Considering the practical restriction of discrete phase shifts, a constant- modulus sensing waveform was designed in [20], in which the multi-user interference was minimized under the sensing CRB constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In [21], the authors considered utilizing the RIS to provide sensing services to the blind-zone target, where the CRB minimization problems were investigated in the cases of point target and extended target, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' However, the direct combination of RIS and ISAC in the aforementioned works inevitably increased the number of reflections experienced by the echo signals, which restricted the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To address this issue, the authors of [22] pioneered a RIS-self-sensing architecture, where the dedicated sensors are deployed at the RIS to carrying out the sensing functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Shortly thereafter, the authors of [23] proposed a two-phase semi-passive RIS-assisted ISAC scheme, where the RIS supported the uplink communications in the first phase while carrying out the multi-user location sensing in the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Exploiting the same semi-passive sensing- at-RIS architecture, the authors of [24] studied the effect of sensing functionality on the commu- nications, where the RIS was employed to sense the user location to facilitate the communication beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Most recently, there was a preliminary exploration of STARS-enabled ISAC networks in [25], where a sensing-at-STARS structure was proposed to achieve the 2-dimension direction-of-arrivals (DOAs) estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Motivations and Contributions Based on the aforementioned RIS-enabled ISAC works [17]–[25], we can obtain following two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Although there have been a few works focusing on the RIS/STARS-enabled ISAC systems, the communication users and/or targets are only considered to be located on one side of the RIS/STARS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Whereas in the practical networks, the users and targets are probably in different geographical positions at different times, even on both sides of the RIS/STARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Apparently, such a problem cannot be coped with by the existing schemes, which thus calls for a more general strategy for the RIS/STARS-enabled ISAC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For the sensor-at-RIS/STARS architecture [22]–[25], a critical endogenous problem has not been answered yet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', should we deploy more sensors or passive elements (PEs) at the RIS/STARS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Given the fixed number of total elements of the RIS/STARS, deploying more sensors can increase the echo sampling resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Whereas deploying more PEs can introduce more spatial DoFs to favor both communication and sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Hence, there may exist a tradeoff between the number of sensors and PEs, which requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Motivated by the above observations, we propose a STARS-enabled ISAC framework, where the communication users and the targets are located on both sides of the STARS, with a particular focus on the tradeoff between the number of sensors and PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Our main contributions are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' We propose a novel bi-directional sensing-STARS architecture, where the micro-sized sen- sors with encapsulated antennas are integrated into the transparent substrate of STARS to provide full-space communication and sensing service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To tackle the energy/signal leakage issue in uplink STARS transmissions, a time switching (TS) protocol based two-phase scheme is proposed, where the STARS periodically switches between the reflection and transmission modes to support different users/targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With this transmission framework, the closed-form CRB expression is derived as the sensing performance metric for estimating the 2-dimension DOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1In the work [25], STARS is employed to divide the whole space into the communication region and sensing region, where the communication users and target are only situated on the corresponding half-space region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5 We first consider a two-user network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' A CRB minimization problem is formulated subject to the communication quality of service (QoS) constraint of the ergodic achievable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To facilitate the optimization, the approximated ergodic rate is derived in the closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we propose an alternating optimization (AO) algorithm, where the optimal sensing waveform, transmit power, and reflection/transmission coefficients are alternately obtained by utilizing the standard semidefinite program (SDP) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To provide further insights, the maximum number of sensors that can be deployed is derived in the closed-form expression, which unveils that the maximum number of sensors is only relevant to the QoS requirements of communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' We further consider a multi-user network, where a new metric of extended CRB (ECRB) is proposed to transform the impact of the number of sensors on the sensing accuracy into an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' We aim to minimize the proposed ECRB under the communication QoS requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To efficiently solve the formulated mixed integer non-linear program (MINLP), a generic decomposing method is devised to transform the non-convex objective function of the ECRB into a weighted linear summation form of the constant ECRB matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Based on the transformation, a penalty-based double-loop (PDL) algorithm is proposed to solve the resultant non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Numerical results demonstrate the effectiveness and convergence of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It is also verified that the proposed PDL can enable a near-optimal allocation of the number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Besides, two insights are observed: 1) with the proposed bi-directional sensing- STARS architecture, reducing the number of receive antennas at the BS does not deteriorate the sensing performance provided that QoS requirements are satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 2) given a fixed total number of STARS elements, deploying more PEs at the STARS is more attractive than sensors in terms of sensing performance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In Section II, we present the system model and performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In Section III, we conceive an AO algorithm to minimize CRB under the given number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Section IV develops a PDL algorithm for the joint optimization of beamforming and the number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The numerical results are illustrated in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Finally, the conclusion is presented in Section VI Notations: we use the boldface capital X and lower-case letter x to represent matrix and vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For any N × M-dimensional matrix X ∈ CN×M, XT and XH denote the transpose and Hermitian conjugate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Similarly, rank(X), Tr(X), ∥X∥, ∥X∥F represent 6 BS T1 UK U1 T2 Sensing element Passive element Sensing element Passive element Communication signal: Probing signal or sensing echo: Radar sensor with encapsulated antennas Tunable element 1 UK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1 1 UK + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The bi-directional sensing-STARS enabled uplink ISAC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' the rank value, trace value, spectral norm operation and Frobenius norm operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' X ⪰ 0 denotes that X is a positive semidefinite matrix, while x ∼ CN (0, X) denotes that x is a circularly symmetric complex Gaussian (CSCG) vector with zero mean and covariance matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For a matrix X, vec(X) and X−1 denote the vectorizing and inverse matrix operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For any vector x, diag(x) denotes a diagonal matrix whose main diagonal elements equal to the elements of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' |x| and ∥x∥ denote the modulus of x and the Euclidean norm of the vector x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' E(·) is the statistical expectation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' IN is a N-dimensional identity matrix, and 0N is a N-dimensional zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ℜ(·) and ℑ(·) denote the real component and imaginary component of the complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For any complex scalar z, ˜z denotes the conjugate of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For any real scalar x,⌊x⌋ and ⌈x⌉ denote the round-down and round-up operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ⊙ denotes the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Network Description As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1, we consider a STARS-enabled uplink ISAC network, where a STARS is deployed to establish reliable LoS links for K blind-zone users {U1, · · · , UK} to communicate with a BS while relaying the probing signal intended for the targets {T1, T2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The whole space is divided by the STARS into two separate region, each of which contains a target requiring estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It is assumed that the direct links of BS-users, users-targets, and BS-targets channels are blocked due to the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To mitigate the full-duplex self-interference at the BS, the BS is assumed to be equipped with an Mt-antenna transmit uniform linear array (ULA) and an Mr-antenna receive ULA [15], and all the users are single-antenna nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The STARS is 7 composed of a uniform planar array (UPA) with N = NvNh sub-wavelength elements, where Nv and Nh denote the number of elements located vertically and horizontally in the x-o-z plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To support the full-space communication and sensing, we propose a bi-directional sensing- STARS architecture, which divides the N STARS elements into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The former N1 PEs are employed to support the uplink communication, and the latter N2 = N − N1 elements (referred to as sensing elements) are equipped with the sensors for targets estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' More specifically, the PEs operate in the reflection or transmission mode for information transmission [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For each sensing element, a micro-sized low-cost sensor with antennas in packages is integrated inside the transparent substrate of STARS, where the adjacent tunable element op- erates in the full transmission mode, with the unit amplitude coefficient and zero phase-shift manipulation for the dual-sided incident signals [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Thus, the sensor is cable of receiving full-space echo waves without suffering penetration attenuation caused by STARS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' All the inter-antenna/element distances are assumed to be sub-wavelength, so the adjacent channel reflected/transmitted by the STARS element can be deemed to be independent channels [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The considered network is assumed to be a narrow-band system, where the BS and users transmit probing signal and communication signal in one coherence block of T consecutive sample frames [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' All the channels are assumed to be static at one coherence block, but vary over different coherence blocks [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The channel coefficients from PEs to the BS and Uk are respectively denoted as Gr ∈ CN1×Mr, Gt ∈ CN1×Mt and hk,S ∈ CN1×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' All the channels are assumed to follow Rician fading model since STARS can be flexibly deployed to favor the LoS links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Hence, the communication channels Hc ∈ {Gr, Gt, hk,S} are modeled as Hc = Lc �� κ 1 + κ ˆHc + � 1 1 + κ ˜Hc � , (1) where κ denotes the Rician factor, ˆHc represents the LoS component, and ˜Hc denotes the non- LoS component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Lc = � L0d−αc c ∈{Lr, Lt, Lk,S} denotes the corresponding path loss, where dc is the communication distance, αc represents the path-loss exponent, and L0 denotes the path loss at the reference distance of 1 meter (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For the sensing process, the probing signal and echo wave undergoes the PEs→target→sensors channels, which is modeled as H[ı],s = α[ı]b(ϕ[ı], φ[ı])aT(ϕ[ı], φ[ı]), 1 ≤ ı ≤ 2, (2) 8 where α[ı] ∈ C is the reflection coefficient containing the radar cross section (RCS) of Tı and the round-trip path-loss, ϕ[ı] denotes the azimuth angle of arrival/departure from the STARS to the target Tı, and φ[ı] denote the elevation angle of arrival/departure from the STARS to the target Tı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Note that a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) denote the steering vectors of PEs and sensors, where the n-th elements of a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) are given by [23] a(ϕ[ı], φ[ı])[n] = e\uf6be[¯nπ cos φ[ı] sin ϕ[ı]+(n−1−Nh¯n)π sin φ[ı]], 1 ≤ n ≤ N1, (3) b(ϕ[ı], φ[ı])[n] = e\uf6be[¯nπ cos φ[ı] sin ϕ[ı]+(n−1−Nh¯n)π sin φ[ı]], N1 + 1 ≤ n ≤ N, (4) where ¯n = ⌊n−1 Nh ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To investigate the fundamental performance limit of the considered network, the perfect channel information state (CSI) is assumed for all the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Signal Model Without loss of generality, we focus on the transmission at the t-th time frame and propose a TS-based framework, which equally divides the each time frame into following two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1) Phase I: The PEs operates in the reflection mode and users K1 ≜ {U1, · · · , UK1} transmit the communication signal c[1](t) = � k∈K1 √Pkck(t) with E{|ck(t)|2} = 1 to the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Meanwhile, the BS exploits the multiple beams to send the dedicated probing signal s[1](t) = �I[1] j=1 ˇsj(t) ∈ CM×1 with a general-rank covariance matrix R[1],s = E{s[1](t)sH [1](t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' On receiving the omnidi- rectional signal, the PEs reflects the communication signal to the BS while combining the c[1](t) and s[1](t) as a new probing signal to perform estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) Phase II: The PEs operates in the transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The users K2 ≜ {UK1+1, · · · , UK} send the communication signal c[2](t) = � k∈K2 √Pkck(t) to the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Meanwhile, the BS transmits probing signal s[2](t) = �I[2] j=1 ˇsj(t) ∈ CM×1 (R[2],s = E{s[2](t)sH [2](t)}) to the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' At the STARS, the PEs is exploited to transmit the c[2](t) to the BS while reconfiguring the probing signal to detect T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, the received signal at the BS in the ı-th phase is given by y[ı](t) = hH k,SΘr/tGr � k∈Kı � Pkck(t) + n[ı](t), (5) where n[ı](t) ∼ CN(0, σ2IMr) denotes the ı-th phase additive white Gaussian noise (AWGN) at the BS, and Θr/t = Θr in case of ı = 1 while denoting Θt in case of ı = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' There- into, the reflection/transmission coefficient matrix is defined as Θr/t = diag(ur/t) with ur/t = 9 [�βr/t,1e\uf6beθr/t,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' , �βr/t,Ne\uf6beθr/t,N]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To recover ck(t), we assume that a unit-norm linear com- bination vector v[ı],k ∈ CMr×1 is employed at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The extracted signal for Uk is given by y[ı](t)v[ı],k = hH k,SΘr/tGrv[ı],k � Pkck(t) � �� � desired signal + hH j,SΘr/tGrv[ı],k � j̸=k,j∈Kı � Pjcj(t) � �� � interference +n[ı](t)v[ı],k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (6) While for the sensing, the equivalent probing signal from PEs at the t-th time frame of the ı-th phase is given by x[ı](t) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 HU,S √¯Pc(t) + Gts[1](t), if ı = 1, Gts[2](t), if ı = 2, (7) where HU,S = [h1,S, · · · , hK1,S], ¯P = diag[P1, · · · , PK1], and c(t) = [c1, · · · , cK1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, the covariance matrix of x[ı](t) is given by R[ı],x = E[x[ı](t)x[ı](t)H] = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 HU,S¯PHH U,S + GtR[1],sGH t , if ı = 1, GtR[2],sGH t , if ı = 2, (8) Thus, the received echo wave at the sensors over T consecutive time frames of the ı-th phase is given by Y[ı],s = H[ı],sΘr/tX[ı] + N[ı], (9) where X[ı] = [x[ı](1), · · · , x[ı](T)] and N[ı] = [n[ı](1), · · · , n[ı](T)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Performance Metric As stated above, the achievable rate at the BS in the ı-th phase to decode ck(t) is expressed as R[ı],k = 1 2 log2 \uf8eb \uf8ec \uf8ed1 + Pk|hH k,SΘr/tGrv[ı],k|2 � j̸=k,j∈Kı Pj|hH j,SΘr/tGrv[ı],k|2 + σ2 \uf8f6 \uf8f7 \uf8f8 , 1 ≤ ı ≤ 2, (10) For the sensing, we focus on the CRB performance with unknown parameters ς[ı] = [ϕ[ı], φ[ı], ℜ(α[ı]), ℑ(α[ı])] ∈ R4×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To facilitate deriving CRB expression, we vectorize equation (9), which can be rewritten as y[ı],s = vec(Y[ı],s) = vec � H[ı],sΘr/tX[ı] � + n[ı], (11) 10 where n[ı] = vec(N[ı]) ∼ CN (0, σ2IMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Let q[ı] = vec � H[ı],sΘr/tX[ı] � , the Fisher information matrix (FIM) F[ı] ∈ R4×4 for estimating ς[ı] can be expressed as a Jacobian matrix with the h-th row and v-th column element being given by [28] F[ı][h, v] = 2ℜ � ∂qH [ı] ∂ς[ı],h R−1 n[ı] ∂q[ı] ∂ς[ı],v � + Tr � R−1 n[ı] ∂Rn[ı] ∂ς[ı],h R−1 n[ı] ∂Rn[ı] ∂ς[ı],v � = 2 σ2ℜ � ∂qH [ı] ∂ς[ı],h ∂q[ı] ∂ς[ı],v � , 1 ≤ h, v ≤ 4, (12) where Rn[ı] = σ2IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, we can repartition F[ı] as F[ı] = \uf8ee \uf8f0JΨ[ı]Ψ[ı] JΨ[ı]α[ı] JT Ψ[ı]α[ı] Jα[ı]α[ı] \uf8f9 \uf8fb , (13) where Ψ[ı] = [ϕ[ı], φ[ı]], α[ı] = [ℜ(αı), ℑ(αı)], while the specific expressions of JΨ[ı]Ψ[ı], JΨ[ı]α[ı] and Jα[ı]α[ı] are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, with the inverse formula of the second order matrix, we can derive the CRB expression of the ı-th phase with regard to Ψ[ı] [29] as CRB(Ψ[ı]) = � JΨ[ı]Ψ[ı] − JΨ[ı]α[ı]J−1 α[ı]α[ı]JT Ψ[ı]α[ı] �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (14) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' BEAMFORMING OPTIMIZATION UNDER FIXED SENSOR NUMBER In this section, we concentrate on joint sensing waveform and communication beamforming optimization with the fixed number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To draw instructive insights for practical system design, we consider a special network setup of two users, and utilize the ergodic rate as the average communication performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The the closed-form approximation expression of ergodic achievable rate is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, an AO algorithm to efficiently solve the non- convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Problem Formulation Since inter-user interference is non-existent in the two-user case, the achievable rate at the BS in the ı-th phase to decode ck(t) is rewritten to R[ı],k = 1 2 log2 � 1 + Pk|hH k,SΘr/tGrv[ı],k|2 σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (15) To provide the generalised insights to the CRB optimization, we employ the ergodic achievable rate as the average performance metric for the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, we target at minimiz- ing the CRB performance for DOA Ψ[ı] at each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Subject to the average QoS requirements of 11 users, the joint optimization of the transmit power at the users, reflection/transmission coefficients of the PEs, the receive beamforming and the sensing waveform at the BS is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The optimization problem in the ı-th phase is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' min P[ı],Θr/t,R[ı],s,v[ı],k Tr(CRB(Ψ[ı])) (16a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Tr(R[ı],s) ≤ PBS,max, (16b) Pk ≤ PU,max, 1 ≤ k ≤ K, (16c) E{Rk} ≥ Rer,t, (16d) ∥v[ı],k∥2 = 1, 1 ≤ k ≤ K, (16e) θr,n, θt,n ∈ [0, 2π], 1 ≤ n ≤ N, (16f) βr,n, βt,n ∈ [0, 1], 1 ≤ n ≤ N, (16g) where P[1] = [P1, · · · , PK1], P[2] = [PK1+1, · · · , PK], Rer,t denotes the ergodic QoS rate of users, PU,max denotes the maximal transmit power at the users, and PBS,max denotes the transmit power budget at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b) and (16c) denote the transmit power constraint at the BS and users’ sides, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16d) represents the ergodic QoS constraint of users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16e) denotes the normalization constraint of the receive beamforming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16f) and (16g) are the phase-shift and amplitude constraints of the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Notably, the intractable expression of CRB and the non- convex constraints (16d) and (16e) make problem (16) non-convex and challenging to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In the following, we consider optimizing the sensing waveform and communication beamforming in an alternating manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Problem Reformulation Before handling the challenging problem, we can observe that v[ı],k only exists in the constraint (16d) and has no direct influence on the CRB performance, which indicates that only the feasible v[ı],k are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For maximum compliance with QoS constraint, the optimal receive beamforming is given by v[ı],k = (hH k,SΘr/tGr)H ∥hH k,SΘr/tGr∥ , to obtain the best communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Hence, (15) can be transformed into R[ı],k = 1 2 log2 � 1 + Pk∥hH k,SΘr/tGr∥2 σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With this in mind, we further rewrite ∥ˆhH k,SΘr/t ˆGr∥2 as Tr( ˆHk,S ˆHH k,SUr/t), with definition of Ur/t = ur/tuH r/t and ˆHk,S = diag(ˆhH k,S) ˆGr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we resort Lemma 1 to obtain the approximation expression of E{Rk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 12 Lemma 1: The ergodic achievable rate in (16d) can be approximated as E{R[ı],k} ≈ 1 2 log2 � 1+ PkL2 k,SL2 r σ2(1+κ)2 � κ2Tr( ˆHk,S ˆHH k,SUr/t)+κTr( ˆGr ˆGH r Ur/t)+ κMrTr � diag(ˆhH k,S)Ur/tdiag(ˆhk,S) � + MrTr(Ur/t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (17) Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The approximation accuracy can be verified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To proceed, we further introduce Lemma 2 to convexify the non-convex objective function (16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Lemma 2: According to [30], we have that for any X ⪰ 0 and Y ⪰ 0, if X ⪰ Y is guaranteed, the inequality of Tr(X−1) ≤ Tr(Y−1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As such, with the fact that FIM matrix � JΨ[ı]Ψ[ı] − JΨ[ı]α[ı]J−1 α[ı]α[ı]JT Ψ[ı]α[ı] � is positive semidefi- nite, we can introduce an auxiliary variable E[ı] ⪰ 0 to equivalently transform (16a) into Tr(E−1 [ı] ), with satisfying following linear matrix inequality (LMI) constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' \uf8ee \uf8f0JΨ[ı]Ψ[ı] − E[ı] JΨ[ı]α[ı] JT Ψ[ı]α[ı] Jα[ı]α[ı] \uf8f9 \uf8fb ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (18) With the transformations above, problem (16) can be reformulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' min E[ı],Ur/t,R[ı],s,P[ı] Tr(E−1 [ı] ) (19a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b), (16c), (18), (19b) PkL2 k,SL2 r (1 + κ)2 � κ2Tr( ˆHk,S ˆHH k,SUr/t)+κMrTr � diag(ˆhH k,S)Ur/tdiag(ˆhk,S) � + κTr( ˆGr ˆGH r Ur/t) + MrTr(Ur/t) � ≥ σ2(22Rer,t − 1), (19c) E[ı] ⪰ 0, Ur/t ⪰ 0, (19d) Ur/t[n, n] ≤ 1, 1 ≤ n ≤ N, (19e) rank(Ur/t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (19f) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Joint Beamforming Optimization 1) Transmit power and sensing waveform optimization: With the fixed Ur/t, the problem (19) is reduced to the following subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' min R[ı],s,E[ı],P[ı] Tr(E−1 [ı] ) (20a) 13 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b), (16c), (18), (19c), (20b) E[ı] ⪰ 0, (20c) which is a SDP and can be optimally solved by the convex toolbox, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) Reflection/Transmission coefficients optimization: With fixed {R[ı],s, P[ı]}, problem (19) is rewritten as min E[ı],Ur/t Tr(E−1 [ı] ) (21a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (18), (19c) − (19f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (21b) To handle the non-convex LMI constraint (18), we adopt the singular value decomposition (SVD) to equivalently convert the quadratic terms {JΨ[ı]Ψ[ı], JΨ[ı]α[ı], Jα[ı]α[ı]} to the tractable forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Specifically, by decomposing Rx[ı] into � q sqdq, we have Θr/tRx[ı]ΘH r/t = � q diag(sq)ur/tuH r/tdiag(dq) = � q Squr/tuH r/tDq = � q SqUr/tDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (22) Then, we substitute (22) into FIM matrix, the constraint (18) becomes convex with respect to Ur/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' While for the non-convex constraint (19f), we employ the penalty-based rank-one relaxation approach [31], which exploits the successive convex approximation (SCA) technique to relax the equivalent rank-one constraint Tr(Ur/t) − ∥Ur/t∥2 = 0 as a convex penalty term in objective function (21a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accordingly, the problem (21) is reformulated as min E[ı],Ur/t Tr(E−1 [ı] ) − 1 2ρ1 � Tr(Ur/t) − ∥U[n−1] r/t ∥2 − Tr(¯u[n−1] r/t (¯u[n−1] r/t )H(Ur/t − U[n−1] r/t )) � , (23a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (18), (19c) − (19e), (23b) where U[n−1] r/t denotes the optimized result in the (n − 1)-th iteration, ¯u[n−1] r/t denotes the leading eigenvector of U[n−1] r/t , and ρ1 represents the penalty factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The resultant problem (23) is a convex program, where the rank-one solution can be optimally obtained when p is sufficiently small [31, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The specific SCA algorithm to optimize Ur/t is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Remark 1: (MLE Validation) In this paper, we consider employing the maximum likelihood estimation (MLE) approach in [21, Appendix E] to obtain the estimated DoA Ψes [ı] under the 14 Algorithm 1 SCA algorithm for rank-one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1: Initialize initial U[n−1] r/t and p1 with n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Set a convergence accuracy ǫ1 and calculate the leading eigenvector ¯u[n−1] r/t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2: repeat 3: update U[n] r/t by solving problem (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4: update the leading eigenvector ¯u[n] r/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5: set n = n + 1 and ρ1 = ρ1 c1 (c1 > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6: until the penalty term in objective function (23a) drops below ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Algorithm 2 AO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1: Initialize initial U[l−1] r/t and Tr(CRB(Ψ[ı]))[l−1] with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Set a convergence accuracy ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2: repeat 3: update the optimal receive beamforming vopt [ı],k = (hH k,SΘr/tGr)H ∥hH k,SΘr/tGr∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4: update optimal {R[l] [ı],s, P[l] [ı]} by solving problem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5: update optimal U[l] r/t by carrying out Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6: set l = l + 1 and calculate Tr(CRB(Ψ[ı]))[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7: until |Tr(CRB(Ψ[ı]))[l] − Tr(CRB(Ψ[ı]))[l−1]| ≤ ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' optimized waveform, where the the correctness of the proposed CRB optimization framework is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Corollary 1: (Maximal Number of Sensor Deployment) For the special case of single receive antenna, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', Mr = 1, the optimal reflection/transmission coefficients and the maximal number of sensors to be deployed can be derived as following closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 θopt r,n = ∠ˆh1,S[n] − ∠ˆgr[n], βopt r,n = 1 θopt t,n = ∠ˆh2,S[n] − ∠ˆgr[n], βopt t,n = 1, (24) Nmax 2 = N − � 1 2κ2 �� (2κ + 1)2 + 4κ2σ2(22Rer,t − 1)(1 + κ)2 PU,maxL2 k,SL2 r − (2κ + 1) �� , (25) where ˆgr is the degenerated channel of ˆGr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Proof: See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Overall Algorithm 15 The overall algorithm is summarized in Algorithm 2, which optimizes the sensing waveform and reflection/transmission coefficients alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' By denoting the CRB value at l-th iteration as a function of R[ı],s and Ur/t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', g(R[ı],s, P[ı], Ur/t), the following inequality always holds g(R[l−1] [ı],s , P[l−1] [ı] , U[l−1] r/t ) (a) ≥ g(R[l] [ı],s, P[l] [ı], U[l−1] r/t ) (b) ≥ g(R[l] [ı],s, P[l] [ı], U[l] r/t), (26) where inequality signs (a) and (b) hold because the optimal sensing waveform and optimal reflection/transmission coefficients are both guaranteed in the step 4 and step 5 at the same AO iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Meanwhile, since g(R[ı],s, P[ı], Ur/t) is continuous over the compact feasible set of problem, there exists a finite positive number that serves as a lower bound on the objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This proves that our proposed AO algorithm remains non-increasing over the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' On the other hand, the computational complexity of AO algorithm mainly relies on solving SDP problems (20) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The overall complexity based on the interior-point method is given by O � log( 1 ǫ2) � (M2 t + 5)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 + log( 1 ǫ1)(N2 1 + 4)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5�� [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' HOW MANY SENSORS DO WE NEED?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In this section, we consider the general multi-user system with the joint optimization of beamforming design and the number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' By modifying the traditional CRB expression, a new metric of ECRB is proposed, which can evaluate the sensing performance while taking the sensors’ deployment into the consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Based on the proposed ECRB, a PDL algorithm is devised to jointly optimize the ISAC waveform, reflection/transmission coefficients and the number of PEs/sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Extended CRB Derivation To facilitate the optimization of sensor number, we define two N-dimensional matrices A = diag[IN1, 0N2] and B = diag[0N1, IN2], where A and B are the element selection matrices for PEs and sensors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With the definition above, we can rewrite the steering vector a(ϕ[ı], φ[ı]) and b(ϕ[ı], φ[ı]) as the extended form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' a(ϕ[ı], φ[ı]) = Aε(ϕ[ı], φ[ı]) b(ϕ[ı], φ[ı]) = Bε(ϕ[ı], φ[ı]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (27) Here, ε(ϕ[ı], φ[ı]) ∈ CN×1 denotes the steering vector of the STARS, n-th element of which is defined in (3) with 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For convenience of denotation, we abbreviate ε(ϕ[ı], φ[ı]) as ε[ı] in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we introduce Proposition 1 to derive the expression of the extended FIM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 16 Proposition 1: The h-th row and v-th column element of extended FIM matrix is given by F[ı][h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' v] = 2TC(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='v) [ı] σ2 Tr � B¯Cς[ı][v]AΘr/tRX[ı]ΘH r/tA¯CH ς[ı][h]B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1 ≤ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' v ≤ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (28) where C(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='j) [ı] = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 |α[ı]|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if 1 ≤ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' j ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ˜α[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if ι = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ι ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' \uf6be˜α[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if ι = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ι ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if 3 ≤ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' j ≤ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ¯Cς[ı][i] = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∂ε[ı] ∂Ψ[ı][i]εT [ı] + ε[ı] ∂εT [ı] ∂Ψ[ı][i] if 1 ≤ i ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ε[ı]εT [ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if 3 ≤ i ≤ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (29) where ι = max{i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' j} and ι = min{i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' j},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and Θr/t ∈ CN×N is the reflection/transmission coefficient matrix of STARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Proof: See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Therefore, the ECRB expression can be derived by substituting the extended FIM matrix expres- sion into (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Similarly, the extended achievable rate expression can be expressed as R[ı],k = 1 2 log2 � 1 + Pk|hH k,SΘr/tAGrv[ı],k|2 � j̸=k,j∈Kı Pj|hH k,SΘr/tAGrv[ı],k|2+σ2 � with hk,S ∈ CN×1 and Gr ∈ CN×Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Problem Formulation With the derivations above, we aim to minimize the ECRB value for estimating Ψ[ı] of each phase under the QoS constraints, by jointly optimizing the transmit power at the users, the reflection/transmission coefficients of the PEs, the number of PEs/sensors at the STARS, the receive beamforming and sensing waveform at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Based on the definitions of Ur/t = ur/tuH r/t and Hk,S = diag(hH k,S)Gr, the problem formulation in the ı-th phase is given by min E[ı],P[ı],Ur/t, V[ı],k,A,B,R[ı],s Tr(E−1 [ı] ) (30a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b), (16c), (18), (19d) − (19f), (30b) PkTr(AHk,SV[ı],kHH k,SAUr/t)≥γt � � j̸=k,j∈Kı PjTr(AHj,SV[ı],kHH j,SAUr/t)+σ2� , (30c) A[n, n] ∈ {0, 1} , B[n, n] ∈ {0, 1}, 1 ≤ n ≤ N, (30d) A + B = IN, (30e) Tr(V[ı],k) = 1, V[ı],k ⪰ 0, (30f) 17 where γt = (22Rt − 1) with Rt representing the QoS requirements of users, and V[ı],k = v[ı],k(v[ı],k)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (30d) and (30e) denote the integer variable constraints for selection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Note the problem (30) is a MINLP that cannot be optimally solved by conventional convex optimization methods, except for exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To strike a balance between optimality and complexity, we propose a PDL algorithm obtain the near-optimal solution of problem (30), which optimizes the constructed augmented Lagrangian (AL) problem in the inner loop while updating the penalty factor in the outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Augmented Lagrangian Problem Construction To convert problem (30) to a tractable form, we introduce the auxiliary variables [p1, · · · , pN−1], which satisfies pn ∈ {0, 1} and �N−1 n=1 pn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we can equivalently rewrite (28) as F[ı][h, v] = 2TC(h,v) [ı] σ2 N−1 � n=1 pnTr � ¯Fn,vΘr/tRX[ı]ΘH r/t¯FH n,h � , (31) where the constant matrix ¯Fn,v = ¯Cς[ı][v]An − An ¯Cς[ı][v]An with An = [In, 0N−n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Also, the QoS constraint (30c) can be transformed into PkHk,k [ı],n ≥ γt � � j̸=k,j∈Kı PjHk,j [ı],n + σ2 � , (32) where Hk,j [ı],n = �N−1 n=1 pnTr(AnHj,SV[ı],kHH j,SAnUr/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Note that when pn = 1 and pm = 0 (m ̸= n) hold, the selection matrix A can be exactly determined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', A = An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Thus, the problem (30) can be converted to the following AL form without selection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' min E[ı],P[ı],Ur/t,V[ı],k,R[ı],s,pn,ρ2 Tr(E−1 [ı] ) + 1 2ρ2 N−1 � n=1 (pn − p2 n) (33a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b), (16c), (18), (19d) − (19f), (30f), (32), (33b) N−1 � n=1 pn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (33c) Thereinto, when ρ2 → ∞, the penalty term pn − p2 n approaches 0, which would be equivalent to integer constraint pn ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In the inner loop, we adopt the AO framework to optimize the transmit power P[ı], the ISAC waveform R[ı],s, the reflection/transmission coefficients Ur/t and the weight coefficient pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Joint Beamforming and Elements Optimization 1) Receive beamforming optimization: With fixed {P[ı], Ur/t, R[ı],s, pn}, problem (33) is equiv- alent to the following feasible detection problem with respect to {V[ı],k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' find V[ı],k (34a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' rank(V[ı],k) = 1, (34b) (30f), (32), (34c) which can be efficiently handled by using penalty-based rank-one relaxation method [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The converted problem is given by min V[ı],k 1 2ρ1 � Tr(V[ı],k) − ∥V[n−1] [ı],k ∥2 − Tr(¯v[n−1] [ı],k (¯v[n−1] [ı],k )H(V[ı],kV[n−1] [ı],k )) � (35a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (30f), (32), (35b) where V[n−1] [ı],k is the given point determined by (n − 1)-th iteration and ¯v[n−1] [ı],k is the leading eigenvector of V[n−1] [ı],k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Note that the optimal v[ı],k can be obtained by carrying out SCA iterations with the accuracy ǫv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) Transmit power and sensing waveform optimization: With fixed {Ur/t, v[ı],k, pn}, problem (33) is reduced to min R[ı],s,E[ı],P[ı] Tr(E−1 [ı] ) (36a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16b), (18), (20c), (32), (36b) The optimal solutions {R[ı],s, P[ı]} can be easily obtained by solving the SDP problem (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 3) Reflection/transmission coefficient optimization: With fixed {R[ı],s, P[ı], v[ı],k, pn} and the transformation of (22), problem (33) can be re-expressed as min E[ı],Ur/t Tr(E−1 [ı] ) (37a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (18), (19d) − (19f), (32), (37b) which can be optimally solved by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 19 Algorithm 3 SCA algorithm for weight coefficient optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1: Initialize p[m−1] n and Tr(CRB(Ψ[ı]))[m−1] with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Set a convergence accuracy ǫ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2: repeat 3: update p[m] n by solving problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4: calculate the Tr(CRB(Ψ[ı]))[m] and set m = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5: until |Tr(CRB(Ψ[ı]))[m] − Tr(CRB(Ψ[ı]))[m−1]| ≤ ǫ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4) Weight coefficient optimization: With the fixed {R[ı],s, P[ı], v[ı],k, Ur/t}, the following AL problem is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' min E[ı],pn Tr(E−1 [ı] ) + 1 2ρ2 N−1 � n=1 (pn − p2 n) (38a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (18), (20c), (32), (33c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (38b) To handle the non-convex penalty term pn − p2 n, we adopt the first-order Taylor expansion to construct the linear upper bound approximation expression at m-th iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', pn − p2 n ≤ pn − (p[m−1] n )2 − 2p[m−1] n (pn − p[m−1] n ), (39) where p[m−1] n denotes the optimized value of pn at (m − 1)-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' By substituting the right-hand side of (39) into the penalty term of the objective function (38a), the problem (38) becomes convex with respect to pn and can be solved over the SCA iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The details of the SCA algorithm to optimize weight coefficient is summarized in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In the outer loop, we consider initializing ρ2 as a large value to make the integer constraint trivial at the beginning, so that we can find a good starting point for original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we gradually decrease ρ2 over the outer loop iterations to obtain the feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Overall Algorithm The overall algorithm is summarized in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Similarly, since the optimal trans- mit power, sensing waveform, reflection/transmission coefficients and weight coefficients are guaranteed at each step, the proposed PDL algorithm theoretically converges to the stationary point solution over the non-increasing iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For the computational complexity, the entire complexity of Algorithm 4 relies on the complexity to solve the SDP problems (35), (36) (37) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' By exploiting the interior-point method, the computational complexity of PDL algorithm at the ı-th phase can be expressed as O � lout log( 1 ǫ2) � log( 1 ǫv)(M2 r )3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 + (M2 t + 4 + 20 Algorithm 4 Penalty-based double-loop algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 1: Initialize {U[m−1] r/t , p[m−1] n , Tr(CRB(Ψ[ı]))[m−1]} with m = 1 and ρ[l−1] 2 with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Set the convergence accuracy {ǫ2, ρth}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2: repeat 3: repeat 4: update v[m] [ı],k by solving problem (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5: update {R[m] [ı],s, P[m] [ı] } by solving problem (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6: update U[m] r/t by carrying out Algorithm 1 to solve problem (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7: update p[m] n by carrying out Algorithm 3 to solve problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 8: calculate the CRB value Tr(CRB(Ψ[ı]))[m]and set m = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 9: until |Tr(CRB(Ψ[ı]))[m] − Tr(CRB(Ψ[ı]))[m−1]| ≤ ρth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 10: ρ[l] 2 = ρ[l] 2 c2 (c2 > 1) and set l = l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 11: until The penalty term �N−1 n=1 (pn − p2 n) drops below ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' BS(0,0,0) x y o 342 10m T1 T2 U1 UK 20m STAR-RIS (25 ,25 ,0) 2 STAR-RIS (25 ,25 ,0) 2 2 o 18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Top view of the simulation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Kı)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 + log( 1 ǫ1)(N2 + 4)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 + log( 1 ǫ3)(N + 3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5�� [32], where Kı = K1 in case of ı = 1, Kı = K − K1 in case of ı = 2, and lout denotes the iteration number required in the outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, the numerical results are provided to demonstrate the effectiveness of the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The top view of a three-dimensional coordinate network simulation setup is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2, where the BS is located at (0, 0, 0) m, and the STARS is located at a distance of 50 m from the BS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', (25 √ 2, 25 √ 2, 0) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Uk is randomly distributed on a circle with a radius of dk m centred on STARS, while T1 and T1 are located at a distance of 10 m from 21 0 10 20 30 40 50 0 1 2 3 4 5 6 7 Ergodic rate (bps/Hz) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Accuracy of the approximated ergodic rate versus κ with Mr = 8, Mt = 16, d1 = 20 m, K = 2, d2 = 10 m, and PU,max = 15 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 0 5 10 15 20 25 2 4 6 8 10 12 14 16 18 20 Maximum number of sensors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The maximum number of sensor deployment ver- sus the transmit power budget at the users with N = 20, Mr = 1, Mt = 4, K = 2, dk = 20 m, and PBS,max = 15 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' the STARS with the directions of (342◦, 30◦) and (18◦, 30◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The path loss at the unit reference distance is set as L0 = −30 dB, the path-loss exponents for communication links are set as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='2, the path-loss exponents of sensing links are set as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5, the noise power is set as −115 dBm, T = 10, Nv = 5, and Nh = N Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The other simulation parameters are listed in the caption of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Moreover, each figure is the average result over the 100 independent Monte-Carlo experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Verification of Lemma 1 and Corollary 1 The accuracy of the approximated ergodic rate expression derived in Lemma 1 is verified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 3, where the identity reflection/transmission coefficients Ur/t = IN1 are adopted and the numerical ergoric rate is calculated based on the 10000 independent channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It is shown that the approximated ergodic rate is accurate to the actual ergodic rate for different N1, and U2 achieves higher ergodic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' These observations are expected since 1) the average non-LoS components follow the complex Gaussian distribution with unit variance due to the law of large numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 2) U1 suffers the severer large-scale path loss than U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It is also observed that the achievable ergodic rate firstly increases with the rise of κ and gradually turns into stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is since that 1) when the κ is relatively small, increasing κ can make the LoS components prodominate and provide the stronger transmission links;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 2) when κ becomes large, the Rician channels tend to be the constant pure LoS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 22 80 60 40 20 0 20 40 60 80 Angle (deg) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 1 Azimuth Elevation 80 60 40 20 0 20 40 60 80 Angle (deg) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 1 Normalized amplitude of MLE spectrum Azimuth Elevation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The estimated azimuth and elevation angles via the MLE method with N = 20, N1 = 5, Mr = Mt = 8, K = 2, dk = 20 m, Rer,t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 bps/Hz, and PU,max = PBS,max = 35 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5 10 15 20 25 30 35 40 45 Number of iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='030202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='030203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='030204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='030205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='030206 Root CRB (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='044566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='044568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='04457 Root CRB (deg) Convergence of AO algorithm R phase T phase 5 10 15 20 25 30 35 40 45 50 Number of iterations 10-2 Root CRB (deg) 10-2 10-1 Root CRB (deg) Convergence of PDL algorithm R phase T phase Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Convergence of the proposed algorithms with the common parameters N = 20, Mr = 8, and dk = 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4, we evaluate the the maximum-number sensor deployment policy derived in Corollary 1 for both the transmission (T) and reflection (R) phases, where the actual maximum number of sensors are obtained by solving the feasible detection problem with different number of sensors for 100 independent channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Firstly, we can observe that the analytical maximum number of sensors is exactly equal to the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Also can be seen, the maximum number of sensors for T phase and R phase are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is intuitive because the maximum- number sensor deployment is only restricted by the corresponding QoS target rates, which are the same for different phases in the considered network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Moreover, when PU,max increases and Rer,t decreases, less PEs would be required to satisfy the QoS constraint, so a significant upward trend in the number of sensors can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Effectiveness of Proposed Algorithms To demonstrate the correctness of the proposed algorithms from the perspective of sensing accuracy, the MLE spectrum lines under the two-user and fixed number of sensors are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 5, where the estimated DOA angle can be determined by the exhaustive search of the highest value point of the MLE spectrum [21, Appendix E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As such, we can observe that the MLE estimates angles are (−17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='9904◦, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='9999◦) and (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='9904◦, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='9999◦), respectively, which perfectly align with the presupposed DOA angles (342◦, 30◦) and (18◦, 30◦) and validate the effectiveness of the optimized sensing waveform and reflection/transmission coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6 plots the convergence performance of the proposed algorithms, where Mt = 8, N1 = 15, K = 2, 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 5 10-2 10-1 100 Root CRB (deg) R phase: STAR-RIS-AO T phase: STAR-RIS-AO R phase: C-RIS-AO T phase: C-RIS-AO R phase: STAR-RIS-COC T phase: STAR-RIS-COC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The root CRB versus the QoS target rate with N = 20, N1 = 10, Mr = Mt = 8, K = 2, dk = 20 m, PU,max = 15 dBm, and PBS,max = 30 dBm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 35 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 40 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 45 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='06 Root CRB (deg) R phase: STAR-RIS-PDL T phase: STAR-RIS-PDL R phase: C-RIS-PDL T phase: C-RIS-PDL R phase: STAR-RIS-exhaustive-search T phase: STAR-RIS-exhaustive-search Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The root CRB versus the transmit power budget at the BS with with N = 20, Mr = 8, Mr = 12, K = 4, dk = 20 m, Rer,t = 1 bps/Hz, and PU,max = 45 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Rer,t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 bps/Hz, PU,max = 25 dBm, PBS,max = 35 dBm are adopted for AO algorithm, while Mt = 12, K = 4, Rer,t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 bps/Hz, PU,max = PBS,max = 45 dBm are set for PDL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Note that for coinciding with the practical DOA angles, we transform the radian-based root CRB into the degree unit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As can be observed, both algorithms are capable of converging to the stationary point solutions within the finite iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is because the optimal solutions at each step for solving the subproblem can be guaranteed in both two algorithms, which ensures a non-increasing trend over the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Meanwhile, we can observe that R phase achieves smaller CRB compared to the T phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It can be explained by the fact that in the R phase, the STARS reconfigure the communication signal from the users and the sensing signal from the BS as a new probing signal for estimation, which introduces more DoFs and enhance the received echo energy, thus improving the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Performance Comparison To verify the performance of the proposed scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' we consider following benchmark schemes for comparison: C-RIS-AO: In the “conventional-RIS-AO” (C-RIS-AO) scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' we consider replacing the STARS by a reflecting-only RIS and a transmitting-only RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' where the proposed AO algorithm is modified to jointly optimize the coefficients of PEs at conventional RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' transmit power at users and the receive beamforming at the BS under the two-user and fixed sensor 24 number setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For fairness comparison, each conventional RIS is assumed to possess N1 2 PEs and N2 2 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' STARS-COC: In the “STARS-communication-oriented coefficients (COC)” scheme, the re- flection/transmission coefficients of PEs is determined by maximizing the minimum ergodic rate of the user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', ΘCOC r/t = max Θr/t min k E{R[ı],k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' C-RIS-PDL: In the “C-RIS-PDL” scheme, we deploy an N 2 -element reflecting-only RIS and an N 2 -element transmitting-only RIS at the same location of STARS, where the proposed PDL algorithm is modified to jointly optimize the number of sensors, the coefficients of PEs, transmit power at users and the receive beamforming at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' STARS-exhaustive-search: In the “STARS-exhaustive-search” scheme, we independently optimize N −1 subproblems with respect to the reflection/transmission coefficients, transmit power and the beamforming at the BS, where pn = 1 (�N−1 n pn = 1) is set for the n-th subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we choose the smallest CRB as the final solution of this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 8, the proposed scheme can always achieve better sens- ing performance compared to the conventional RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This can be expected since the STARS enables the double number of elements, which introduces more spatial DoFs for supporting the communication and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7, it is also observed that the sensing performance of the considered network deteriorates with the increasing QoS target rate, and the STARS-COC scheme achieves the worst performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' An intuitive explanation for the phenomenon is: 1) both the sensing and communication relies on the coefficients setup of PEs, so when the QoS rate increases, STARS has to prioritise support for communications to satisfy the QoS constraints, which leads to a reduction in sensing performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 2) since the STARS-COC scheme only focuses on the communication performance improvement, which would inevitably sacrifice the accuracy of sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Moreover, we can observe that the CRB achieved by the STARS-COC scheme does not vary with the QoS changes from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is because the impact of QoS target rate on the sensing performance can only be realized by affecting the coefficients of PEs, which indicates that the sensing performance is independent of the QoS constraint under any fixed reflection/transmission coefficients setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' For the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 8, we can observe that the proposed PDL algorithm can achieves the comparable performance of the optimal solution via the exhaustive search, which significantly validates the equivalence of the derived ECRB expression in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 25 12 14 16 18 20 22 24 26 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='055 Root CRB (deg) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The root CRB versus the number of STARS elements with K = 4, dk = 20 m, Rer,t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 bps/Hz, and PU,max = PBS,max = 45 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 4 6 8 10 12 14 16 Number of sensors 2 4 6 8 10 12 14 Root CRB (deg) 10-3 R phase: K = 4 T phase: K = 4 R phase: K = 8 T phase: K = 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Benchmarks under 2-user setup with N = 20, Mr = 8, Mt = 12, dk = 20 m, Rer,t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='5 bps/Hz, PU,max = PBS,max = 45 dBm, and Nv = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Impact of STARS Elements Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 9 investigates the influence of STARS element number on the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Firstly, It can be observed that the root CRB monotonically decreases as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is because:1)a larger N can provide a higher array gains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' and 2) under the near-optimal allocation of number of sensors, increasing N also increases the number of sensors, which improves the sensing reception ability at the sensing array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' We can also observe an interesting result that increasing number of transmit antennas at the BS significantly reduces the root CRB, but increasing number of receive antennas at the BS has almost no effect on the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This is resulted from the bi- directional sensing-STARS architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Specifically, by exploiting this architecture, the receive antenna at the BS is only responsible for receiving the communication signal, which implies that we only need the number of receive antennas that meets the QoS requirements in the considered network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' However, increasing transmit antenna can expand the DoFs of the sensing waveform, which provides a more flexible design of the sensing waveform for supporting the sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 10, we illustrate the impact of the number of sensors under the fixed total number of STARS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It can be seen that when the number of sensors is less than 7, deploying more sensors than PEs are preferable, and when the number of sensors is larger than 7, increasing the number of sensors can hardly bring the constructive impact to the network anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It reveals that there exists a trade-off between the the number of sensors and PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In detail, with a small number of sensors, it is required to deploy more sensors to obtain sufficient echo sampling resolution to extract target information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Whereas, in the case of large number of sensors, the information loss 26 of the sensing targets caused by insufficient echo sampling resolution is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Thus, deploying more PEs to provide more DoFs for both communication and sensing becomes the dominant factor in the sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Furthermore, we observe that increasing number of users degrades the sensing performance, which is expected since the reflection/transmission coefficients of PEs needs to be more oriented towards enhancing communications for satisfying the QoS requirements of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' CONCLUSION A new STARS-empowered ISAC scheme was proposed, where a bi-directional sensing-STARS architecture was devised to support the full-space communication and sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Two effi- cient algorithms were developed to obtain the near-optimal solutions of the CRB minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' The correctness and effectiveness of proposed scheme was demonstrated by the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' It was also unveiled that: 1) STARS was capable of providing superior performance compared to the conventional RIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 2) under the unique design of bi-directional sensing-STARS architecture, the number of receive antennas at the BS has little impact on the sensing performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 3) increasing number of PEs is more appealing than sensors for sensing performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This work validated the potential of STARS in supporting full-space dual-functional transmissions, and revealed an endogenous tradeoff that how do we determine the number of sensors to be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Both of them provided useful guidance for practical system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' APPENDIX A: DERIVATION OF FIM MATRIX According (11), the element matrix in FIM matrix F[i] can be expressed as JΨ[ı]Ψ[ı] = 2 σ2 \uf8ee \uf8ef\uf8ef\uf8f0 ℜ � ∂qH [ı] ∂ϕ[ı] ∂q[ı] ∂ϕ[ı] � ℜ � ∂qH [ı] ∂ϕ[ı] ∂q[ı] ∂φ[ı] � ℜ � ∂qH [ı] ∂φ[ı] ∂q[ı] ∂ϕ[ı] � ℜ � ∂qH [ı] ∂φ[ı] ∂q[ı] ∂φ[ı] � \uf8f9 \uf8fa\uf8fa\uf8fb , (A-1) where ∂q[ı] ∂Ψ[ı][h] = vec � α[ı] � ∂b(ϕ[ı],φ[ı]) ∂Ψ[ı][h] aT(ϕ[ı], φ[ı])+b(ϕ[ı], φ[ı]) ∂aT (ϕ[ı],φ[ı]) ∂Ψ[ı][h] � Θr/tX[ı] � (1 ≤ h ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With the derivative chain rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' we have ∂ǫ(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı]) ∂Ψ[ı][h] = ǫ(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı]) ⊙ ˙ǫΨ[ı][h],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ǫ(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı]) ∈ {a(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' b(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı])},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (A-2) 27 where the n-th elements of ˙ǫΨ[ı][h] are given by ˙ǫΨ[ı][h][n] = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \uf6be¯nπ cos φ[ı] cos ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if h = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' \uf6be(−¯nπ sin φ[ı] sin ϕ[ı] + (n − 1 − Nh¯n)π cos φ[ı]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' if h = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (A-3) Let ˙QΨ[ı][h] = ∂b(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='φ[ı]) ∂Ψ[ı][h] aT (ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı])+b(ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' φ[ı]) ∂aT (ϕ[ı],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='φ[ı]) ∂Ψ[ı][h] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' we can rewrite JΨ[ı]Ψ[ı] as JΨ[ı]Ψ[ı] = 2T|α[ı]|2 σ2 ℜ \uf8eb \uf8ed \uf8ee \uf8f0Tr( ˙Qϕ[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) Tr( ˙Qϕ[ı]Θr/tRx[ı]ΘH r/t ˙QH φ[ı]) Tr( ˙Qφ[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) Tr( ˙Qφ[ı]Θr/tRx[ı]ΘH r/t ˙QH φ[ı]) \uf8f9 \uf8fb \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (A-4) where RX[ı] ≈ 1 T X[ı]XH [ı].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Similarly, let Q[ı] = b(ϕ[ı], φ[ı])aT(ϕ[ı], φ[ı]), we can obtain JΨ[ı]α[ı] = 2T σ2 ℜ \uf8eb \uf8ed \uf8ee \uf8f0˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) \uf6be˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) ˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) \uf6be˜α[ı]Tr(Q[ı]Θr/tRx[ı]ΘH r/t ˙QH ϕ[ı]) \uf8f9 \uf8fb \uf8f6 \uf8f8 , (A-5) Jα[ı]α[ı] = 2T σ2 Tr(Q[ı]Θr/tRx[ı]ΘH r/tQH [ı])I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (A-6) This completes the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' APPENDIX B: DERIVATION OF ERGODIC RATE With the results in [33, Lemma 1], the ergodic rate is approximated as E{R[ı],k} ≈ 1 2 log2 � 1 + PkE{∥hH k,SΘr/tGr∥2} σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (B-1) Substituting (1) into (B-1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' we can obatin E{∥hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/tGr∥2}=E \uf8f1 \uf8f2 \uf8f3 ������ �� κL2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S 1 + κ ˆhH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S+ � L2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S 1 + κ ˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S � Θr/t �� κL2 r 1 + κ ˆGr+ � L2 r 1 + κ ˜Gr ������� 2\uf8fc \uf8fd \uf8fe ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (a) = L2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SL2 r (1 + κ)2 � κ2E{∥ˆhH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t ˆGr∥2} + κE{∥ˆhH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t ˜Gr∥2}+ κE{∥˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t ˆGr∥2} + E{∥˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t ˜Gr∥2} � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (b)= � κ2∥ˆhH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t ˆGr∥2+κMr∥ˆhH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SΘr/t∥2+κ∥Θr/t ˆGr∥2 F+Mr∥Θr/t∥2 F � (1 + κ)2/L2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='SL2 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (B-2) where equality (a) holds because the ˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S and ˜Gr are independent of each other,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' while equality (b) holds since all the elements in ˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='S and ˜Gr follow the complex Gaussian distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With the result of (B-2), we can easily obtain the approximated ergodic rate expression in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' 28 APPENDIX C: PROOF OF COROLLARY 1 For the considered problem (16), we can readily know that achieving the maximum-number sensor deployment is tantamount to search for the minimum-dimensional reflection/transmission coefficients and the feasible transmit power that satisfy the QoS constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', find {Θmin r/t , P} (C-1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (16d), (16f), (16g), (C-2b) where Θmin r/t denotes the minimum-dimensional coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' By substituting Mr = 1 to the ergodic rate expression in Lemma 1, it is readily to show that when the reflection/transmission co- efficients of PEs are aligned to the cascaded channels ˆhH k,SΘr/tˆgr, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', the reflection/transmission coefficients derived in (24), and Pk = PU,max holds, it achieves the best communication perfor- mance with the least N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' As such, constraint (19c) can be transformed into κ2 (1 + κ)2N2 1 + 2κ + 1 (1 + κ)2N1 − (22Rer,t − 1)σ2 PU,maxL2 k,SL2 r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (C-3) Resorting the standard quadratic-root formula and the non-negativity of N1, the minimum Nmin 1 can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Thus, the maximum Nmax 2 can be derived as shown in (25) based on Nmin 1 + Nmax 2 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' APPENDIX D: DERIVATION OF EXTENDED FIM MATRIX Firstly, we can rewrite (11) as y[ı],s = vec � α[ı]Bε[ı]εT [ı]AΘr/tX[ı] � �� � q[ı] � + n[ı], (D-1) where X[ı] ∈ CN×T is the equivalent signal matrix with regarding the whole STARS as PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Similar to Appendix A, we have ∂q[ı] ∂Ψ[ı][h] = vec � α[ı]B � ∂ε[ı] ∂Ψ[ı][h]εT [ı] + ε[ı] ∂εT [ı] ∂Ψ[ı][h] � AΘr/tX[ı] � , 1 ≤ h ≤ 2, (D-2) where ∂ε[ı] ∂Ψ[ı][h] is given in (A-2) and (A-3) with 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Hence, the h-th row and v-th column element of JΨ[ı]Ψ[ı] is given by ∂qH [ı] ∂Ψ[ı][h] ∂q[ı] ∂Ψ[ı][v] = vec � α[ı]B ˙CΨ[ı][h]AΘr/tX[ı] �Hvec � α[ı]B ˙CΨ[ı][v]AΘr/tX[ı] � , = T|α[ı]|2Tr � B ˙CΨ[ı][v]AΘr/tRX[ı]ΘH r/tA ˙CH Ψ[ı][h]B � , 1 ≤ h, v ≤ 2, (D-3) 29 where ˙CΨ[ı][h] = ∂ε[ı] ∂Ψ[ı][h]εT [ı] + ε[ı] ∂εT [ı] ∂Ψ[ı][h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' In the same way, the h-th row and v-th column element of JΨ[ı]α[ı] and the h-th element on the diagonal of Jα[ı]α[ı] can be rewritten as ∂qH [ı] ∂Ψ[ı][h] ∂q[ı] ∂α[ı][v] = T(\uf6be)v−1˜α[ı]Tr � BC[ı]AΘr/tRX[ı]ΘH r/tA ˙CH Ψ[ı][h]B � , 1 ≤ h, v ≤ 2, (D-4) ∂qH [ı] ∂α[ı][h] ∂q[ı] ∂α[ı][h] = TTr � BC[ı]AΘr/tRX[ı]ΘH r/tACH [ı]B � , 1 ≤ h ≤ 2, (D-5) where C[ı] = ε[ı]εT [ı].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Substituting (D-3)–(D-5) into (A-4)–(A-6), we can obtain the extended FIM matrix in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Then, we prove the equivalence between the extended FIM matrix F[ı] and the original FIM matrix Fo [ı] under any given A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' To elaborate, let bp ∈ CN2×1 and ap ∈ CN1×1 denote the practical steering vectors at the sensors and PEs, the received echo signal can be expressed as Ep [ı] = α[ı]bp(ap)TΘr/tX[ı].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' With this in hand, it is easy to obtain the following identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' ���� ∂q[ı] ς[ı][i] ���� = ������ ∂vec �� 0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Ep [ı], 0 �� ς[ı][i] ������ = ����� ∂qp [ı] ς[ı][i] ����� , 1 ≤ i ≤ 4, (D-6) where qp [ı] = vec(Ep [ı]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Therefore, the h-th row and v-th column element of the extended FIM matrix is exactly equivalent to that of the original FIM matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=', F[ı][h, v] = ∂qH [ı] ς[ı][h] ∂q[ı] ∂ς[ı][v] = ∂(qp [ı])H ς[ı][h] ∂qp [ı] ∂ς[ı][v] = Fo [ı][h, v], 1 ≤ h, v ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' (D-7) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Letaief, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfywUm/content/2301.03436v1.pdf'} +page_content=' Chen, Y.' metadata={'source': 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+NOVEL-RESULT +Connecting theory of plasmoid-modulated reconnection to +observations of solar flares +Andrew Hillier1,* +and Shinsuke Takasao2 +1Department of Mathematics and Statistics, University of Exeter, Exeter EX4 4QE, United Kingdom, and 2Department of Earth +and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan +*Corresponding author. Email: a.s.hillier@exeter.ac.uk +(Received 19 August 2022; Revised 25 October 2022; Accepted 27 October 2022) +Abstract +The short timescale of the solar flare reconnection process has long proved to be a puzzle. Recent studies +suggest the importance of the formation of plasmoids in the reconnecting current sheet, with quantifying the +aspect ratio of the width to length of the current sheet in terms of a negative power α of the Lundquist +number, that is, S�α, being key to understanding the onset of plasmoids formation. In this paper, we make the +first application of theoretical scalings for this aspect ratio to observed flares to evaluate how plasmoid +formation may connect with observations. For three different flares that show plasmoids we find a range of α +values of α ¼ 0:26 to 0:31. The values in this small range implies that plasmoids may be forming before the +theoretically predicted critical aspect ratio (α ¼ 1=3) has been reached, potentially presenting a challenge for +the theoretical models. +Key words: instabilities; magnetic reconnection; MHD; solar flares; sun +Introduction +Solar flares, large releases of energy from the solar corona, are driven by a process called magnetic +reconnection (e.g., Priest, 2014). This is where free energy stored in the magnetic field is released +as thermal and kinetic energy through the magnetic field changing its connectivity (e.g., Yamada et al., +2010). Observations of flares show that the energy release takes place on a timescale of hours (e.g., Fletcher +et al., 2011; Shibata & Magara, 2011). However, this timescale is much shorter than the timescale to +diffuse the magnetic field in the solar corona of 106 years (e.g., Shibata & Magara, 2011). +Understanding the short timescales, compared to the diffusion time, of solar flares has presented a +theoretical challenge for many years. The first major step forward in explaining flare energy release was +the Sweet–Parker reconnection model (Parker, 1957; Sweet, 1958), a steady-state model where flows +bring magnetic field into a region of high current (called a current sheet), where it is annihilated, leading +to heating and driving jets of material ejected from the reconnection region. In this model, the +reconnection rate scales as S�1=2, where S is the Lundquist number defined as S � LVA=η ¼ τη=τA with +L the current sheet half-length, VA the Alfvén speed, η the magnetic diffusivity, and τη and τA the +diffusion and Alfvén times. With timescales for reconnection in this model scaling as the inverse root of +the Lundquist number, with S > 1012 in the solar corona, this model is still unable to explain the short +timescales of solar flares. +To bridge this gap in timescales, it was proposed that the development of plasmoids in a +reconnecting current sheet through the tearing instability (Furth et al., 1963) could play an important +© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative +Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, +provided the original article is properly cited. +Experimental Results (2022), 3, e26, 1–10 +doi:10.1017/exp.2022.23 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +role in breaking up the current sheet and driving fast reconnection (e.g., Loureiro et al., 2007; Shibata & +Tanuma, 2001). This would be either through creating a turbulent current sheet or through plasmoid +dynamics locally thinning the current sheet to scales where kinetic effects can anomalously enhance the +magnetic diffusivity (e.g., Zweibel & Yamada, 2009). Subsequently, observations have shown what +appear to be plasmoids developing in a flare current sheet (e.g., Takasao et al., 2012) supporting this +idea. Figure 1a shows the flare observed by Takasao et al. (2012) using the Atmospheric Imaging +Assembly (AIA; Lemen et al., 2012) on the Solar Dynamics Observatory with a zoomed image of the +current sheet showing the plasma blobs interpreted as plasmoids. Panel (b) of that figure gives a +schematic diagram that explains how the observations connect to the formation of plasmoids in a +reconnecting current sheet. +To understand how the tearing instability develops in a reconnecting current sheet, it is standard +practice to rescale the growth rates and wavenumbers to calculate them in terms of the half-length of the +current sheet and not the half-width (which is used normally when calculating the growth rate of the +instability; e.g., Furth et al., 1963). To do this, the aspect ratio a=L (with a the half-width of the current +sheet and L the half-length of the current sheet) is given to scale as S�α (e.g., MacTaggart, 2020). After +performing this rescaling, the instability is often known as the plasmoid instability. In a Sweet–Parker +current sheet (α ¼ 1=2), the maximum growth rate of the instability scales as S1=4 (e.g., Loureiro et al., +2007). However, as explained by Pucci and Velli (2014), the current sheet would become unstable to +plasmoid formation before it has thinned/stretched to the Sweet–Parker aspect ratio. They proposed that +α ¼ 1=3 is the correct scaling to expect as this is the aspect ratio where the tearing timescale becomes equal +to the timescale for the ejection of a plasmoid (i.e., the Alfvén time). +The aspect ratio of � S�1=3 has been found to be important for triggering the onset of plasmoid +formation in a number of analytical and numerical studies (e.g., Comisso et al., 2016; Huang et al., 2017), +but the question still remains as to what is happening in observed astrophysical systems. In this paper, we +will investigate whether the magnetic reconnection behind observed solar flares can be understood +through the plasmoid instability paradigm. We use some simple scaling laws to investigate the value of α +required to explain the development of plasmoids that have been observed in solar flares, and connect +these values with the current theoretical understanding. +Scaling laws and their application to observations +For the tearing instability, there are well-established derivations of the most unstable mode of the +instability (e.g., Tajima & Shibata, 2002). These give +1.5× 10 cm +9 +(a) +Solar +surface +Plasmoids +arrows: plasma flows +94Å +193Å +~3” +(b) +Figure 1. Image of a solar flare observed on August 18, 2010 with the Atmospheric Imaging Assembly (panel [a]). The zoomed +image shows plasma blobs formed in the plasma sheet. Panel (b) presents a schematic diagram of these observations where +the plasma sheet is understood as a current sheet with the plasma blobs interpreted as plasmoids. +2 +Andrew Hillier and Shinsuke Takasao +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +a +VA +σ max ¼ C1S∗�1=2, +(1) +akmax ¼ C2S∗�1=4, +(2) +where kmax is the most unstable wavenumber of the system, σ max is the corresponding growth rate, +S∗ � aVA=η (the Lundquist number defined by the current sheet half-width a), and C1 and C2 are +constants of order 1. For example, for a Harris current sheet, they become C1≈0:62 and C2≈1:36. +Following the arguments for the plasmoid instability (e.g., MacTaggart, 2020), we set that the half- +width and half-length of the current sheet are connected by a=L ¼ S�α. We can rescale the maximum +growth rate and the most unstable wavelength to be in terms of S and not S∗. For the growth rate, we have +σ max ¼ VA +a C1S∗�1=2 ¼ C1 +VA +L +L +aS�1=2 a +L +� ��1=2 +¼ C1 +VA +L S 3α�1 +ð +Þ=2: +(3) +For the corresponding wave number, we have +kmax ¼ 1 +aC2S∗�1=4 ¼ C2 +1 +L +L +aS�1=4 a +L +� ��1=4 +¼ C2 +1 +LS 5α�1 +ð +Þ=4: +(4) +Using Equation (4), and taking that C2 ≈ 1, we can rearrange to solve for α, that is, +α ¼ 4 +5 +log 10 kmaxL +ð +Þ +log 10 S +ð Þ +þ1 +4 +� +� +: +(5) +It is these relations that we will apply to the flare observations to determine the value of α. +The hypothesis we will test is whether, as expected from reconnection theory, solar flare observations +present a consistent value of α. If found, this would provide further evidence of the importance of the +tearing instability in solar flare reconnection. +Application to observed plasmoids +In this section, we analyze the plasmoids observed in three separate flares (displaying notably different +scales). The data for the three flares we use are presented in Takasao et al. (2012), Milligan et al. (2010), +and Patel et al. (2020), respectively. Below, we detail the key characteristics of these different observations, +and then summarize the key quantities in Table 1, where the estimated α value is also presented. +The observations of Takasao et al. (2012) show the development of a long, thin current sheet, in which +plasma blobs develop and are ejected. The outflow velocity (a good proxy for the Alfvén speed, e.g., +Parker, 1957) was observed to be 220–460 km/s. Here, we take the largest value 4:6�105 m/s to be the +Table 1. Key characteristics of the current sheet and plasmoids including the half-length of the current sheet, the +estimated Alfvén speed, the characteristic plasmoid size, and the Lundquist number and α value calculated from these +measurements for the different observed flares +Obs. date +Current sheet +half-length (m) +Alfvén speed +(m/s) +Plasmoid +size (m) +S +α +August 18, 2010a +7 � 106 +4.6 � 105 +2.9 � 106 +3.2 � 1012 +0.28 +January 25, 2007b +4.4 � 107 to 1.1 � 108 4.6 � 105 to 1.3 � 106 +2.5 � 107 +2.2 � 1013 to 1.4 � 1014 +0.26–0.28 +September 10, 2017c +5.4 � 107 +4.3 � 105 +5.65 � 106 +2.3 � 1013 +0.31 +aData extracted from Takasao et al. (2012). +bData extracted from Milligan et al. (2010). +cData extracted from Patel et al. (2020). +Experimental Results +3 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +representative value of the Alfvén speed. The half-length of the current sheet was observed to be at least +7�106 m. The typical size of the observed plasma blobs, interpreted to be plasmoids due to their +movement along the current sheet, was 2:9�106 m (the interpretation as plasmoids was further +supported by radio observations that detected the signatures of electron acceleration associated with +their movement [Takasao et al., 2016]). This implies a wavenumber of 3�10�6/m. Taking the tempera- +ture to be 106 K, we expect the magnetic diffusivity to be 1 m2/s, meaning that we have S ¼ 3:2�1012. +The observations of Milligan et al. (2010) show a plasmoid observed in hard X-ray by RHESSI (Reuven +Ramaty High Energy Solar Spectroscopic Imager). From the observations presented in Figure 3 of +Milligan et al. (2010), it seems reasonable to take a plasmoid (Source A in that figure) to be of size +� 35 arcsec (� 2:5�107 m), which becomes the wavelength used for our analysis. We can also make an +estimate of the half-length of the current sheet. First, we can consider as a lower estimate the distance +between the center of the two hard X-ray sources (again as shown in Figure 3 of Milligan et al., 2010), +which is � 60 arcsec (� 4:4�107 m). Alternatively, we can take as an upper limit of the length the +distance between the flare arcade and the inner edge of the COR2 Coronagraph on the STEREO B satellite +(Kaiser et al., 2008), which is � 300 arcsec, which gives an upper estimate of the half-length of 150 arcsec +(� 1:1�108 m). From these observations, it is not possible to directly estimate the Alfvén speed. We can +give an upper estimate for the Alfvén speed from the CME speed, which is 1:3�106 m/s (the SOHO/ +LASCO CME Catalog; Yashiro et al., 2004), although we also look at the effect of using the slower speed of +4:6�105 m/s as found in the study of Takasao et al. (2012). Again, we take η ¼ 1 m2/s. These values give a +range of Lundquist numbers between S ¼ 2:2�1013 and 1:4�1014. +The observations of Patel et al. (2020) show an evolving current sheet, which produces many +plasmoids of varying size traveling at varying speeds. These plasmoids are observed in the lower solar +corona by AIA and further out by COR2. To distill the observations into the key set of numbers we +require, we focus on the observed plasmoids as seen by AIA. Taking the Alfvén speed as the approximate +upper limit of the observed plasmoid velocity, we use a value of 4:3�105 m/s. The observed plasmoid +width of � 5:65�106 m is used as the wavelength of the tearing instability. The estimate for the half- +length of the current sheet can be estimated from their Figure 11, where the stagnation height of the +plasmoids can be determined. Subtracting the height of the flare arcade leads to a height of � 75 arcsec +(5:4�107 m). Again, we take η ¼ 1 m2/s. +The α values found for these observations are displayed in the rightmost column of Table 1, giving a +range between 0:26 and 0:31. Although there is naturally some uncertainty in the number of the +parameters used to calculate the α values, and this is potentially responsible for some of the spread +observed, in general the widely different scales of the observations are presenting α values that are in a +relatively small range. Considering the range of Lundquist numbers by one to two orders of magnitude, as +well as the order of magnitude range in plasmoid size and current sheet lengths, this provides evidence +that the theoretical understanding of how plasmoids are formed in flaring current sheets is consistent +with the observations. +To highlight how robust the values calculated for α actually are, we can look at what happens if we take +into account some of the uncertainties in our estimates for various quantities and apply these to the α +value for the flare observed by Takasao et al. (2012). First, we can take C1 to be 1:36. In this case, we find a +value of α ¼ 0:27. Alternatively, we can assume that the half-length of our current sheet has been +underestimated due to projection effects. Making L to be 30% larger (and with it S to be 1:32 larger due to +the effect of the projection effects on the estimate of the Alfvén speed), we find α ¼ 0:28. Finally, for the +diffusion, we had assumed that the temperature of the medium was 106 K, but η∝T�3=2 and the +temperature in the current sheet is likely to be up to one order of magnitude hotter than the temperature +assumed here. Taking T ¼ 107 K, we find α ¼ 0:27. Further uncertainties still exist. The question of +accurate determination of the current sheet length (where the observed length may also contain the +reconnection jets as well as the current sheet) also leads to uncertainty. However, for the flare observed by +Takasao et al. (2012), reducing the length by a factor of 2 results in a small reduction of α to α ¼ 0:26. +Moreover, the observed plasmoid size may be overestimated as the observations are likely to show the +later state of a plasmoid once it has accumulated more flux. Taking a plasmoid to initially be only half the +4 +Andrew Hillier and Shinsuke Takasao +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +observed size results in α ¼ 0:29. Therefore, we can conclude that reasonable levels of uncertainty do not +result in large variation in the calculated value of α. +Determining an α value has a particular consequence, and it implicitly makes a prediction for the +thickness of the flare current sheet. Looking at the value obtained for the flare studied by Takasao et al. +(2012), the calculated value of α implies that the actual thickness of the flare current sheet is +a ≈ 2:2�103 m. This is approximately 2.5 orders of magnitude smaller than the observed half-thickness +of the plasma sheet in the observations of 7�105 m (Takasao et al., 2012). However, this can be easily +explained if the current sheet is not exactly aligned with the line of sight, meaning that its depth is +projected to look like its width, or as a result of a thermal halo formed around the current sheet through +heat conduction (Forbes & Malherbe, 1991; Takasao et al., 2015; Yokoyama & Shibata, 2001). In this case, +the measured value of α would imply a timescale for the growth of the tearing instability of ≈150 s. This is +much shorter than the timeframe of the flare observations, meaning that for the α value we find it would +be reasonable for plasmoids to develop during the course of the observations. +Discussion +There have been many attempts in recent years to understand the connection between the fast, bursty +reconnection observed in astrophysical systems through the plasmoid instability. These studies have +made great progress through analytic theory and through numerical modeling. In this paper, we have +extended these studies by looking at observational data of three solar flares and showing that it is +consistent with an aspect ratio of a=L ¼ S�0:26 to S�0:31. +The key finding for the solar flares we have studied is that the value for α (α ¼ 0:26 to 0:31) is relatively +close to the theoretical predictions of Pucci and Velli (2014) (e.g., α ¼ 1=3). However, it is important to +note that this small difference is in fact somewhat difficult to reconcile through observational error due to +the lack of sensitivity of α to reasonable estimates of the errors in the parameters used. There will always +be some uncertainty with the value of α that cannot be quantified; for example, Huang et al. (2017) found +in their numerical simulations that the wavenumber that ultimately grew was a factor of 3 to 6 smaller +than the most unstable mode. If we take that the measured plasmoids are from a wavenumber six times +smaller than the most unstable mode, we then find α ¼ 0:31 to 0:35, that is, it gives the predicted scaling of +α ¼ 1=3 by Pucci and Velli (2014), but it is not possible to prove with current observations that this +process is happening in solar flare reconnection. +If we consider how a plasmoid may be formed when the timescale for its growth is longer than the +expected timescale of the ejection, there may be some aspect of the reconnection flow that allows this to +happen. For example, this may be occurring through plasmoid formation around the stagnation point of +the flows into and out of the current sheet, allowing the first plasmoid to form in the current sheet where it +takes significantly longer to eject allowing them to grow. This is an area for future investigation. +Magnetic reconnection is an important physical process in many astrophysical and space systems for +driving the quick release of energy stored in magnetic fields. Therefore, quantifying how observed +magnetic reconnection fits into current models of magnetic reconnection is an important topic of +research. The methods laid out in this paper should be applicable for any observed reconnection region +where the plasmoids have been observed. Finding further observations, in any system, to see if a similar +value of α is consistently found would be an important future step. +Acknowledgment. A.H. would like to acknowledge the lectures of Dr. David MacTaggart at the Advanced Topics in MHD +Summer School held in CISM, Udine, which led to the idea for this paper. +Data availability statement. The data used in this paper are available already in other published works. +Author contributions. A.H. designed the study and wrote the manuscript. S.T. provided feedback and guidance on the flare +observations and application of the theory. +Funding statement. A.H. is supported by STFC Research Grant No. ST/V000659/1. S.T. is supported by JSPS KAKENHI +Grant Nos. JP22H00134, JP22K14074, and JP21H04487. +Experimental Results +5 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +Conflict of interest. Both authors have no conflicts of interest to declare. +References +Comisso, L., Lingam, M., Huang, Y. M., & Bhattacharjee, A. (2016). General theory of the plasmoid instability. Physics of +Plasmas, 23, 100702. +Fletcher, L., Dennis, B. R., Hudson, H. S., Krucker, S., Phillips, K., Veronig, A., Battaglia, M., Bone, L., Caspi, A., Chen, Q., +Gallagher, P., Grigis, P. T., Ji, H., Liu, W., Milligan, R. O., & Temmer, M. (2011). An observational overview of solar flares. +Space Science Reviews, 159, 19–106. +Forbes, T. G., & Malherbe, J. M. (1991). 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Avalon Publishing. +Takasao, S., Asai, A., Isobe, H., & Shibata, K. (2012). Simultaneous observation of reconnection inflow and outflow associated +with the 2010 August 18 solar flare. The Astrophysical Journal Letters, 745, L6. +Takasao, S., Asai, A., Isobe, H., & Shibata, K. (2016). Observational evidence of particle acceleration associated with plasmoid +motions. The Astrophysical Journal, 828, 103. +Takasao, S., Matsumoto, T., Nakamura, N., & Shibata, K. (2015). Magnetohydrodynamic shocks in and above post-flare +loops: Two-dimensional simulation and a simplified model. The Astrophysical Journal, 805, 135. +Yamada, M., Kulsrud, R., & Ji, H. (2010). Magnetic reconnection. Reviews of Modern Physics, 82, 603–664. +Yashiro, S., Gopalswamy, N., Michalek, G., St. Cyr, O. C., Plunkett, S. P., Rich, N. B., & Howard, R. A. (2004). A catalog of +white light coronal mass ejections observed by the SOHO spacecraft. Journal of Geophysical Research: Space Physics, 109, +A07105. +Yokoyama, T., & Shibata, K. (2001). Magnetohydrodynamic simulation of a solar flare with chromospheric evaporation effect +based on the magnetic reconnection model. The Astrophysical Journal, 549, 1160–1174. +Zweibel, E. G., & Yamada, M. (2009). Magnetic reconnection in astrophysical and laboratory plasmas. Annual Review of +Astronomy and Astrophysics, 47, 291–332. +Cite this article: Hillier A, Takasao S (2022). Connecting theory of plasmoid-modulated reconnection to observations of +solar flares. Experimental Results, 3, e26, 1–10. https://doi.org/10.1017/exp.2022.23 +6 +Andrew Hillier and Shinsuke Takasao +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +Peer Reviews +Reviewing editor: Prof. Stefano Camera, PhD +Universita degli Studi di Torino, Physics, Via Pietro Giuria, 1, Torino, Italy, 10124 +Minor revisions requested. +doi:10.1017/exp.2022.23.pr1 +Review 1: Connecting Theory of Plasmoid-modulated Reconnection to Observations of Solar +Flares +Reviewer: Dr. PengFei Chen +Date of review: 22 September 2022 +© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of +the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, +distribution and reproduction, provided the original article is properly cited. +Conflict of interest statement. Reviewer declares none. +Comment +Comments to the Author: In this manuscript the authors tried to connect observational parameters of +solar flares to the scaling law of magnetic reconnection, and it seems that the scaling law is quite similar to +what proposed earlier via numerical simulations. If the topic fits into the theme of the journal, I’d like to +recommend it to be published after revisions. +Major issues: +1. Personally I tend to think that the 1.5*10^9 cm in Fig. 1 is not the length of the current sheet. It +might consists of a much shorter current sheet in the middle and long bidirectional outflows both upward +and downward. In this case L might be much smaller. Its effect on alpha should also be discussed on p.4. +2. In this paper, the magnetic diffusivity is taken to be the classical value. However, it has been +proposed that anomalous resistivity is required for MR to occur, and particle simulations indeed showed +that the diffusivity is enhanced by ~6 orders of magnitude during reconnection. In this case, S would be +substantially smaller, and alpha would be much larger according to Eq. (5). +Minor issues: +1. In several places, “magnetic diffusion” should be “magnetic diffusivity”; +2. In several places, “Arcsec” should be “arcsec”; +3. P. 1: “is unable to still” --> “is still unable to”; +4. Figure1 panel (a) --> Figure 1(a); +5. table 1 -->Table 1; +6. The X8.2-class solar flare on 2017-09-10 can be used as the 4th example if possible. +Score Card +Presentation +3.7 +/5 +Is the article written in clear and proper English? (30%) +● +4/5 +Is the data presented in the most useful manner? (40%) +● +4/5 +Does the paper cite relevant and related articles appropriately? (30%) +● +3/5 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +Context +3.8 +/5 +Does the title suitably represent the article? (25%) +● +5/5 +Does the abstract correctly embody the content of the article? (25%) +● +4/5 +Does the introduction give appropriate context? (25%) +● +4/5 +Is the objective of the experiment clearly defined? (25%) +● +2/5 +Analysis +3.8 +/5 +Does the discussion adequately interpret the results presented? (40%) +● +4/5 +Is the conclusion consistent with the results and discussion? (40%) +● +4/5 +Are the limitations of the experiment as well as the contributions of +the experiment clearly outlined? (20%) +● +3/5 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +doi:10.1017/exp.2022.23.pr2 +Review 2: Connecting Theory of Plasmoid-modulated Reconnection to Observations of Solar +Flares +Reviewer: Dr. Yi-Min Huang +Princeton University, United States +Date of review: 26 August 2022 +© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of +the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, +distribution and reproduction, provided the original article is properly cited. +Conflict of interest statement. Reviewer declares none. +Comment +Comments to the Author: The authors’ approach of using the observed sizes of plasmoids and the +rescaled fastest-growing wavenumber to infer the aspect ratio of the current sheet is quite ingenious. The +authors have also discussed several possible uncertainties in their method. In addition to those +uncertainties, I would like to point out another cause of errors that the authors have not addressed. +Simulations have shown that small plasmoids tend to coalesce and form larger plasmoids. Moreover, +plasmoid size grows over time as they chew magnetic flux through reconnection. Plasmoids could be +smaller than the observed ones when they were first born but only become visible when they grow larger +through coalescence or reconnection. Therefore, the observed plasmoid sizes may be viewed as an upper +bound for the plasmoid instability wavelength. Future solar imagers with higher resolution may help to +resolve this problem. +Minor issues: +(a) Page 2, last paragraph: The reference Comisso et al. (2016) is a purely analytic paper and contains +no simulations. It is true that S^-1/3 appears in the aspect ratio derived by Comisso et al., but there is an +additional logarithmic factor that depends on S and the initial perturbation amplitude. +(b) Second paragraph of Discussion, line 7: “siz” appears to be a typo. +Score Card +Presentation +5.0 +/5 +Is the article written in clear and proper English? (30%) +● +5/5 +Is the data presented in the most useful manner? (40%) +● +5/5 +Does the paper cite relevant and related articles appropriately? (30%) +● +5/5 +Context +5.0 +/5 +Does the title suitably represent the article? (25%) +● +5/5 +Does the abstract correctly embody the content of the article? (25%) +● +5/5 +Does the introduction give appropriate context? (25%) +● +5/5 +Is the objective of the experiment clearly defined? (25%) +● +5/5 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + +Analysis +4.8 +/5 +Does the discussion adequately interpret the results presented? (40%) +● +5/5 +Is the conclusion consistent with the results and discussion? (40%) +● +5/5 +Are the limitations of the experiment as well as the contributions of +the experiment clearly outlined? (20%) +● +4/5 +https://doi.org/10.1017/exp.2022.23 Published online by Cambridge University Press + diff --git a/m9E1T4oBgHgl3EQfhQSd/content/tmp_files/load_file.txt b/m9E1T4oBgHgl3EQfhQSd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18d56be6b2fc6748f41dfa33764c2433723d8771 --- /dev/null +++ b/m9E1T4oBgHgl3EQfhQSd/content/tmp_files/load_file.txt @@ -0,0 +1,631 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf,len=630 +page_content='PHYSICS AND ASTRONOMY NOVEL-RESULT Connecting theory of plasmoid-modulated reconnection to observations of solar flares Andrew Hillier1,* and Shinsuke Takasao2 1Department of Mathematics and Statistics, University of Exeter, Exeter EX4 4QE, United Kingdom, and 2Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='hillier@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='uk (Received 19 August 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Revised 25 October 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Accepted 27 October 2022) Abstract The short timescale of the solar flare reconnection process has long proved to be a puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Recent studies suggest the importance of the formation of plasmoids in the reconnecting current sheet, with quantifying the aspect ratio of the width to length of the current sheet in terms of a negative power α of the Lundquist number, that is, S�α, being key to understanding the onset of plasmoids formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this paper, we make the first application of theoretical scalings for this aspect ratio to observed flares to evaluate how plasmoid formation may connect with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' For three different flares that show plasmoids we find a range of α values of α ¼ 0:26 to 0:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The values in this small range implies that plasmoids may be forming before the theoretically predicted critical aspect ratio (α ¼ 1=3) has been reached, potentially presenting a challenge for the theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Key words: instabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' magnetic reconnection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' MHD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' solar flares;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' sun Introduction Solar flares, large releases of energy from the solar corona, are driven by a process called magnetic reconnection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Priest, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is where free energy stored in the magnetic field is released as thermal and kinetic energy through the magnetic field changing its connectivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Observations of flares show that the energy release takes place on a timescale of hours (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Shibata & Magara, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, this timescale is much shorter than the timescale to diffuse the magnetic field in the solar corona of 106 years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Shibata & Magara, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Understanding the short timescales, compared to the diffusion time, of solar flares has presented a theoretical challenge for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The first major step forward in explaining flare energy release was the Sweet–Parker reconnection model (Parker, 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Sweet, 1958), a steady-state model where flows bring magnetic field into a region of high current (called a current sheet), where it is annihilated, leading to heating and driving jets of material ejected from the reconnection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this model, the reconnection rate scales as S�1=2, where S is the Lundquist number defined as S � LVA=η ¼ τη=τA with L the current sheet half-length, VA the Alfvén speed, η the magnetic diffusivity, and τη and τA the diffusion and Alfvén times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' With timescales for reconnection in this model scaling as the inverse root of the Lundquist number, with S > 1012 in the solar corona, this model is still unable to explain the short timescales of solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' To bridge this gap in timescales, it was proposed that the development of plasmoids in a reconnecting current sheet through the tearing instability (Furth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 1963) could play an important © The Author(s), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Published by Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Experimental Results (2022), 3, e26, 1–10 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press role in breaking up the current sheet and driving fast reconnection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Loureiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Shibata & Tanuma, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This would be either through creating a turbulent current sheet or through plasmoid dynamics locally thinning the current sheet to scales where kinetic effects can anomalously enhance the magnetic diffusivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Zweibel & Yamada, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Subsequently, observations have shown what appear to be plasmoids developing in a flare current sheet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2012) supporting this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Figure 1a shows the flare observed by Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012) using the Atmospheric Imaging Assembly (AIA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2012) on the Solar Dynamics Observatory with a zoomed image of the current sheet showing the plasma blobs interpreted as plasmoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Panel (b) of that figure gives a schematic diagram that explains how the observations connect to the formation of plasmoids in a reconnecting current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' To understand how the tearing instability develops in a reconnecting current sheet, it is standard practice to rescale the growth rates and wavenumbers to calculate them in terms of the half-length of the current sheet and not the half-width (which is used normally when calculating the growth rate of the instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Furth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' To do this, the aspect ratio a=L (with a the half-width of the current sheet and L the half-length of the current sheet) is given to scale as S�α (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', MacTaggart, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' After performing this rescaling, the instability is often known as the plasmoid instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In a Sweet–Parker current sheet (α ¼ 1=2), the maximum growth rate of the instability scales as S1=4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Loureiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, as explained by Pucci and Velli (2014), the current sheet would become unstable to plasmoid formation before it has thinned/stretched to the Sweet–Parker aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' They proposed that α ¼ 1=3 is the correct scaling to expect as this is the aspect ratio where the tearing timescale becomes equal to the timescale for the ejection of a plasmoid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', the Alfvén time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The aspect ratio of � S�1=3 has been found to be important for triggering the onset of plasmoid formation in a number of analytical and numerical studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Comisso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2017), but the question still remains as to what is happening in observed astrophysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this paper, we will investigate whether the magnetic reconnection behind observed solar flares can be understood through the plasmoid instability paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' We use some simple scaling laws to investigate the value of α required to explain the development of plasmoids that have been observed in solar flares, and connect these values with the current theoretical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Scaling laws and their application to observations For the tearing instability, there are well-established derivations of the most unstable mode of the instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Tajima & Shibata, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' These give 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='5× 10 cm 9 (a) Solar surface Plasmoids arrows: plasma flows 94Å 193Å ~3” (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Image of a solar flare observed on August 18, 2010 with the Atmospheric Imaging Assembly (panel [a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The zoomed image shows plasma blobs formed in the plasma sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Panel (b) presents a schematic diagram of these observations where the plasma sheet is understood as a current sheet with the plasma blobs interpreted as plasmoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 2 Andrew Hillier and Shinsuke Takasao https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press a VA σ max ¼ C1S∗�1=2, (1) akmax ¼ C2S∗�1=4, (2) where kmax is the most unstable wavenumber of the system, σ max is the corresponding growth rate, S∗ � aVA=η (the Lundquist number defined by the current sheet half-width a), and C1 and C2 are constants of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' For example, for a Harris current sheet, they become C1≈0:62 and C2≈1:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Following the arguments for the plasmoid instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', MacTaggart, 2020), we set that the half- width and half-length of the current sheet are connected by a=L ¼ S�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' We can rescale the maximum growth rate and the most unstable wavelength to be in terms of S and not S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' For the growth rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' we have σ max ¼ VA a C1S∗�1=2 ¼ C1 VA L L aS�1=2 a L � ��1=2 ¼ C1 VA L S 3α�1 ð Þ=2: (3) For the corresponding wave number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' we have kmax ¼ 1 aC2S∗�1=4 ¼ C2 1 L L aS�1=4 a L � ��1=4 ¼ C2 1 LS 5α�1 ð Þ=4: (4) Using Equation (4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' and taking that C2 ≈ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' we can rearrange to solve for α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' α ¼ 4 5 log 10 kmaxL ð Þ log 10 S ð Þ þ1 4 � � : (5) It is these relations that we will apply to the flare observations to determine the value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The hypothesis we will test is whether, as expected from reconnection theory, solar flare observations present a consistent value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' If found, this would provide further evidence of the importance of the tearing instability in solar flare reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Application to observed plasmoids In this section, we analyze the plasmoids observed in three separate flares (displaying notably different scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The data for the three flares we use are presented in Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012), Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2010), and Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2020), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Below, we detail the key characteristics of these different observations, and then summarize the key quantities in Table 1, where the estimated α value is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The observations of Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012) show the development of a long, thin current sheet, in which plasma blobs develop and are ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The outflow velocity (a good proxy for the Alfvén speed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Parker, 1957) was observed to be 220–460 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Here, we take the largest value 4:6�105 m/s to be the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Key characteristics of the current sheet and plasmoids including the half-length of the current sheet, the estimated Alfvén speed, the characteristic plasmoid size, and the Lundquist number and α value calculated from these measurements for the different observed flares Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' date Current sheet half-length (m) Alfvén speed (m/s) Plasmoid size (m) S α August 18, 2010a 7 � 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='6 � 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='9 � 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2 � 1012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='28 January 25, 2007b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='4 � 107 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1 � 108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='6 � 105 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='3 � 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='5 � 107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2 � 1013 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='4 � 1014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='26–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='28 September 10, 2017c 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='4 � 107 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='3 � 105 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='65 � 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='3 � 1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='31 aData extracted from Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' bData extracted from Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' cData extracted from Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Experimental Results 3 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press representative value of the Alfvén speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The half-length of the current sheet was observed to be at least 7�106 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The typical size of the observed plasma blobs, interpreted to be plasmoids due to their movement along the current sheet, was 2:9�106 m (the interpretation as plasmoids was further supported by radio observations that detected the signatures of electron acceleration associated with their movement [Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This implies a wavenumber of 3�10�6/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Taking the tempera- ture to be 106 K, we expect the magnetic diffusivity to be 1 m2/s, meaning that we have S ¼ 3:2�1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The observations of Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2010) show a plasmoid observed in hard X-ray by RHESSI (Reuven Ramaty High Energy Solar Spectroscopic Imager).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' From the observations presented in Figure 3 of Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2010), it seems reasonable to take a plasmoid (Source A in that figure) to be of size � 35 arcsec (� 2:5�107 m), which becomes the wavelength used for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' We can also make an estimate of the half-length of the current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' First, we can consider as a lower estimate the distance between the center of the two hard X-ray sources (again as shown in Figure 3 of Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2010), which is � 60 arcsec (� 4:4�107 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Alternatively, we can take as an upper limit of the length the distance between the flare arcade and the inner edge of the COR2 Coronagraph on the STEREO B satellite (Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2008), which is � 300 arcsec, which gives an upper estimate of the half-length of 150 arcsec (� 1:1�108 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' From these observations, it is not possible to directly estimate the Alfvén speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' We can give an upper estimate for the Alfvén speed from the CME speed, which is 1:3�106 m/s (the SOHO/ LASCO CME Catalog;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yashiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2004), although we also look at the effect of using the slower speed of 4:6�105 m/s as found in the study of Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Again, we take η ¼ 1 m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' These values give a range of Lundquist numbers between S ¼ 2:2�1013 and 1:4�1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The observations of Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2020) show an evolving current sheet, which produces many plasmoids of varying size traveling at varying speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' These plasmoids are observed in the lower solar corona by AIA and further out by COR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' To distill the observations into the key set of numbers we require, we focus on the observed plasmoids as seen by AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Taking the Alfvén speed as the approximate upper limit of the observed plasmoid velocity, we use a value of 4:3�105 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The observed plasmoid width of � 5:65�106 m is used as the wavelength of the tearing instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The estimate for the half- length of the current sheet can be estimated from their Figure 11, where the stagnation height of the plasmoids can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Subtracting the height of the flare arcade leads to a height of � 75 arcsec (5:4�107 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Again, we take η ¼ 1 m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The α values found for these observations are displayed in the rightmost column of Table 1, giving a range between 0:26 and 0:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Although there is naturally some uncertainty in the number of the parameters used to calculate the α values, and this is potentially responsible for some of the spread observed, in general the widely different scales of the observations are presenting α values that are in a relatively small range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Considering the range of Lundquist numbers by one to two orders of magnitude, as well as the order of magnitude range in plasmoid size and current sheet lengths, this provides evidence that the theoretical understanding of how plasmoids are formed in flaring current sheets is consistent with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' To highlight how robust the values calculated for α actually are, we can look at what happens if we take into account some of the uncertainties in our estimates for various quantities and apply these to the α value for the flare observed by Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' First, we can take C1 to be 1:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this case, we find a value of α ¼ 0:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Alternatively, we can assume that the half-length of our current sheet has been underestimated due to projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Making L to be 30% larger (and with it S to be 1:32 larger due to the effect of the projection effects on the estimate of the Alfvén speed), we find α ¼ 0:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Finally, for the diffusion, we had assumed that the temperature of the medium was 106 K, but η∝T�3=2 and the temperature in the current sheet is likely to be up to one order of magnitude hotter than the temperature assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Taking T ¼ 107 K, we find α ¼ 0:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Further uncertainties still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The question of accurate determination of the current sheet length (where the observed length may also contain the reconnection jets as well as the current sheet) also leads to uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, for the flare observed by Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012), reducing the length by a factor of 2 results in a small reduction of α to α ¼ 0:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Moreover, the observed plasmoid size may be overestimated as the observations are likely to show the later state of a plasmoid once it has accumulated more flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Taking a plasmoid to initially be only half the 4 Andrew Hillier and Shinsuke Takasao https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press observed size results in α ¼ 0:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Therefore, we can conclude that reasonable levels of uncertainty do not result in large variation in the calculated value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Determining an α value has a particular consequence, and it implicitly makes a prediction for the thickness of the flare current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Looking at the value obtained for the flare studied by Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012), the calculated value of α implies that the actual thickness of the flare current sheet is a ≈ 2:2�103 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='5 orders of magnitude smaller than the observed half-thickness of the plasma sheet in the observations of 7�105 m (Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, this can be easily explained if the current sheet is not exactly aligned with the line of sight, meaning that its depth is projected to look like its width, or as a result of a thermal halo formed around the current sheet through heat conduction (Forbes & Malherbe, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Takasao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yokoyama & Shibata, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this case, the measured value of α would imply a timescale for the growth of the tearing instability of ≈150 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is much shorter than the timeframe of the flare observations, meaning that for the α value we find it would be reasonable for plasmoids to develop during the course of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Discussion There have been many attempts in recent years to understand the connection between the fast, bursty reconnection observed in astrophysical systems through the plasmoid instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' These studies have made great progress through analytic theory and through numerical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this paper, we have extended these studies by looking at observational data of three solar flares and showing that it is consistent with an aspect ratio of a=L ¼ S�0:26 to S�0:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The key finding for the solar flares we have studied is that the value for α (α ¼ 0:26 to 0:31) is relatively close to the theoretical predictions of Pucci and Velli (2014) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', α ¼ 1=3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, it is important to note that this small difference is in fact somewhat difficult to reconcile through observational error due to the lack of sensitivity of α to reasonable estimates of the errors in the parameters used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' There will always be some uncertainty with the value of α that cannot be quantified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' for example, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2017) found in their numerical simulations that the wavenumber that ultimately grew was a factor of 3 to 6 smaller than the most unstable mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' If we take that the measured plasmoids are from a wavenumber six times smaller than the most unstable mode, we then find α ¼ 0:31 to 0:35, that is, it gives the predicted scaling of α ¼ 1=3 by Pucci and Velli (2014), but it is not possible to prove with current observations that this process is happening in solar flare reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' If we consider how a plasmoid may be formed when the timescale for its growth is longer than the expected timescale of the ejection, there may be some aspect of the reconnection flow that allows this to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' For example, this may be occurring through plasmoid formation around the stagnation point of the flows into and out of the current sheet, allowing the first plasmoid to form in the current sheet where it takes significantly longer to eject allowing them to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is an area for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Magnetic reconnection is an important physical process in many astrophysical and space systems for driving the quick release of energy stored in magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Therefore, quantifying how observed magnetic reconnection fits into current models of magnetic reconnection is an important topic of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The methods laid out in this paper should be applicable for any observed reconnection region where the plasmoids have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Finding further observations, in any system, to see if a similar value of α is consistently found would be an important future step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' would like to acknowledge the lectures of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' David MacTaggart at the Advanced Topics in MHD Summer School held in CISM, Udine, which led to the idea for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Data availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The data used in this paper are available already in other published works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Author contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' designed the study and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' provided feedback and guidance on the flare observations and application of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Funding statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' is supported by STFC Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' ST/V000659/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' is supported by JSPS KAKENHI Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' JP22H00134, JP22K14074, and JP21H04487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Experimental Results 5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press Conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Both authors have no conflicts of interest to declare.' metadata={'source': 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+page_content=', Asai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Isobe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Shibata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Simultaneous observation of reconnection inflow and outflow associated with the 2010 August 18 solar flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The Astrophysical Journal Letters, 745, L6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Takasao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Asai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Isobe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Shibata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Observational evidence of particle acceleration associated with plasmoid motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The Astrophysical Journal, 828, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Takasao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Matsumoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Nakamura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Shibata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Magnetohydrodynamic shocks in and above post-flare loops: Two-dimensional simulation and a simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The Astrophysical Journal, 805, 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yamada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Kulsrud, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Ji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Reviews of Modern Physics, 82, 603–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yashiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Gopalswamy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Michalek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Cyr, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Plunkett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', Rich, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' A catalog of white light coronal mass ejections observed by the SOHO spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Journal of Geophysical Research: Space Physics, 109, A07105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yokoyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Shibata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Magnetohydrodynamic simulation of a solar flare with chromospheric evaporation effect based on the magnetic reconnection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The Astrophysical Journal, 549, 1160–1174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Zweibel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', & Yamada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Magnetic reconnection in astrophysical and laboratory plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Annual Review of Astronomy and Astrophysics, 47, 291–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Cite this article: Hillier A, Takasao S (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Connecting theory of plasmoid-modulated reconnection to observations of solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Experimental Results, 3, e26, 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 6 Andrew Hillier and Shinsuke Takasao https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press Peer Reviews Reviewing editor: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Stefano Camera, PhD Universita degli Studi di Torino, Physics, Via Pietro Giuria, 1, Torino, Italy, 10124 Minor revisions requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='pr1 Review 1: Connecting Theory of Plasmoid-modulated Reconnection to Observations of Solar Flares Reviewer: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' PengFei Chen Date of review: 22 September 2022 © The Author(s), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Published by Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Conflict of interest statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Reviewer declares none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Comment Comments to the Author: In this manuscript the authors tried to connect observational parameters of solar flares to the scaling law of magnetic reconnection, and it seems that the scaling law is quite similar to what proposed earlier via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' If the topic fits into the theme of the journal, I’d like to recommend it to be published after revisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Major issues: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Personally I tend to think that the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='5*10^9 cm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 1 is not the length of the current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' It might consists of a much shorter current sheet in the middle and long bidirectional outflows both upward and downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this case L might be much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Its effect on alpha should also be discussed on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this paper, the magnetic diffusivity is taken to be the classical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' However, it has been proposed that anomalous resistivity is required for MR to occur, and particle simulations indeed showed that the diffusivity is enhanced by ~6 orders of magnitude during reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In this case, S would be substantially smaller, and alpha would be much larger according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Minor issues: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In several places, “magnetic diffusion” should be “magnetic diffusivity”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In several places, “Arcsec” should be “arcsec”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 1: “is unable to still” --> “is still unable to”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Figure1 panel (a) --> Figure 1(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' table 1 -->Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The X8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2-class solar flare on 2017-09-10 can be used as the 4th example if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Score Card Presentation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='7 /5 Is the article written in clear and proper English?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (30%) 4/5 Is the data presented in the most useful manner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 4/5 Does the paper cite relevant and related articles appropriately?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (30%) 3/5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press Context 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='8 /5 Does the title suitably represent the article?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 5/5 Does the abstract correctly embody the content of the article?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 4/5 Does the introduction give appropriate context?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 4/5 Is the objective of the experiment clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 2/5 Analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='8 /5 Does the discussion adequately interpret the results presented?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 4/5 Is the conclusion consistent with the results and discussion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 4/5 Are the limitations of the experiment as well as the contributions of the experiment clearly outlined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (20%) 3/5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='pr2 Review 2: Connecting Theory of Plasmoid-modulated Reconnection to Observations of Solar Flares Reviewer: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Yi-Min Huang Princeton University, United States Date of review: 26 August 2022 © The Author(s), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Published by Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Conflict of interest statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Reviewer declares none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Comment Comments to the Author: The authors’ approach of using the observed sizes of plasmoids and the rescaled fastest-growing wavenumber to infer the aspect ratio of the current sheet is quite ingenious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' The authors have also discussed several possible uncertainties in their method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' In addition to those uncertainties, I would like to point out another cause of errors that the authors have not addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Simulations have shown that small plasmoids tend to coalesce and form larger plasmoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Moreover, plasmoid size grows over time as they chew magnetic flux through reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Plasmoids could be smaller than the observed ones when they were first born but only become visible when they grow larger through coalescence or reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Therefore, the observed plasmoid sizes may be viewed as an upper bound for the plasmoid instability wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Future solar imagers with higher resolution may help to resolve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Minor issues: (a) Page 2, last paragraph: The reference Comisso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (2016) is a purely analytic paper and contains no simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' It is true that S^-1/3 appears in the aspect ratio derived by Comisso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=', but there is an additional logarithmic factor that depends on S and the initial perturbation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (b) Second paragraph of Discussion, line 7: “siz” appears to be a typo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' Score Card Presentation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='0 /5 Is the article written in clear and proper English?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (30%) 5/5 Is the data presented in the most useful manner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 5/5 Does the paper cite relevant and related articles appropriately?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (30%) 5/5 Context 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='0 /5 Does the title suitably represent the article?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 5/5 Does the abstract correctly embody the content of the article?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 5/5 Does the introduction give appropriate context?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 5/5 Is the objective of the experiment clearly defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (25%) 5/5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press Analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='8 /5 Does the discussion adequately interpret the results presented?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 5/5 Is the conclusion consistent with the results and discussion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (40%) 5/5 Are the limitations of the experiment as well as the contributions of the experiment clearly outlined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content=' (20%) 4/5 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='1017/exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} +page_content='23 Published online by Cambridge University Press' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQfhQSd/content/2301.03239v1.pdf'} diff --git a/mNAyT4oBgHgl3EQf_vq9/content/2301.00915v1.pdf 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+(Dated: January 23, 2023) +We investigate electrostatic traps as a novel source of positron beams for accelerator physics +applications. +Penning-Malmberg (PM) traps are commonly employed in low-energy antimatter +experiments. Positrons contained in the trap are cooled to room temperature or below. We calculate +the thermal emittance of the positrons in the trap and show that it is comparable to or better than +the performance of state-of-the-art photocathode guns. +We propose a compact positron source +comprised of a PM trap, electrostatic compressor, and rf accelerator that can be built and operated +at a fraction of the cost and size of traditional target-based positron sources, albeit at a reduced +repetition rate. We model the acceleration of a positron bunch up to an energy of 17.6 MeV with +a final thermal emittance of 0.60 µm-rad and bunch length of 190 µm. This system may be useful +for acceleration physics studies, such as investigations of flat-beam sources for linear colliders and +positron plasma wakefield acceleration. +I. +INTRODUCTION +Positron beams are traditionally produced by sending +high-energy electron beams into a high-Z target, captur- +ing positrons from the resulting electromagnetic shower, +and cooling the positrons in a damping ring before reac- +celeration [1]. +This process requires significant experi- +mental infrastructure and hardware. As a result, there +are relatively few laboratories producing positron beams +for accelerator physics experiments [2]. Research into ad- +vanced positron sources has been recognized as an area- +of-need for future accelerator R&D [3]. +One research area impacted by the lack of positron +beam sources is Plasma Wakefield Acceleration (PWFA). +PWFA is a promising technique for accelerating charged +particles at high gradients. +Preserving the quality of +positron beams while accelerating them in plasma is an +unsolved challenge [4–9]. The question of how best to +accelerate a positron beam in plasma can only be re- +solved by committing significant experimental and com- +putational resources to the task. New types of positron +sources will expand access to positron beams which can +be used for these experiments. +We propose a novel, compact, electrostatic positron +source for accelerator physics research. Previous research +has explored electron beams from ulta-cold plasmas +(UCP) [10, 11] and magneto-optical traps (MOT) [12, +13]. Our concept is the first to examine this possibility +for positron beams. The positron source is based on the +electrostatic Penning-Malmberg (PM) trap, commonly +employed in low-energy antimatter experiments [14]. +These traps have the advantage of providing cold, low- +emittance beams, although the repetition rate of these +∗ rafimah@stanford.edu +† sgess@slac.stanford.edu +devices is too low to be useful for High Energy Physics ap- +plications such as Linear Colliders. The trap is combined +with a short linac to compress and accelerate the beam +such that the final energy and bunch length is suitable for +injection in a plasma wake. While positron PWFA exper- +iments are the motivation for this concept, the compact +positron source would be of great interest to any facility +that desires positron beams for physics studies. +II. +OVERVIEW OF THE ELECTROSTATIC +POSITRON BEAM SOURCE +In this section, we provide a description of the elec- +trostatic beam source and explain how properties of the +electrostatic trap impact beam parameters like bunch +length and emittance. The review of electrostatic traps +by Danielson, et. +al. +provides a detailed overview of +these systems [14]. +A. +Positron Sources +Positrons for electrostatic traps are typically produced +by β-decay emitters such as 22Na. The emitters are sold +as small encapsulated sources that can be attached to +a vacuum beamline. The primary limitation of the en- +capsulated source is that they contain a limited amount +of radioactive material for safe handling and produce at +most 109 positrons per second [15]. +An alternative method for generating positrons for the +compact source employs a small, 9 MeV electron accel- +erator and impacts the beam on a high-Z target [16]. +This creates low-energy positrons from an electromag- +netic shower, but the initial beam energy is low enough as +to not activate the target material which reduces shield- +ing requirements. This approach is being pursued by the +arXiv:2301.08368v1 [physics.acc-ph] 20 Jan 2023 + +2 +GBAR experiment at CERN with a goal 1010 positrons +per second from the target [16]. +For both the encapsulated radioactive source and the +compact accelerator-based source, the positrons have a +large kinetic energy relative to the depth of the electro- +static trap and a large energy spread. In order to trap the +positrons, the beam must first be sent through a moder- +ator which slows the positrons. A commonly employed +moderator is solid neon with an efficiency of 10−2 [17]. +Therefore, the flux of slow positrons into the trap is about +107 positrons per second for an encapsulated radioactive +source and 108 positrons per second for the accelerator- +based source. +B. +The Electrostatic Trap +The positrons enter an electrostatic trap consisting +of a series of ring electrodes surrounded by a solenoid +magnet. The ring electrodes create the axial potential +well that traps the positrons longitudinally, while the +solenoid provides radial confinement. The depth of the +well needs to be greater than the space charge potential +of the positrons in the trap, given by +∆φ = enr2 +p +4ε0 +� +1 + 2 ln +�rw +rp +�� +, +(1) +for positron density n, plasma radius rp, and trap radius +rw [14]. +The properties of the beam inside the electrostatic trap +are defined by the trap’s parameters. In particular, the +radial extent of the positrons in the trap, and therefore +the density of the positrons in the trap are defined by +the magnetic field and the rotation rate of the positron +plasma. The rotation rate is a free parameter which can +be imposed upon the positron plasma through a “rotating +wall” electrode [18]. In this scenario, the positron plasma +is a uniform cylinder of charge extending to radius rp with +the density given by +n = 2ε0Bωr +e +, +(2) +where B is the solenoid field and ωr is the rotation rate +of the positron plasma. +For our calculations and simulations, we selected for +the desired output beam parameters and designed a hy- +pothetical trap around those values, consistent with pa- +rameters achieved by traps utilized in existing experi- +ments. The trap parameters and beam parameters used +in the simulation are shown in Table I. We note that the +parameters we chose for our simulation are conservative. +For example, we assume a solenoid field of 1 T whereas +the GBAR experiment employs a 5 T magnet [19], and a +trap temperature of 273 K whereas GBAR’s cryo-cooled +trap can produce positron plasmas as cold as 10 K via +cyclotron radiation cooling. The trap temperature in our +simulation is achieved using room-temperature nitrogen +FIG. 1. Depiction of the beamline used in the simulation. The +end of the trap is denoted by A, the ends of the electrostatic +accelerator are denoted by B and C, and the ends of the 3 +GHz linac are denoted by D and E. +buffer gas for cooling [20]. The externally imposed ωr +is roughly the same as GBAR’s at around 3 MHz. The +only constraint on ωr is that it is much less than the +cyclotron frequency Ωc. Since the Debye length of the +positron plasma is much smaller than the plasma radius, +the positron plasma is well-approximated as a uniform +density cylinder with radius rp and length lp [21]. +Parameter +Symbol +Value +Trap radius +rw +4 cm +Trap length +lw +10 cm +Magnetic field +B +1 T +e+ plasma radius +rp +1.3 mm +e+ plasma length +rl +5 cm +Temperature +T +273 K +Number of positrons +N +108 +Space charge potential ∆φ +22.4 V +Debye length +λD +60.6 µm +Cyclotron frequency +Ωc +175.6 GHz +Rotation frequency +ωr +3.2 MHz +Transverse emittance +εx,y +0.11 µm-rad +TABLE I. Parameters used to define the initial plasma distri- +bution inside the trap. +III. +ANALYTIC EQUATION FOR TRAP +EMITTANCE +Starting from the standard equation for normalized +emittance +ϵn = +1 +mc +� +⟨x2⟩⟨p2x⟩ − ⟨xpx⟩2 +(3) +we derive an analytic expression for the transverse emit- +tance in a single plane (here we consider the x-plane) of a +positron beam at rest in the electrostatic trap. The only +coherent motion of the positron plasma is the rotation +about the axis, but since the thermal velocity is much + +1 meter-long 3 GHz Cavity +E +100 kV Electrostatic +Accelerator +D +1 T Solenoid +Positron +c +B +A +Trap3 +-20 +-10 +0 +10 +20 +Z (mm) +-400 +-200 +0 +200 +400 +Pz (eV/c) +-20 +-10 +0 +10 +20 +Z (mm) +20 +22 +24 +26 +28 +Pz (keV/c) +-10 +-5 +0 +5 +10 +Z (mm) +22 +24 +26 +28 +Pz (keV/c) +-5 +0 +5 +Z (mm) +0.28 +0.29 +0.3 +0.31 +Pz (MeV/c) +-2 +0 +2 +Z (mm) +0.285 +0.29 +0.295 +0.3 +0.305 +0.31 +Pz (MeV/c) +-0.2 +-0.1 +0 +0.1 +Z (mm) +17.4 +17.5 +17.6 +17.7 +17.8 +Pz (MeV/c) +Initial Distribution +A +B +D +E +C +FIG. 2. Longitudinal phasespace at demarcated positions along the beamline. The initial distribution corresponds to the beam +inside the trap. Positions A through E correspond to the start and end of accelerator components described in Figure 1. +greater than the rotational velocity vth >> ωrrp, we can +safely ignore x − px correlations. This assumption holds +down to positron beam temperatures of a few Kelvin for +the trap parameters considered here. The single-plane +transverse emittance reduces to ϵx = σxσpx/mc and it +remains to calculate σpx and σx. +The momentum spread is purely thermal +σpx = +� +mkBT, +(4) +while σx is derived from the uniform positron density +extending out to the edge of the plasma cylinder rp +σ2 +x = ⟨x2n(r)⟩ +⟨n(r)⟩ += r2 +p +4 , +(5) +with n(r) = n, the constant beam density, cancelling +out of the equation. Utilizing Equation 2 and the finite +plasma length Lp, we can rewrite rp purely in terms of +trap parameters +rp = +� +qN +2πωrϵ0BLp +, +(6) +which gives +σ2 +x = +qN +8πϵ0BωrLp +. +(7) +Combining equations 7, 4, and 3, we derive an equa- +tion for the normalized, thermal beam emittance defined +solely in terms of trap parameters and bunch charge +ϵth = +1 +mc +� +qNmkBT +8πϵ0BωrLp +. +(8) +For the parameters in our simulation, we find a single- +plane thermal emittance of 0.11 µm-rad, which is compa- +rable to or better than the performance of state-of-the-art +photocathode guns. +The single-plane, thermal beam emittance results are +encouraging, but do not describe the full dynamics of the +beam in the trap. The positron beam is cooled in a strong +magnetic field which generates correlations in the beam +phase space that create angular momentum-dominated +beams. Following the formalism in Ref [22], we define +the transverse beam Σ matrix as +Σ = +� +⟨X ˜X⟩ ⟨X ˜Y ⟩ +⟨Y ˜X⟩ ⟨Y ˜Y ⟩ +� +, +(9) +with +⟨X ˜X⟩ = +� +⟨x2⟩ +⟨xpx⟩ +⟨xpx⟩ +⟨p2 +x⟩ +� +, +(10) +and +⟨X ˜Y ⟩ = +� +⟨xy⟩ +⟨xpy⟩ +⟨ypx⟩ ⟨pxpy⟩ +� +. +(11) +The transverse emittance ε4D describes all four dimen- +sions of the transverse phase space and is given by +ε4D = det(Σ) = ε2 +eff − L2, +(12) +where εeff is the effective emittance in one plane and +angular momentum L = +1 +2mc⟨xpy − ypx⟩. +The thermal emittance is related to the full transverse +emittance by εth = √ε4D, and the effective single-plane +emittance is +εeff = +� +ε2 +th + L2. +(13) + +4 +The effective single-plane emittance will be dominated +by angular momentum when L ≫ εth. Intuitively, this +means that although the volume of the beam in phase +space εth is small, there are no projections of the beam +phase space into the x − y plane such that εx = εth and +εy = εth. However, it is possible to manipulate the beam +to minimize either εx or εy and produce a flat beam [22]. +The amplitude of the angular momentum L is given by +L = eBσ2 +r +2mc . +(14) +For our parameters of B = 1 T and σr = 0.65 mm, we +find L ≈ 250 µm-rad. This is over 3 orders of magnitude +greater than the thermal emittance, implying that this +is indeed an angular-momentum dominated beam with +L ≫ εth. Such beams may be useful for tests of Linear +Collider transport systems which employ flat beams from +damping rings. +IV. +BEAMLINE DESIGN AND SIMULATION +Figure 1 illustrates the beamline used to longitudinally +compress and accelerate the beam. The entire beamline +is encapsulated by a 1 T solenoid. The simulations of +the beamline were performed with the General Particle +Tracer (GPT) code [23]. The beam begins in the elec- +trostatic trap with zero longitudinal energy. The initial +bunch distribution is a uniform cylinder [14], and the lon- +gitudinal extent of the beam is defined by the position of +the trap electrodes. The beam in the trap has a bunch +length σz = 14.4 mm (50 mm uniform distribution). +The bunch length is long compared to millimeter-scale +bunches produced by photocathodes, and much longer +than the micron-scale bunches required for PWFA ex- +periments. Therefore, the beam must be longitudinally +compressed as it is accelerated. Figure 2 shows the evolu- +tion of the longitudinal phase space along the beamline. +Initial +compression +and +acceleration +of +the +long +positron bunch is accomplished with a low-field electro- +static buncher inside the trap. +A harmonic bunching +potential is applied by ring electrodes, such that they +provide an accelerating field that decreases linearly along +the bunch from the tail to the head [24]. The bunching +potential is 10 cm long and the bunch initially occupies +the central portion of the potential (2.5 cm to 7.5 cm). +The voltage drop across the buncher is 2 kV. Figure 3 +shows the longitudinal field Ez as a function of position +in the accelerator. +The buncher creates a longitudinal focus 7 cm beyond +the end of the trap at a longitudinal position of 17 cm +in the simulation, immediately after position B in Fig. 3. +A pulsed, 100 kV electrostatic accelerator extends from +16.4 cm to 26.4 cm (positions B to C). The high volt- +age pulse is provided by a nanosecond pulse generator. +The accelerating pulse is timed with the beam such that +the field is applied when the beam is between the two +FIG. 3. +Plot of the longitudinal field along the length of +the beamline. The trap extends from z = 0 cm to z = 10 +cm (Position A), the electrostatic accelerator extends from +z = 16.4 cm (position B) to z = 26.4 cm (Position C), and +the 3 GHz linac extends from z = 50 cm (Position D) to +z = 1.547 cm (Position E). +accelerating plates. The beam experiences a uniform ac- +celerating field, but positrons at the back of the bunch +experience the field for a longer period of time and gain +energy relative to particles at the head of the bunch. The +beam exits the electrostatic accelerator traveling roughly +half the speed of light and undergoes velocity bunching +as it travels toward the rf cavity. The second longitudi- +nal focus is at z = 0.50 m with σz = 1.3 mm (position +D). At this point, the bunch is short enough for injection +into the RF cavity. +FIG. 4. Bunch length and emittance along the beamline. The +trap extends from z = 0 cm to z = 10 cm (Position A), the +electrostatic accelerator extends from z = 16.4 cm (position +B) to z = 26.4 cm (Position C), and the 3 GHz linac extends +from z = 50 cm (Position D) to z = 1.547 cm (Position E). +The entrance to the s-band accelerator structure is lo- +cated at z = 0.50 (position D). The capture phase of +the s-band structure is set to both accelerate and longi- +tudinally compress the beam to the final bunch length +σz = 190 µm and energy of 17.6 MeV. Figure 4 shows +the bunch length and emittance along the accelerator. +There is an abrupt increase in the emittance from 0.11 +µm-rad to 0.60 µm-rad at the start of the s-band cavity +due to defocusing rf fields. Further studies will exam- +ine the possibility of tailoring the solenoidal magnet field + +15 +20 +(MV/m) +(kV/m) +10 +AB +15 +N +N +E +E +5 +10 +0 +0.05 +0.1 +(u) Z +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +(w) Z-Bunch Length +10 +0.8 +(μm-rad) +一Emittance +0.6 +AB +E +Emittance +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Z(m)5 +to suppress emittance growth at this location. Table II +shows the output beam parameters. These parameters +are comparable to those achieved by the AWAKE elec- +tron accelerator [25] for injection in a proton beam-driven +plasma wakefield. +Beam parameter +Value +Beam energy +17.6 MeV +Beam charge +15.43 pC +Bunch length (rms) +190 µm +Energy spread (rms) +0.76% +Transverse emittance 0.60 µm-rad +TABLE II. Beam parameters at the end of the simulation. +V. +CONCLUSIONS AND FUTURE WORK +The electrostatic trap and beamline described here is +capable of producing useful positron beams in a compact +footprint. Such a device will enable access to positron +beams for accelerator physics studies at universities and +national laboratories that currently lack infrastructure +for positron beam generation. Although the repetition +rate of this positron source is too low for High Energy +Physics applications, it is sufficient for studies at PWFA +facilities, including the AWAKE facility which produces +an experimental shot once every thirty seconds [26]. +Further studies will be undertaken to explore tai- +lored solenoidal magnetic fields that suppress emittance +growth at the start of the rf cavity. +We also plan to +study remoderation of the positron beam to remove in- +trinsic angular momentum at the cost of reduced bunch +charge [27]. The brighter positron beams produced by +remoderation may prove useful as a compliment to Ultra- +fast Electron Diffraction (UED) experiments [28] where +the positive beam charge can be used to reduce sys- +tematics when used in tandem with electron beams. +The ultimate application of this technology would be +a positron source for a damping ring-free collider [29]. +This would require multiplexing of the compact positron +source. Multiplexing of positron sources has been previ- +ously considered to meet the demands of the NLC collider +concept [30]. However, given the repetition rate of exist- +ing compact positron sources, this would require thou- +sands of sources operating simultaneously, so research in +this direction should focus on increasing the repetition +rate of a single source. +VI. +ACKNOWLEDGEMENTS +Many individuals helped to provide background on +positron sources for this project. +We thank Dirk Pe- +ter Van Der Werf, Samuel Niang, and Laszlo Liszkay +for showing us the GBAR experiment at CERN. Thank +you to David Cooke, David Cassidy, Allen Mills, and +Cliff Surko for background on positrons from electrostatic +traps. Thank you to Pietro Musumeci for background on +UED systems. Klaus Floettman and Bas van der Geer +provided input on simulations in ASTRA and GPT, re- +spectively. Thank you to the AWAKE electron source +group Seongyeol Kim, Mohsen Dayyani Kelisani, Steffen +Doebert, and Edda Gschwendtner from CERN for their +useful discussions and support. +[1] I. Chaikovska, R. Chehab, V. Kubytskyi, S. Ogur, +A. 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Yeremian, in Proceedings Particle Accelerator +Conference (IEEE, 1995), URL https://doi.org/10. +1109/pac.1995.505120. + diff --git a/mtE_T4oBgHgl3EQf7Bz-/content/tmp_files/load_file.txt b/mtE_T4oBgHgl3EQf7Bz-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..425c4d1b14398b6eaf27bf22e92c5825251563b5 --- /dev/null +++ b/mtE_T4oBgHgl3EQf7Bz-/content/tmp_files/load_file.txt @@ -0,0 +1,599 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf,len=598 +page_content='A Compact Source of Positron Beams with Small Thermal Emittance Rafi Hessami∗ Applied Physics Department, Stanford University, Stanford, USA and SLAC National Accelerator Laboratory, Menlo Park, USA Spencer Gessner† SLAC National Accelerator Laboratory, Menlo Park, USA (Dated: January 23, 2023) We investigate electrostatic traps as a novel source of positron beams for accelerator physics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Penning-Malmberg (PM) traps are commonly employed in low-energy antimatter experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Positrons contained in the trap are cooled to room temperature or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We calculate the thermal emittance of the positrons in the trap and show that it is comparable to or better than the performance of state-of-the-art photocathode guns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We propose a compact positron source comprised of a PM trap, electrostatic compressor, and rf accelerator that can be built and operated at a fraction of the cost and size of traditional target-based positron sources, albeit at a reduced repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We model the acceleration of a positron bunch up to an energy of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 MeV with a final thermal emittance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='60 µm-rad and bunch length of 190 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This system may be useful for acceleration physics studies, such as investigations of flat-beam sources for linear colliders and positron plasma wakefield acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' INTRODUCTION Positron beams are traditionally produced by sending high-energy electron beams into a high-Z target, captur- ing positrons from the resulting electromagnetic shower, and cooling the positrons in a damping ring before reac- celeration [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This process requires significant experi- mental infrastructure and hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' As a result, there are relatively few laboratories producing positron beams for accelerator physics experiments [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Research into ad- vanced positron sources has been recognized as an area- of-need for future accelerator R&D [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' One research area impacted by the lack of positron beam sources is Plasma Wakefield Acceleration (PWFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' PWFA is a promising technique for accelerating charged particles at high gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Preserving the quality of positron beams while accelerating them in plasma is an unsolved challenge [4–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The question of how best to accelerate a positron beam in plasma can only be re- solved by committing significant experimental and com- putational resources to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' New types of positron sources will expand access to positron beams which can be used for these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We propose a novel, compact, electrostatic positron source for accelerator physics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Previous research has explored electron beams from ulta-cold plasmas (UCP) [10, 11] and magneto-optical traps (MOT) [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Our concept is the first to examine this possibility for positron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The positron source is based on the electrostatic Penning-Malmberg (PM) trap, commonly employed in low-energy antimatter experiments [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' These traps have the advantage of providing cold, low- emittance beams, although the repetition rate of these ∗ rafimah@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='edu † sgess@slac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='edu devices is too low to be useful for High Energy Physics ap- plications such as Linear Colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The trap is combined with a short linac to compress and accelerate the beam such that the final energy and bunch length is suitable for injection in a plasma wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' While positron PWFA exper- iments are the motivation for this concept, the compact positron source would be of great interest to any facility that desires positron beams for physics studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' OVERVIEW OF THE ELECTROSTATIC POSITRON BEAM SOURCE In this section, we provide a description of the elec- trostatic beam source and explain how properties of the electrostatic trap impact beam parameters like bunch length and emittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The review of electrostatic traps by Danielson, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' provides a detailed overview of these systems [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Positron Sources Positrons for electrostatic traps are typically produced by β-decay emitters such as 22Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The emitters are sold as small encapsulated sources that can be attached to a vacuum beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The primary limitation of the en- capsulated source is that they contain a limited amount of radioactive material for safe handling and produce at most 109 positrons per second [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' An alternative method for generating positrons for the compact source employs a small, 9 MeV electron accel- erator and impacts the beam on a high-Z target [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This creates low-energy positrons from an electromag- netic shower, but the initial beam energy is low enough as to not activate the target material which reduces shield- ing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This approach is being pursued by the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='08368v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='acc-ph] 20 Jan 2023 2 GBAR experiment at CERN with a goal 1010 positrons per second from the target [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' For both the encapsulated radioactive source and the compact accelerator-based source, the positrons have a large kinetic energy relative to the depth of the electro- static trap and a large energy spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' In order to trap the positrons, the beam must first be sent through a moder- ator which slows the positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' A commonly employed moderator is solid neon with an efficiency of 10−2 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Therefore, the flux of slow positrons into the trap is about 107 positrons per second for an encapsulated radioactive source and 108 positrons per second for the accelerator- based source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The Electrostatic Trap The positrons enter an electrostatic trap consisting of a series of ring electrodes surrounded by a solenoid magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The ring electrodes create the axial potential well that traps the positrons longitudinally, while the solenoid provides radial confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The depth of the well needs to be greater than the space charge potential of the positrons in the trap, given by ∆φ = enr2 p 4ε0 � 1 + 2 ln �rw rp �� , (1) for positron density n, plasma radius rp, and trap radius rw [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The properties of the beam inside the electrostatic trap are defined by the trap’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' In particular, the radial extent of the positrons in the trap, and therefore the density of the positrons in the trap are defined by the magnetic field and the rotation rate of the positron plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The rotation rate is a free parameter which can be imposed upon the positron plasma through a “rotating wall” electrode [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' In this scenario, the positron plasma is a uniform cylinder of charge extending to radius rp with the density given by n = 2ε0Bωr e , (2) where B is the solenoid field and ωr is the rotation rate of the positron plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' For our calculations and simulations, we selected for the desired output beam parameters and designed a hy- pothetical trap around those values, consistent with pa- rameters achieved by traps utilized in existing experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The trap parameters and beam parameters used in the simulation are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We note that the parameters we chose for our simulation are conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' For example, we assume a solenoid field of 1 T whereas the GBAR experiment employs a 5 T magnet [19], and a trap temperature of 273 K whereas GBAR’s cryo-cooled trap can produce positron plasmas as cold as 10 K via cyclotron radiation cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The trap temperature in our simulation is achieved using room-temperature nitrogen FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Depiction of the beamline used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The end of the trap is denoted by A, the ends of the electrostatic accelerator are denoted by B and C, and the ends of the 3 GHz linac are denoted by D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' buffer gas for cooling [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The externally imposed ωr is roughly the same as GBAR’s at around 3 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The only constraint on ωr is that it is much less than the cyclotron frequency Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Since the Debye length of the positron plasma is much smaller than the plasma radius, the positron plasma is well-approximated as a uniform density cylinder with radius rp and length lp [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Parameter Symbol Value Trap radius rw 4 cm Trap length lw 10 cm Magnetic field B 1 T e+ plasma radius rp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='3 mm e+ plasma length rl 5 cm Temperature T 273 K Number of positrons N 108 Space charge potential ∆φ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 V Debye length λD 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 µm Cyclotron frequency Ωc 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 GHz Rotation frequency ωr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 MHz Transverse emittance εx,y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='11 µm-rad TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Parameters used to define the initial plasma distri- bution inside the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' ANALYTIC EQUATION FOR TRAP EMITTANCE Starting from the standard equation for normalized emittance ϵn = 1 mc � ⟨x2⟩⟨p2x⟩ − ⟨xpx⟩2 (3) we derive an analytic expression for the transverse emit- tance in a single plane (here we consider the x-plane) of a positron beam at rest in the electrostatic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The only coherent motion of the positron plasma is the rotation about the axis, but since the thermal velocity is much 1 meter-long 3 GHz Cavity E 100 kV Electrostatic Accelerator D 1 T Solenoid Positron c B A Trap3 20 10 0 10 20 Z (mm) 400 200 0 200 400 Pz (eV/c) 20 10 0 10 20 Z (mm) 20 22 24 26 28 Pz (keV/c) 10 5 0 5 10 Z (mm) 22 24 26 28 Pz (keV/c) 5 0 5 Z (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='31 Pz (MeV/c) 2 0 2 Z (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='31 Pz (MeV/c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='1 Z (mm) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='8 Pz (MeV/c) Initial Distribution A B D E C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Longitudinal phasespace at demarcated positions along the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The initial distribution corresponds to the beam inside the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Positions A through E correspond to the start and end of accelerator components described in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' greater than the rotational velocity vth >> ωrrp, we can safely ignore x − px correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This assumption holds down to positron beam temperatures of a few Kelvin for the trap parameters considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The single-plane transverse emittance reduces to ϵx = σxσpx/mc and it remains to calculate σpx and σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The momentum spread is purely thermal σpx = � mkBT, (4) while σx is derived from the uniform positron density extending out to the edge of the plasma cylinder rp σ2 x = ⟨x2n(r)⟩ ⟨n(r)⟩ = r2 p 4 , (5) with n(r) = n, the constant beam density, cancelling out of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Utilizing Equation 2 and the finite plasma length Lp, we can rewrite rp purely in terms of trap parameters rp = � qN 2πωrϵ0BLp , (6) which gives σ2 x = qN 8πϵ0BωrLp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' (7) Combining equations 7, 4, and 3, we derive an equa- tion for the normalized, thermal beam emittance defined solely in terms of trap parameters and bunch charge ϵth = 1 mc � qNmkBT 8πϵ0BωrLp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' (8) For the parameters in our simulation, we find a single- plane thermal emittance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='11 µm-rad, which is compa- rable to or better than the performance of state-of-the-art photocathode guns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The single-plane, thermal beam emittance results are encouraging, but do not describe the full dynamics of the beam in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The positron beam is cooled in a strong magnetic field which generates correlations in the beam phase space that create angular momentum-dominated beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Following the formalism in Ref [22], we define the transverse beam Σ matrix as Σ = � ⟨X ˜X⟩ ⟨X ˜Y ⟩ ⟨Y ˜X⟩ ⟨Y ˜Y ⟩ � , (9) with ⟨X ˜X⟩ = � ⟨x2⟩ ⟨xpx⟩ ⟨xpx⟩ ⟨p2 x⟩ � , (10) and ⟨X ˜Y ⟩ = � ⟨xy⟩ ⟨xpy⟩ ⟨ypx⟩ ⟨pxpy⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' (11) The transverse emittance ε4D describes all four dimen- sions of the transverse phase space and is given by ε4D = det(Σ) = ε2 eff − L2, (12) where εeff is the effective emittance in one plane and angular momentum L = 1 2mc⟨xpy − ypx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The thermal emittance is related to the full transverse emittance by εth = √ε4D, and the effective single-plane emittance is εeff = � ε2 th + L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' (13) 4 The effective single-plane emittance will be dominated by angular momentum when L ≫ εth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Intuitively, this means that although the volume of the beam in phase space εth is small, there are no projections of the beam phase space into the x − y plane such that εx = εth and εy = εth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' However, it is possible to manipulate the beam to minimize either εx or εy and produce a flat beam [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The amplitude of the angular momentum L is given by L = eBσ2 r 2mc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' (14) For our parameters of B = 1 T and σr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='65 mm, we find L ≈ 250 µm-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This is over 3 orders of magnitude greater than the thermal emittance, implying that this is indeed an angular-momentum dominated beam with L ≫ εth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Such beams may be useful for tests of Linear Collider transport systems which employ flat beams from damping rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' BEAMLINE DESIGN AND SIMULATION Figure 1 illustrates the beamline used to longitudinally compress and accelerate the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The entire beamline is encapsulated by a 1 T solenoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The simulations of the beamline were performed with the General Particle Tracer (GPT) code [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The beam begins in the elec- trostatic trap with zero longitudinal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The initial bunch distribution is a uniform cylinder [14], and the lon- gitudinal extent of the beam is defined by the position of the trap electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The beam in the trap has a bunch length σz = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 mm (50 mm uniform distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The bunch length is long compared to millimeter-scale bunches produced by photocathodes, and much longer than the micron-scale bunches required for PWFA ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Therefore, the beam must be longitudinally compressed as it is accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Figure 2 shows the evolu- tion of the longitudinal phase space along the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Initial compression and acceleration of the long positron bunch is accomplished with a low-field electro- static buncher inside the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' A harmonic bunching potential is applied by ring electrodes, such that they provide an accelerating field that decreases linearly along the bunch from the tail to the head [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The bunching potential is 10 cm long and the bunch initially occupies the central portion of the potential (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='5 cm to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='5 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The voltage drop across the buncher is 2 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Figure 3 shows the longitudinal field Ez as a function of position in the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The buncher creates a longitudinal focus 7 cm beyond the end of the trap at a longitudinal position of 17 cm in the simulation, immediately after position B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' A pulsed, 100 kV electrostatic accelerator extends from 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm (positions B to C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The high volt- age pulse is provided by a nanosecond pulse generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The accelerating pulse is timed with the beam such that the field is applied when the beam is between the two FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Plot of the longitudinal field along the length of the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The trap extends from z = 0 cm to z = 10 cm (Position A), the electrostatic accelerator extends from z = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm (position B) to z = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm (Position C), and the 3 GHz linac extends from z = 50 cm (Position D) to z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='547 cm (Position E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' accelerating plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The beam experiences a uniform ac- celerating field, but positrons at the back of the bunch experience the field for a longer period of time and gain energy relative to particles at the head of the bunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The beam exits the electrostatic accelerator traveling roughly half the speed of light and undergoes velocity bunching as it travels toward the rf cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The second longitudi- nal focus is at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='50 m with σz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='3 mm (position D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' At this point, the bunch is short enough for injection into the RF cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Bunch length and emittance along the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The trap extends from z = 0 cm to z = 10 cm (Position A), the electrostatic accelerator extends from z = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm (position B) to z = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 cm (Position C), and the 3 GHz linac extends from z = 50 cm (Position D) to z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='547 cm (Position E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The entrance to the s-band accelerator structure is lo- cated at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='50 (position D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The capture phase of the s-band structure is set to both accelerate and longi- tudinally compress the beam to the final bunch length σz = 190 µm and energy of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Figure 4 shows the bunch length and emittance along the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' There is an abrupt increase in the emittance from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='11 µm-rad to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='60 µm-rad at the start of the s-band cavity due to defocusing rf fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Further studies will exam- ine the possibility of tailoring the solenoidal magnet field 15 20 (MV/m) (kV/m) 10 AB 15 N N E E 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='1 (u) Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 (w) Z-Bunch Length 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='8 (μm-rad) 一Emittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 AB E Emittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='4 Z(m)5 to suppress emittance growth at this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Table II shows the output beam parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' These parameters are comparable to those achieved by the AWAKE elec- tron accelerator [25] for injection in a proton beam-driven plasma wakefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Beam parameter Value Beam energy 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='6 MeV Beam charge 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='43 pC Bunch length (rms) 190 µm Energy spread (rms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='76% Transverse emittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='60 µm-rad TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Beam parameters at the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK The electrostatic trap and beamline described here is capable of producing useful positron beams in a compact footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Such a device will enable access to positron beams for accelerator physics studies at universities and national laboratories that currently lack infrastructure for positron beam generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Although the repetition rate of this positron source is too low for High Energy Physics applications, it is sufficient for studies at PWFA facilities, including the AWAKE facility which produces an experimental shot once every thirty seconds [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Further studies will be undertaken to explore tai- lored solenoidal magnetic fields that suppress emittance growth at the start of the rf cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We also plan to study remoderation of the positron beam to remove in- trinsic angular momentum at the cost of reduced bunch charge [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The brighter positron beams produced by remoderation may prove useful as a compliment to Ultra- fast Electron Diffraction (UED) experiments [28] where the positive beam charge can be used to reduce sys- tematics when used in tandem with electron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' The ultimate application of this technology would be a positron source for a damping ring-free collider [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' This would require multiplexing of the compact positron source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Multiplexing of positron sources has been previ- ously considered to meet the demands of the NLC collider concept [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' However, given the repetition rate of exist- ing compact positron sources, this would require thou- sands of sources operating simultaneously, so research in this direction should focus on increasing the repetition rate of a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Many individuals helped to provide background on positron sources for this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' We thank Dirk Pe- ter Van Der Werf, Samuel Niang, and Laszlo Liszkay for showing us the GBAR experiment at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Thank you to David Cooke, David Cassidy, Allen Mills, and Cliff Surko for background on positrons from electrostatic traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Thank you to Pietro Musumeci for background on UED systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Klaus Floettman and Bas van der Geer provided input on simulations in ASTRA and GPT, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Thank you to the AWAKE electron source group Seongyeol Kim, Mohsen Dayyani Kelisani, Steffen Doebert, and Edda Gschwendtner from CERN for their useful discussions and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Chaikovska, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content=' Chehab, V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} +page_content='505120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE_T4oBgHgl3EQf7Bz-/content/2301.08368v1.pdf'} diff --git a/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/2301.11343v1.pdf.txt b/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/2301.11343v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8acc687821fbabb3eb181858c6c7cf63a07e6086 --- /dev/null +++ b/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/2301.11343v1.pdf.txt @@ -0,0 +1,147 @@ +The Stern-Gerlach Experiment +Translation of: “Der experimentelle Nachweis der +Richtungsquantelung im Magnetfeld” +Martin Bauer +Institute for Particle Physics Phenomenology, Department of Physics, +Durham University, Durham, DH1 3LE, United Kingdom +The following is a translation of the paper by Walther Gerlach and Otto +Stern1) that reported the first evidence for the quantisation of atoms in a +magnetic field. The atoms have quantum states corresponding to a limited +number of possible angles between the directions of the angular momenta of +the atoms and the magnetic field, also called space quantisation. Wording and +layout have been chosen to be as close to the original as possible. For context +we recommend the recent review2). +I thank Phillip Helbig for substantial help in preparing this translation and +Chanda Prescod-Weinstein for bringing to my attention that there is no +available english translation of the original paper by Gerlach and Stern. +1W. Gerlach u. O. Stern, Zeitschrift f¨ur Physik 9, 349–352, 1922. +2H. Schmidt-B¨ocking, L. Schmidt, H. L¨udde, W. Trageser, A. Templeton and T. Sauer, +Eur. Phys. J. H 41, 327–364, 2016. +1 +arXiv:2301.11343v1 [physics.hist-ph] 26 Jan 2023 + +Experimental Evidence for Space Quantisation in a +Magnetic Field. +By Walther Gerlach in Frankfurt a.M. and Otto Stern in Rostock. +Including 7 figures. (Recieved March 1, 1922.) +Recently3) this journal published a possible method to experimentally +answer the question whether space quantisation in a magnetic field exists. +It was shown in a second communication4) that the normal silver atom has +a magnetic moment. We allow ourselves to report in the following that the +continuation of these investigations has led to est ab l is h s p ace q uant i - +s at i o n i n a m a g n e t i c fi el d as a f act . +E x p e r i m e nt a l s e t u p . Method and experimental setup are generally the +same as in our earlier experiments, but substantial improvements have been +made to some parts of it5), which we will describe here in addition to the +information provided earlier. The beam of silver atoms emer- +ges from an electrically heated small chamotte oven with a +steel insert and a cover with a 1 mm2 circular hole. The +distance between the hole in the oven and the first beam +aperture was increased to 2.5 cm to prevent clogging by oc- +casional small silver droplets sprayed from the oven as well +as precipation from the atomic beam. This first aperture is +almost circular and its surface measures 3·10−3 mm2. 3.3 cm +behind this circular aperture, the silver beam passes through +a slit aperture with a length of 0.8 mm and a width between 0.03 mm and +0.04 mm. Both apertures are made from platinum sheet. The slit aperture is +positioned where the magnetic field begins. The opening of the slit aperture +is right above the knife-edge S (see Fig. 1) and is adjusted with respect to +the circular aperture and the hole in the oven in such a way that the silver +beam travels in parallel along the 3.5 cm long knife-edge. Precisely at the end +of this knife-edge the silver beam hits a glass plate where it condenses. +The two apertures, the poles of the magnet and the glass plate are in a +brass housing with wall of thickness 1 cm, which are rigidly connected so that +3O. Stern, ZS. f. Phys. 7, 249, 1921 +4W. Gerlach u. O. Stern, ibid. 8, 110, 1921. +5These were worked out and tested collaboratively. The final experiments had to be +performed by one of us alone (G.) due to the departure from Frankfurt of one of us (St.). +2 + +Fig.1.pressure from the poles of the electromagnet doesn’t result in a deformation +of the housing nor cause a shift in the relative position of the apertures, the +poles, and the glass plate. +Two Volmer diffusion pumps and a Gaede Hg-pump as pre-pump are used +for evacuation. Through continuous pumping and cooling with solid carbon +dioxide a vacuum of 10−5mm Hg was achieved and maintained. +The “exposure time” was increased to eight hours without interruption. +But as a result of the very narrow apertures and the long beam, the silver +film on the glass plate was so thin that it had to be developed —as reported +previously— even after eight hours of vaporisation. +R e s u l t s . First, Fig. 2 shows a photograph after 41/2 hours of exposure time +without a magnetic field; it’s magnified by a factor of 20. Measurements of +the original under the microscope using an ocular micrometer resulted in the +following dimensions: Length 1.1 mm, width at the narrowest point 0.06 mm, +at the widest point 0.10 mm. We see that the slit isn’t exactly parallel. It +should be noted, however, that the figure shows the slit magnified by a factor +of 40, since the “silver image” of the slit is already twice the size; it is difficult +to make such a slit in a frame only a few millimetres in size. +3 + +30020 +Fig. 2. +Fig. 3.Fig. 3 shows a photograph after eight hours of exposure time magnified +by a factor 20 (20 scale divisions of the imaged scale = 1 mm). This is the +best photograph we took. Two other photographs show the same result in all +relevant aspects, but don’t show this complete symmetry. It has to be said +that adjustment of these small apertures by optical methods is very difficult, +so that it takes some luck to obtain a perfectly symmetric photograph as +shown in Fig.3; errors in adjusting an aperture by just a few hundredths of a +millimeter are enough to completely ruin a photograph. +The results of the two other experiments are shown schematically in Fig. +4a and 4b. In Fig. 4a the silver beam ran intentionally at a slightly greater +distance past the knife-edge than in the experiment of Fig.3. The slit aperture +wasn’t completely “filled” here. Fig. 4b shows the deposit of an experiment +both with and without a field on the same plate; the beam ran very close to +the knife-edge, but was moved 0.3 mm perpendicular to the field (Fig. 4c). +Regarding clarity of the pictures, the complete splitting, and all other details, +these pictures are in no way inferior to those shown in Fig. 3. +The photographs show that the silver-atom beam is split in an inhomoge- +neous magnetic field in two directions in the direction of the inhomogeneity, +of which one is attracted by the knife-edge pole and the other is repelled by +the knife-edge pole. The deposits show the following details (see the schematic +Fig. 5) +a) T h e d i m e n s i o n s of the original were measured in a microscope: +Length 1.1 mm, width a 0.11 mm, width b 0.20 mm. +b) T h e a t o m i c b e a m sp l i t s i nt o two d i scr et e b eams i n a +m a g n e t i c fi e l d . We f ou n d n o ev i d en ce f or n on -d efl ec - +4 + +28mm +a +S +Fig.4a. +Fig. 4 b. +Fig. 4c. +Fig. 5.t e d a t o m s . +c) T h e a t t r a c t i o n i s sl i ght l y st r on ger t h an t h e r ep u l si on . +The attracted atoms get closer to the pole and therefore to the regi- +on of largest inhomogeneity, so that the deflection while flying past is +stronger. Fig. 3 and 4b show the significantly larger deflection directly +at the knife-edge of this one magnetic pole. In the immediate vicinity of +the knife-edge the attraction becomes very large so that a bulge arises, +with a sharp edge pointing towards the knife-edge. +d) T h e w i d t h o f t h e d efl ect e d b an d s i s l ar ger t h an t h e +w i d t h o f t h e u n d e fl ect ed i mage. The latter is simply the image +of aperture B2 projected by aperture B1 onto the glass plate. The de- +flected band is broadened following the velocity distribution of the silver +atoms. +e) T h i s f a c t s t r e n g t h en s t h e case f or t h er e b ei n g f ew i f +a ny u n d e fl e c t e d a t oms [ see b ) ] . The detection of undeflected +atoms in a small area is much more sensitive than that of deflected atoms +in a large area. There appears to be no magnetic axis perpendicular to +the field direction. +We v i e w t h e s e r e s u l t s as d i r ect , ex p er i ment al ev i d en ce +f o r s p a c e q u a nt i s a t i o n i n a magn et i c fi el d . +A detailed account of the experiment and results in our thus far short +communications will be published in the Annalen der Physik, as soon as we +have precise measurements of the inhomogeneity of the magnetic field and +can provide quantitative information about the size of the magneton. +The electromagnet necessary for these experiments was procured with +funds from the foundation of the Kaiser Wilhelm Institute; to the director, +Mr. A. Einstein, we would like to express our heartfelt thanks. We further +thank the Association of Friends and Sponsors of the University of Frankfurt +sincerely for the abundant resources they gladly made available to fund the +continuation of the experiments. +Frankfurt a. M. and Rostock i. M., February 1922. +5 + diff --git a/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/load_file.txt b/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..120d58dab9d57de9e6e50e18d27e5fed43a5960d --- /dev/null +++ b/ntFIT4oBgHgl3EQfuSvs/content/tmp_files/load_file.txt @@ -0,0 +1,143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf,len=142 +page_content='The Stern-Gerlach Experiment Translation of: “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld” Martin Bauer Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham, DH1 3LE, United Kingdom The following is a translation of the paper by Walther Gerlach and Otto Stern1) that reported the first evidence for the quantisation of atoms in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The atoms have quantum states corresponding to a limited number of possible angles between the directions of the angular momenta of the atoms and the magnetic field, also called space quantisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Wording and layout have been chosen to be as close to the original as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' For context we recommend the recent review2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' I thank Phillip Helbig for substantial help in preparing this translation and Chanda Prescod-Weinstein for bringing to my attention that there is no available english translation of the original paper by Gerlach and Stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 1W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Gerlach u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Stern, Zeitschrift f¨ur Physik 9, 349–352, 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Schmidt-B¨ocking, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Schmidt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' L¨udde, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Trageser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Templeton and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Sauer, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' H 41, 327–364, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='11343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='hist-ph] 26 Jan 2023 Experimental Evidence for Space Quantisation in a Magnetic Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' By Walther Gerlach in Frankfurt a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' and Otto Stern in Rostock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Including 7 figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' (Recieved March 1, 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=') Recently3) this journal published a possible method to experimentally answer the question whether space quantisation in a magnetic field exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' It was shown in a second communication4) that the normal silver atom has a magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' We allow ourselves to report in the following that the continuation of these investigations has led to est ab l is h s p ace q uant i - s at i o n i n a m a g n e t i c fi el d as a f act .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' E x p e r i m e nt a l s e t u p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Method and experimental setup are generally the same as in our earlier experiments, but substantial improvements have been made to some parts of it5), which we will describe here in addition to the information provided earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The beam of silver atoms emer- ges from an electrically heated small chamotte oven with a steel insert and a cover with a 1 mm2 circular hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The distance between the hole in the oven and the first beam aperture was increased to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='5 cm to prevent clogging by oc- casional small silver droplets sprayed from the oven as well as precipation from the atomic beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' This first aperture is almost circular and its surface measures 3·10−3 mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='3 cm behind this circular aperture, the silver beam passes through a slit aperture with a length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='8 mm and a width between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='03 mm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='04 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Both apertures are made from platinum sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The slit aperture is positioned where the magnetic field begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The opening of the slit aperture is right above the knife-edge S (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 1) and is adjusted with respect to the circular aperture and the hole in the oven in such a way that the silver beam travels in parallel along the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='5 cm long knife-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Precisely at the end of this knife-edge the silver beam hits a glass plate where it condenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The two apertures, the poles of the magnet and the glass plate are in a brass housing with wall of thickness 1 cm, which are rigidly connected so that 3O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Stern, ZS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 7, 249, 1921 4W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Gerlach u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Stern, ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 8, 110, 1921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 5These were worked out and tested collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The final experiments had to be performed by one of us alone (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=') due to the departure from Frankfurt of one of us (St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='pressure from the poles of the electromagnet doesn’t result in a deformation of the housing nor cause a shift in the relative position of the apertures, the poles, and the glass plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Two Volmer diffusion pumps and a Gaede Hg-pump as pre-pump are used for evacuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Through continuous pumping and cooling with solid carbon dioxide a vacuum of 10−5mm Hg was achieved and maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The “exposure time” was increased to eight hours without interruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' But as a result of the very narrow apertures and the long beam, the silver film on the glass plate was so thin that it had to be developed —as reported previously— even after eight hours of vaporisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' R e s u l t s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' First, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 2 shows a photograph after 41/2 hours of exposure time without a magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' it’s magnified by a factor of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Measurements of the original under the microscope using an ocular micrometer resulted in the following dimensions: Length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='1 mm, width at the narrowest point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='06 mm, at the widest point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='10 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' We see that the slit isn’t exactly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' It should be noted, however, that the figure shows the slit magnified by a factor of 40, since the “silver image” of the slit is already twice the size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' it is difficult to make such a slit in a frame only a few millimetres in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3 30020 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3 shows a photograph after eight hours of exposure time magnified by a factor 20 (20 scale divisions of the imaged scale = 1 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' This is the best photograph we took.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Two other photographs show the same result in all relevant aspects, but don’t show this complete symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' It has to be said that adjustment of these small apertures by optical methods is very difficult, so that it takes some luck to obtain a perfectly symmetric photograph as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' errors in adjusting an aperture by just a few hundredths of a millimeter are enough to completely ruin a photograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The results of the two other experiments are shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4a and 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4a the silver beam ran intentionally at a slightly greater distance past the knife-edge than in the experiment of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The slit aperture wasn’t completely “filled” here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4b shows the deposit of an experiment both with and without a field on the same plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' the beam ran very close to the knife-edge, but was moved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='3 mm perpendicular to the field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Regarding clarity of the pictures, the complete splitting, and all other details, these pictures are in no way inferior to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The photographs show that the silver-atom beam is split in an inhomoge- neous magnetic field in two directions in the direction of the inhomogeneity, of which one is attracted by the knife-edge pole and the other is repelled by the knife-edge pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The deposits show the following details (see the schematic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 5) a) T h e d i m e n s i o n s of the original were measured in a microscope: Length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='1 mm, width a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='11 mm, width b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' b) T h e a t o m i c b e a m sp l i t s i nt o two d i scr et e b eams i n a m a g n e t i c fi e l d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' We f ou n d n o ev i d en ce f or n on -d efl ec - 4 28mm a S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content='t e d a t o m s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' c) T h e a t t r a c t i o n i s sl i ght l y st r on ger t h an t h e r ep u l si on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The attracted atoms get closer to the pole and therefore to the regi- on of largest inhomogeneity, so that the deflection while flying past is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 3 and 4b show the significantly larger deflection directly at the knife-edge of this one magnetic pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' In the immediate vicinity of the knife-edge the attraction becomes very large so that a bulge arises, with a sharp edge pointing towards the knife-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' d) T h e w i d t h o f t h e d efl ect e d b an d s i s l ar ger t h an t h e w i d t h o f t h e u n d e fl ect ed i mage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The latter is simply the image of aperture B2 projected by aperture B1 onto the glass plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The de- flected band is broadened following the velocity distribution of the silver atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' e) T h i s f a c t s t r e n g t h en s t h e case f or t h er e b ei n g f ew i f a ny u n d e fl e c t e d a t oms [ see b ) ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The detection of undeflected atoms in a small area is much more sensitive than that of deflected atoms in a large area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' There appears to be no magnetic axis perpendicular to the field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' We v i e w t h e s e r e s u l t s as d i r ect , ex p er i ment al ev i d en ce f o r s p a c e q u a nt i s a t i o n i n a magn et i c fi el d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' A detailed account of the experiment and results in our thus far short communications will be published in the Annalen der Physik, as soon as we have precise measurements of the inhomogeneity of the magnetic field and can provide quantitative information about the size of the magneton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' The electromagnet necessary for these experiments was procured with funds from the foundation of the Kaiser Wilhelm Institute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' to the director, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Einstein, we would like to express our heartfelt thanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' We further thank the Association of Friends and Sponsors of the University of Frankfurt sincerely for the abundant resources they gladly made available to fund the continuation of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' Frankfurt a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' and Rostock i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=', February 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFIT4oBgHgl3EQfuSvs/content/2301.11343v1.pdf'} diff --git a/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/2301.01167v1.pdf.txt b/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/2301.01167v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cd34fe07af94bcb6913e5578b68b5348727486b --- /dev/null +++ b/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/2301.01167v1.pdf.txt @@ -0,0 +1,1155 @@ +1 +Consensus-based Distributed Intentional Controlled +Islanding of Power Grids +Francesco Lo Iudice, Ricardo Cardona-Rivera, Antonio Grotta, Marco Coraggio, Mario di Bernardo† +Abstract—The problem of partitioning a power grid into a +set of islands can be a solution to restore power dispatchment +in sections of a grid affected by an extreme failure. Current +solutions to this problem usually involve finding the partition of +the grid into islands that minimizes the sum of their absolute +power imbalances. This combinatorial problem is often solved +through heuristic offline methods. In this paper, we propose +instead a distributed online algorithm through which nodes can +migrate among islands, self-organizing the network into a suitable +partition. We prove that, under a set of appropriate assumptions, +the proposed solution yields a partition whose absolute power +imbalance falls within a given bound of the optimal solution. We +validate our analytical results by testing our partitioning strategy +on the IEEE 118 and 300 benchmark problems. +I. Introduction +The penetration of renewable and distributed generation, e.g., +[1]–[3], and the possible occurrence of cascading failures [4] +have made the problem of devising control strategies to govern +the operation of power grids of crucial concern. Examples of +problems of interest include those reported in [5]–[7], [8], [9] +and [10]–[12]. +When the control architecture fails to guarantee reliable +operation of the transmission grid, last resort strategies have +been devised so as to ensure power dispatchment across at +least some of its sections. Intentional Controlled Islanding (ICI) +strategies address this issue [13]–[17] by identifying sections of +the grid (or islands) that can isolate and operate independently +from the rest of the network. Recently, intentional islanding has +also been proposed in the framework of distribution networks, +see [18] and references therein, as the presence of storage +devices and renewable energy generation allows these grids to +be partitioned into networks of microgrids, e.g., [19]–[21]. +The problem of partitioning a grid into a set of islands is +usually mathematized as a combinatorial problem—see for +example [19], [22]–[24]—and sometimes it is recast as a graph +optimization problem [15]–[17]. Often, solving these problems +numerically is cumbersome or inefficient, so that heuristic +strategies are frequently used to seek a suboptimal solution, +while meeting the required computational time that allows the +network to stabilize after a contingency [13], [25]–[27]. +In this paper, we propose a different distributed approach +to solve the ICI problem, where network nodes can migrate +from an island to another so as to self-organise into a partition +†Corresponding author; e-mail: mario.dibernardo@unina.it. +Francesco Lo Iudice, Ricardo Cardona-Rivera, and Mario di Bernardo are +with the Department of Electrical Engineering and Information Technology, +University of Naples Federico II, 80125, Naples, Italy. Antonio Grotta, +Marco Coraggio and Mario di Bernardo are (also) with the Scuola Superiore +Meridionale, School for Advanced Studies, Naples, Italy. +minimizing the power imbalance between different islands and +avoiding large amounts of load shedding. Specifically, starting +from some initial partition of the grid, we endow the nodes +with the ability of locally estimating the power imbalance of +their island and of those neighboring it, so as to decide whether +to migrate or not to a different island from their own. +The estimation strategy is completely distributed and de- +centralized and relies on nodes running a virtual consensus +dynamics parameterized so that the consensus equilibrium the +nodes reach is proportional to the power imbalance of the island +of interest. Under suitable assumptions, we analytically show +that our migration strategy generates a sequence of partitions +that converge in finite time to a configuration whose average +absolute power imbalance falls within a certain bound of the +minimal one. We validate our strategy by partitioning the +IEEE 118 and IEEE 300 test systems, comparing the viable +partitions we obtain to others suggested in previous papers in +the Literature. +II. Preliminaries and problem statement +Notation: Given a set Q, we denote by |Q| its cardinality; +1 is the column vector of ones, with appropriate dimension. +Power grid: We model a power grid as an undirected +connected graph G = (V, E), where V is the set of 𝑛 ∈ N>0 +grid nodes (loads or generators) and E is the set of edges +representing transmission lines interconnecting them. Without +loss of generality, the 𝑛g ∈ N>0 generators are labeled as nodes +1, . . . , 𝑛g, while the 𝑛l ∈ N>0 loads as nodes 𝑛g + 1, . . . , 𝑛. We +let 𝑝𝑖 ∈ R be the active power generated or consumed at node +𝑖; 𝑝𝑖 > 0 if 𝑖 is a generator, while 𝑝𝑖 ≤ 0 if 𝑖 is a load. We +let 𝐴 be the (symmetric) adjacency matrix associated to the +graph G; its (𝑖, 𝑗)-th element 𝑎𝑖 𝑗 being 1 if {𝑖, 𝑗} ∈ E or 0 +otherwise. +Islands and neighbours: We define an island as a connected +subgraph M𝑙 = (V𝑙, E𝑙) of G, where V𝑙 ⊆ V and E𝑙 = +(V𝑙 × V𝑙) ∩ E. Given a set of nodes V𝑙, we denote by N (V𝑙) +the set of neighbours of the nodes in V𝑙, i.e., N (V𝑙) � {𝑖 ∈ +V \ V𝑙 | ∃𝑗 ∈ V𝑙 : {𝑖, 𝑗} ∈ E}. We say that island M𝑚 is +a neighbor of island M𝑙 if and only if N (V𝑚) ∩ V𝑙 ≠ ∅. +Moreover, we denote by N𝑖 the set of neighbours of node 𝑖. +Grid partitions: We say that the grid is partitioned into 𝑛𝜇 ∈ +N>0 islands, described by the subgraphs M𝑙, . . . , M𝑛𝜇, with +corresponding node sets V1, . . . , V𝑛𝜇, if Π = {V1, . . . , V𝑛𝜇} is +a partition of V. Additionally, a node, say 𝑖, in an island, say +M𝑙, is a boundary node if N𝑖 ∩ (V \ V𝑙) ≠ ∅. Furthermore, +we define the condensed graph GΠ = (VΠ, EΠ) induced by the +partition Π, where node 𝑙 in VΠ is associated to V𝑙 in Π, and +an edge {𝑙, 𝑚} exists in EΠ if and only if V𝑙 ∩ N (V𝑚) ≠ ∅. +arXiv:2301.01167v1 [eess.SY] 3 Jan 2023 + +2 +Power imbalance: The power imbalance of an island M𝑙 is +𝑃𝑙 � +∑︁ +𝑖∈V𝑙 +𝑝𝑖; +(1) +the overall grid’s power imbalance is +𝑃tot � +𝑛 +∑︁ +𝑖=1 +𝑝𝑖 = +𝑛𝜇 +∑︁ +𝑙=1 +𝑃𝑙. +(2) +The power imbalance in (1) is associated to the synchronous +frequency deviation of the island from its nominal value, which +in turn is related to the frequency’s stability [9], [28]. Indeed, +if the generated power exceeds loads’ demand, the frequency +increases, and vice-versa. Excessively large variations in the +operating frequency with respect to the nominal one can cause +faults. +Control problem: The problem we study is to find a partition +of the power grid G into 𝑛𝜇 ≥ 2 microgrids so as to minimize +the average absolute power imbalance, defined as +𝐽 � 1 +𝑛𝜇 +𝑛𝜇 +∑︁ +𝑙=1 +|𝑃𝑙|. +(3) +Note that, as �𝑛𝜇 +𝑙=1 |𝑃𝑙| ≥ +���𝑛𝜇 +𝑙=1 𝑃𝑙 +�� = |𝑃tot|, then +𝐽 ≥ 𝐽∗ � +���� +𝑃tot +𝑛𝜇 +���� . +(4) +The cost function in (3) has been used in previous work in +the literature on grid partitioning, e.g. [13], [19], [25], as an +indicator of the ability of a power system to satisfy the loads’ +demand, which is also known as adequacy [29]. +III. A consensus based partitioning strategy +We propose a strategy that, given an initial partition +Π(0) of the power grid into 𝑛𝜇 islands, uses a consensus +algorithm to let the nodes self-organise into a new partition +that minimizes 𝐽, as defined in (3). In particular, at each step +𝑘 of the algorithm, one node can migrate between islands. +We denote by Π(𝑘) the partition after 𝑘 migrations have +occurred; M𝑙(𝑘) = (V𝑙(𝑘), E𝑙(𝑘)), 𝑙 ∈ {1, . . . , 𝑛𝜇} being +the corresponding islands, 𝑃𝑙(𝑘), 𝑙 ∈ {1, . . . , 𝑛𝜇} their power +imbalances, and 𝐽(𝑘) the corresponding value of the cost +function. +Our strategy is based on two fundamental ingredients: +1) a distributed dynamic estimator based on average consen- +sus dynamics that nodes can use to estimate the power +imbalance in their island and in those of their neighbors; +2) a migration condition according to which a boundary +node can decide whether to migrate from its island to a +neighboring one. +Next, we describe the two elements above. +A. Distributed power imbalance estimation +At any step 𝑘, each node, say 𝑖, can obtain an estimate of +the power imbalance, say 𝑃𝑙(𝑘), of the island it belongs to or +of an island neighboring it, say M𝑙(𝑘) = (V𝑙(𝑘), E𝑙(𝑘)), by +running a consensus based estimation strategy. +Specifically, let us define the auxiliary graph � +M𝑙(𝑘) � +( � +V𝑙(𝑘), �E𝑙(𝑘)) with +� +V𝑙(𝑘) � +� +V𝑙(𝑘) \ 𝑖, +if 𝑖 ∈ V𝑙(𝑘), +V𝑙(𝑘) ∪ 𝑖, +if 𝑖 ∉ V𝑙(𝑘), +(5) +and �E𝑙(𝑘) � ( � +V𝑙(𝑘) × � +V𝑙(𝑘)) ∩ E. To estimate 𝑃𝑙(𝑘), node +𝑖 must trigger the distributed solution of the two virtual +continuous-time consensus dynamics given by +�𝑥ℎ(𝑡) = 𝑝ℎ + +∑︁ +{ 𝑗,ℎ}∈E𝑙 (𝑘) +(𝑥 𝑗 (𝑡) − 𝑥ℎ(𝑡)), +∀ℎ ∈ V𝑙(𝑘), (6a) +��𝑥ℎ(𝑡) = 𝑝ℎ + +∑︁ +{ 𝑗,ℎ}∈�E𝑙 (𝑘) +(�𝑥 𝑗 (𝑡) − �𝑥ℎ(𝑡)), +∀ℎ ∈ � +V𝑙(𝑘), (6b) +starting from null initial conditions. Here, 𝑥ℎ(𝑡) and �𝑥ℎ(𝑡) +are the virtual states associated to each node ℎ ∈ V𝑙(𝑘) and +ℎ ∈ � +V𝑙(𝑘), respectively. +Remark 1. To run the consensus dynamics (6) in a distributed +manner, we assume the virtual states 𝑥ℎ and �𝑥ℎ are broadcast +to all neighboring nodes Nℎ ∩ V𝑙(𝑘). +Now, dynamics (6a) can be recast in matrix form as +�𝑥𝑥𝑥(𝑡) = p − 𝐿𝑥𝑥𝑥(𝑡), +(7) +where 𝑥𝑥𝑥 is the stack vector of the virtual states 𝑥ℎ, p is the +stack vector of the power values 𝑝ℎ, and 𝐿 is the (symmetric) +Laplacian matrix associated to M𝑙(𝑘). Let us recall that 1T +is an eigenvector of the symmetric Laplacian 𝐿, with 0 as an +associated eigenvalue. To obtain the asymptotic behaviour of +(7), we premultiply (7) by 1T, obtaining, for all time 𝑡, +1T�𝑥𝑥𝑥(𝑡) = 1Tp = 𝑃𝑙(𝑘). +(8) +Moreover, differentiating (7), we obtain the dynamical system +�𝑥𝑥𝑥(𝑡) = −𝐿�𝑥𝑥𝑥(𝑡), whose dynamics, determined by the spectral +properties of 𝐿, are such that +lim +𝑡→∞ �𝑥𝑥𝑥(𝑡) ∈ span(1). +(9) +Altogether, (8) and (9) imply that lim𝑡→∞ �𝑥𝑥𝑥(𝑡) = 1𝜔𝑙, where +𝜔𝑙 � +𝑃𝑙(𝑘) +|V𝑙(𝑘)| , +(10) +Similarly, from (6b), we obtain that lim𝑡→∞ ��𝑥𝑥𝑥(𝑡) = 1�𝜔𝑙, with +�𝜔𝑙 � +1 +| � +V𝑙(𝑘)| +∑︁ +𝑗 ∈ � +V𝑙 (𝑘) +𝑝 𝑗. +(11) +Exploting (5), (11) can be recast as +�𝜔𝑙 = +� +1 +|V𝑙 (𝑘) |−1 (𝑃𝑙(𝑘) − 𝑝𝑖) , +if 𝑖 ∈ V𝑙(𝑘), +1 +|V𝑙 (𝑘) |+1 (𝑃𝑙(𝑘) + 𝑝𝑖) , +if 𝑖 ∉ V𝑙(𝑘). +(12) +Then, (10) and (12) can be solved together for the unknowns +𝑃𝑙(𝑘) and |V𝑙(𝑘)|, obtaining +𝑃𝑙(𝑘) = 𝑎𝑙𝜔𝑙 +𝑝𝑖 − �𝜔𝑙 +�𝜔𝑙 − 𝜔𝑙 +, +(13a) +|V𝑙(𝑘)| = 𝑎𝑙 +𝑝𝑖 − �𝜔𝑙 +�𝜔𝑙 − 𝜔𝑙 +, +(13b) + +3 +ℳ! +!! +!" +!# +!$ +!% +!& +ℳ" +"#! +#! +#" +"#" +Distributed estimation +$ +ℳ! +"!! +"!" +"!# +$ +ℳ" +"!& +"!% +"!$ +1 +2 +3 +4 +5 +6 +ℳ! +ℳ" +ℳ! +ℳ" +#" = − &$ − '(" +'(" − (" +## = &$ − '(# +'(# − (# +Migration +rule (13) +Migration strategy +"#! +#! +#" +"#" +(a) +(c) +(b) +(d) +Fig. 1: (a) Initial partition of the power network, with dashed +lines representing the communication links among nodes; the +topology being equal to that of the power network itself. (b) +Boundary node 3 triggers the distributed simulation of the +virtual consensus dynamics in (6) for both islands M1 and +M2. (c) The migration rule (14a) is used to decide whether +to migrate from island M1 to M2 and (d) a new partition is +eventually generated. +with +𝑎𝑙 = +� +−1, +if 𝑖 ∈ V𝑙(𝑘), +1, +if 𝑖 ∉ V𝑙(𝑘). +From (13a), to estimate 𝑃𝑙(𝑘), node 𝑖 needs to compute +𝜔𝑙 and �𝜔𝑙. To do so in a distributed fashion, node 𝑖 starts +the distributed computation of the consensus dynamics (6a) +and (6b) by broadcasting its virtual states 𝑥𝑖 and �𝑥𝑖 to the +nodes in N𝑖 ∩ V𝑙(𝑘). In turn, each of these starts sharing +its virtual state with its neighbors (within V𝑙(𝑘)), until all +nodes in V𝑙(𝑘) join the distributed simulation. Note that the +aforementioned procedure can be conducted through one-hop +communication if each node ℎ has knowledge of the index +𝑙 ∈ {1, ..., 𝑛𝜇} of the island it belongs to, and of its consumed or +generated power 𝑝ℎ. Obviously, in a practical implementation, +the grid nodes should be equipped with sufficient computational +and communication capabilities to run the virtual consensus +dynamics on a timescale that is compatible with the grid +requirements. +In what follows, we will show how the network nodes can +exploit this estimation process to self-organise into a partition +of the power network whose power imbalance (3) is rendered +minimal. +B. Migration Condition +A boundary node (see § II), say 𝑖, in island M𝑚(𝑘), can +decide whether to migrate to a neighboring island M𝑙(𝑘) (see +Figure 1) by assessing the power imbalances 𝑃𝑙(𝑘) and 𝑃𝑚(𝑘) +(computed through our estimation strategy in § III-A). +Specifically, at step 𝑘, node 𝑖 will migrate from M𝑚(𝑘) to +M𝑙(𝑘) if and only if +� min(𝑃𝑙(𝑘), 𝑃𝑚(𝑘)) < min(𝑃𝑙(𝑘 + 1), 𝑃𝑚(𝑘 + 1)), +(14a) +M𝑚(𝑘 + 1) is connected, +(14b) +with +𝑃𝑙(𝑘 + 1) = 𝑃𝑙(𝑘) + 𝑝𝑖, +(15a) +𝑃𝑚(𝑘 + 1) = 𝑃𝑚(𝑘) − 𝑝𝑖, +(15b) +V𝑙(𝑘 + 1) = V𝑙(𝑘) ∪ 𝑖, +(15c) +V𝑚(𝑘 + 1) = V𝑚(𝑘) \ {𝑖}. +(15d) +Remark 2. Condition (14b) concerning connectivity can be +ensured using the estimation strategy in § III-A. Indeed, given +an island M𝑙(𝑘), if there exists a node 𝑖 ∈ V𝑙(𝑘) such that +M𝑙(𝑘) \ {𝑖} is not connected, the virtual derivatives ��𝑥ℎ of its +neighbors in � +M𝑙 (see (6b)) will in general converge to different +values, thus providing a warning signal. +C. Migration Algorithm +According to our decentralised partitioning strategy, starting +from some initial partition at step 𝑘 = 0, each boundary node +must trigger the distributed estimation of the power imbalance +of the island it belongs to and of its neighboring islands by +running the virtual consensus dynamics (6). Then, depending +on these power imbalances, exploiting the migration condition +(14a), the boundary nodes will decide whether to migrate or +not to neighboring islands. +For the sake of clarity, we illustrate the process by referring +to the exemplary situation depicted in Figure 1, where a grid +with 𝑛 = 6 nodes is initially partitioned in 𝑛𝜇 = 2 islands, +M1(0) and M2(0) (Figure 1(a)). Then, each boundary node, +as for instance node 3 ∈ V1(0), must decide whether to migrate +to the other island (M2) or not. To this aim, node 3 triggers +the distributed estimation of the power imbalances 𝑃1(0) and +𝑃2(0) in both the islands M1(0) and M2(0) (see Figure 1(b)), +by running two virtual consensus processes of the form (6) +involving all nodes belonging to each of the islands. Once +a steady state in the distributed simulation of (6) has been +reached, node 3 uses the pairs (𝜔1, ˆ𝜔1) and (𝜔2, ˆ𝜔2) to estimate +𝑃1(0) and 𝑃2(0), which it then uses to evaluate the migration +condition (14a) (Figure 1(c)) to assess whether to migrate from +M1 to M2. Once this decision is taken, a new partition is +generated (Figure 1(d)). +In general, our strategy prescribes that the grid nodes get +involved in all the (possibly multiple) distributed consensus +processes invoked according to (6) by the boundary nodes of +their island or of neighboring ones so as to allow the estimation +of the power imbalances of interest. Hence, at any time, each +node will have a number of virtual states corresponding to +the number of estimation processes it is asked to contribute +to. These steps are summarized in Algorithm 1. Additionally, +as soon as a node becomes a boundary node (see § II), it +must trigger additional virtual dynamics to decide whether +to migrate or not from its island to a neighboring one. This +additional procedure is summarized in Algorithm 2. +The following Lemma and Theorem, whose proofs are given +in Section V, state that the migration process governed by rule + +oL +.. +.. +.. +.. +.. +.. +.. +..4 +Algorithm 1: Default routine for any node ℎ. +1 Broadcast all virtual states to neighboring nodes +2 Obtain virtual states from neighboring nodes +3 Integrate (6) for all simulations where ℎ is involved +Algorithm 2: Additional steps for a boundary node ℎ ∈ +V𝑚. +1 Communicate with the nodes in Nℎ ∩ V𝑚 to trigger a +distributed simulation of (6) +2 for 𝑙 : Nℎ ∩ V𝑙 ≠ ∅ do +3 +Communicate with the nodes in Nℎ ∩ V𝑙 to trigger a +distributed simulation of (6) +4 +Wait for steady state in such simulations +5 +Estimate 𝑃𝑚 and 𝑃𝑙, ∀𝑙 : Nℎ ∩ V𝑙 ≠ ∅ using (13) +6 +Decide whether to migrate from M𝑚 to M𝑙 via (14a) +(14a) generates a finite sequence {Π(𝑘)}𝑘 ∈{0,...,𝐾 } of 𝐾 ∈ N +migration steps, and give a bound on the difference between +the cost 𝐽(𝐾) of the final partition and the optimal cost 𝐽∗ +computed in (4). +Lemma 1. If +|𝑃𝑙(𝑘) − 𝑃𝑚(𝑘)| ≤ ¯𝑝 +∀𝑙, 𝑚 : N (V𝑚(𝑘)) ∩ V𝑙(𝑘) ≠ ∅, (16) +where ¯𝑝 � max𝑖∈V |𝑝𝑖|, then +𝐽(𝑘)−𝐽∗ ≤ 2 +𝑛𝜇 +� +𝑛𝜇 +∑︁ +𝑙=𝑙∗+1 +𝑝∗ + ¯𝑝 +� +𝑙 − 𝑛𝜇 + 1 +2 +�� +−(𝑝∗+|𝑝∗|), (17) +with +𝑙∗ = +� +− 𝑝∗ +¯𝑝 + 𝑛𝜇 + 1 +2 +� +, +(18) +and 𝑝∗ � 𝑃tot/𝑛𝜇. +Theorem III.1. Assume that at each step 𝑘 there exist a node +𝑖 and islands M𝑙(𝑘) and M𝑚(𝑘) (that is a triplet (𝑙, 𝑚, 𝑖)) +such that +��� +��� +𝑖 ∈ {V𝑚(𝑘) ∩ N (V𝑙(𝑘))} +∧ +M𝑚(𝑘) \ 𝑖 is connected +(19a) +and +��� +��� +𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) ∧ 𝑝𝑖 < 0 +∨ +𝑃𝑙(𝑘) < 𝑃𝑚(𝑘) ∧ 𝑝𝑖 > 0. +(19b) +Then, the sequence Π(𝑘) obtained under the migration rule +(14a) is finite and converges in 𝐾 < +∞ steps to a partition +Π(𝐾) such that 𝐽(𝑘) fulfills (17) at 𝑘 = 𝐾. +In the following section, we validate the strategy numerically. +A formal proof of convergence is provided later in Section V. +IV. Numerical Validation +We demonstrate the effectiveness of our algorithm by +deploying it to partition the IEEE 118 and 300 testbed cases +[30]. The nodal power values 𝑝𝑖 are computed by solving an +Optimal Power Flow (OPF) problem, leveraging Matpower +6.0 [31]. As the test cases include nodes with null nodal power +𝑝𝑖 = 0, we allow for these nodes to migrate from their island, +say M𝑚(𝑘), to a neighboring island, say M𝑙(𝑘), as long as +(i) their migration does not render M𝑚(𝑘) disconnected and +(ii) 𝑃𝑙(𝑘) ≠ 𝑃𝑙(𝑘′), ∀𝑘′ < 𝑘 : 𝑖 ∈ V𝑙(𝑘′). +To apply our partitioning strategy (Algorithms 1, 2), we need +some initial partitions Π(0), and to test our algorithm under +different conditions, we considered multiple possible Π(0). In +some cases, we took as Π(0) some selected partitions from [13], +[26], [32]. In other cases, we used what we call the SSRP+BFS +approach to generete Π(0). Namely, we first employ the Search +Space Reduction Procedure [26], which generates a spanning +tree connecting groups of coherent generators (these are taken +from [26]). Then, the remaining nodes are aggregated to the +tree using the Breadth-First Search algorithm [33]. +Remark 3. Throughout our numerical analysis, whenever a +node, say 𝑖 ∈ V𝑚(𝑘), can choose to migrate to more than one +island, it will select the one maximizing the difference +Δ𝑃𝑙 = min{𝑃𝑙(𝑘) + 𝑃𝑖, 𝑃𝑚(𝑘) − 𝑃𝑖} − min{𝑃𝑙(𝑘), 𝑃𝑚(𝑘)}. +This choice ensures that the average absolute power imbalance +is improved the most after the migration. +A. IEEE 118 bus system +We used our Algorithm 1-2 to partition the IEEE 118 +test system in 𝑛𝜇 = 2 and 𝑛𝜇 = 3 islands, considering only +𝑛g = 19 generators (excluding the reactive compensators). We +assume that the migration process is triggered by a three phase +solid ground fault at bus 15 forcing line 14-15 to disconnect. +With 𝑛𝜇 = 2, we considered as initial partition Π(0) the one +generated by SSRP+BFS and the final partition reported in [13]; +with 𝑛𝜇 = 3, we considered as Π(0) the partition generated +by SSRP+BFS and the final one reported in [26]. All relevant +information and the results are reported in Table I. +We observe that the proposed algorithm is indeed capable of +converging in all cases towards partitions that minimize 𝐽, as +𝐽(𝐾) = 𝐽∗. As a representative example, we depict in Figure +2 the case that 𝑛𝜇 = 2 and Π(0) is generated by SSPR+BFS; +namely, Figure 2a portrays the power imbalances 𝑃1(𝑘) and +𝑃2(𝑘) at the various steps, while the final partition Π(𝐾) is +reported in Figure 2b. Note that from the OPF results we have +max𝑖 |𝑝𝑖| = 542.78 MW and 𝐽∗ = 58.25 MW and thus the +bound given in Theorem III.1 is satisfied as |𝐽(𝐾) − 𝐽∗| = 0 +(see Table I). +B. IEEE 300 bus system +We used Algorithms 1 and 2 to partition the IEEE 300 test +system in 𝑛𝜇 = 3 and 𝑛𝜇 = 4 islands, assuming a failure affects +line 194-195. With 𝑛𝜇 = 3, as Π(0) we consider the SSRP+BFS +partition and an arbitrary partition reported in Table II; with +𝑛𝜇 = 4, as Π(0) we consider the SSRP+BFS partition and that +from [26]. In both cases, the groups of coherent generators +were selected as in Table II of [26]. All relevant information +and the results are reported in Table II. +Again, in all cases, our algorithm is capable of finding an +optimal partition, as 𝐽(𝐾) = 𝐽∗; notably, the SSRP+BFS initial + +5 +Case +𝑛𝜇 +𝐾 +Cut-set at Π(0) +Π(0) +Cut-set at Π(𝐾) +𝐽 (0) +𝐽 (𝐾) +𝐽 ∗ +𝑃𝑙 (0) +𝑃𝑙 (𝐾) +Bound (17) +IEEE 118 +2 +10 +{24-70, 34-43, +37-40, 38-65, +39-40, 71-72} +SSRP+BFS +{15-19, 18-19, +19-34, 23-25, +23-32, 30-38, +37-38, 37-39, +37-40, 43-44} +120.5 +58.25 +58.25 +{−74.26, +190.75} +{53.74, +62.75} +213.14 +IEEE 118 +2 +9 +{1-2, 3-12, +5-8, 6-7, +11-12, 15-17, +15-19, 24-70, +30-38, 34-36, +44-45, 70-71} +[13] +{4-5, 5-11, +11-12, 15-17, +15-19, 30-38, +34-37, 35-37, +43-44, 69-70, +70-75, 74-75} +265.5 +58.25 +58.25 +{−258.25, +374.74} +{65.75, +50.74} +213.14 +IEEE 118 +3 +7 +{24-70, 34-43, +37-40, 38-65, +39-40, 68-81, +69-77, 71-72, +75-77, 76-118} +SSRP+BFS +{19-34, 21-22, +23-25, 23-32, +30-38, 34-36, +34-37, 37-38, +37-39, 37-40, +68-81, 69-77 +75-77, 76-118} +80.34 +38.83 +38.83 +{−74.26, +1.98, +188.77} +{53.74, +1.98, +60.77} +335.97 +IEEE 118 +3 +8 +{24-70, 24-72, +38-65, 40-42, +41-42, 44-45, +69-77, 75-77, +81-80, 118-76} +[26] +{24-70, 42-49, +44-45, 61-64, +63-64, 65-66, +65-68, 69-77, +71-72, 75-77, +76-118, 80-81} +147 +38.83 +38.83 +{−199.26 +313.77 +1.98} +{83.66, +30.86, +1.98} +335.97 +TABLE I: Results after applying Algorithms 1 and 2 to the IEEE 118 test case, considering different initial partitions Π(0). +Power values are reported in MW. Note that bound (17) is computed for 𝑘 = 𝐾. +partitions and that in [26] are already optimal, but our algorithm +is able to further decrease the standard deviation between the +power imbalances of the three islands (See Table II). +In Figure 3, we report the representative case that 𝑛𝜇 = 3 +and Π(0) is the arbitrary one. The power imbalances 𝑃1(𝑘), +𝑃2(𝑘), 𝑃3(𝑘) are depicted in Figure 3a, while the final partition +Π(𝐾) is portrayed in Figure 3b. Interestingly, across all our +numerical experiments, not only does our algorithm ensure +fulfillment of the bound given in Theorem III.1, but it also +always ensures 𝐽(𝐾) = 𝐽∗, and in all cases it succeeds in +reducing the standard deviation among the power imbalances +of the islands with respect to that of the initial partition (see +Table II). +Finally, we note that, as shown in Table II, for a given test +case and a desired number of islands 𝑛𝜇, there are multiple +optimal solutions minimizing 𝐽. This opens the possibility of +developing a multi-objective partitioning strategy, which might +be the subject of future study. +V. Proof of Convergence +To prove Lemma 1 and Theorem III.1 we first need to define +the stack vector P(𝑘) � [𝑃1(𝑘) · · · 𝑃𝑛𝜇 (𝑘)]T and P∗ � 𝑝∗1, +and then give the following Lemma. +Proof. From (3), we have that +𝐽(𝑘) = 1 +𝑛𝜇 +�� +� +∑︁ +𝑙:𝑃𝑙 (𝑘)>0 +𝑃𝑙(𝑘) − +∑︁ +𝑙:𝑃𝑙 (𝑘) ≤0 +𝑃𝑙(𝑘)�� +� +. +(20) +Moreover, as +∑︁ +𝑙:𝑃𝑙 (𝑘)>0 +𝑃𝑙(𝑘) + +∑︁ +𝑙:𝑃𝑙 (𝑘) ≤0 +𝑃𝑙(𝑘) = 𝑃tot = 𝑛𝜇𝑝∗, +we can recast (20) as +𝐽(𝑘) = 1 +𝑛𝜇 +�� +� +2 +∑︁ +𝑙:𝑃𝑙 (𝑘)>0 +𝑃𝑙(𝑘) − 𝑛𝜇𝑝∗�� +� +. +Hence, as 𝐽∗ = |𝑝∗| [from (4)], we obtain +𝐽(𝑘) − 𝐽∗ = 2 +𝑛𝜇 +∑︁ +𝑙:𝑃𝑙 (𝑘)>0 +𝑃𝑙(𝑘) − (𝑝∗ + |𝑝∗|). +(21) +Without loss of generality, let us relabel the islands so that +𝑃1(𝑘) ≤ 𝑃2(𝑘) ≤ · · · ≤ 𝑃𝑛𝜇 (𝑘). Then, as the graph G (defined +in § II) and all the islands remain connected for all 𝑘, at each +step also the graph GΠ(𝑘) (defined in § II) will be connected +and thus (16) implies that +𝑃𝑙+1(𝑘) ≤ 𝑃𝑙(𝑘) + max +𝑖∈V |𝑝𝑖|, +∀𝑙 ∈ {1, . . . , 𝑛𝜇 − 1}. +(22) +Note that, from (2), �𝑛𝜇 +𝑙=1 𝑃𝑙(𝑘) = 𝑃tot = 𝑛𝜇𝑝∗, and hence from +(22) we obtain +𝑃𝑙(𝑘) ≤ 𝑝∗ + ¯𝑝 +� +𝑙 − 𝑛𝜇 + 1 +2 +� +, +∀𝑙 ∈ {1, . . . , 𝑛𝜇}, +(23) +with ¯𝑝 � max𝑖∈V |𝑝𝑖|. From (21), 𝐽(𝑘) − 𝐽∗ is maximized +(worst case) when (23) is an equality. In such a case, to compute +𝐽(𝑘) − 𝐽∗ by leveraging (21), we must first find +𝑙∗ : +𝑃𝑙(𝑘) ≥ 0, ∀𝑙 ∈ {𝑙∗, . . . , 𝑛𝜇}. +(24) +Hence, to find 𝑙∗ we must find the smallest integer 𝑙 such that +𝑝∗ + ¯𝑝 +� +𝑙 − 𝑛𝜇 + 1 +2 +� +≥ 0, +(25) +yielding (18). Then, from (24), (23), and (21), we obtain (17) +and the Lemma is proved. +□ + +6 +Case +𝑛𝜇 +𝐾 +Cut-set at Π(0) +Π(0) +Cut-set at Π(𝐾) +𝐽 (0) +𝐽 (𝐾) +𝐽 ∗ +𝑃𝑙 (0) +𝑃𝑙 (𝐾) +Bound (17) +IEEE 300 +3 +3 +{3-129, 7-110, +40-68, 54-123, +57-66, 66-190, +67-190, 68-73, +185-186} +SSRP+BFS +{3-129, 40-68, +54-123, 57-66, +64-67, 66-190, +68-73, 109-110, +184-185, 185-187} +102.92 +102.92 +102.92 +{6.11, +129.98, +172.65} +{6.11, +145.98, +156.65} +1254.95 +IEEE 300 +3 +12 +{40-68, 57-66, +66-190, 67-190, +68-73, 106-113, +112-116, 122-123, +185-186} +Arbitrary +{36-40, 39-40, +61-66, 64-67, +65-66, 68-73, +105-106, 106-107, +106-147, 112-116, +119-121, 121-154, +122-124, 122-128, +127-157, 154-158, +157-158, 168-189, +172-187, 177-188, +184-185} +529.49 +102.92 +102.92 +{−639.87, +775.96, +172.65} +{129.21, +18.89, +160.65} +1254.95 +IEEE 300 +4 +5 +{3-129, 7-110, +40-68, 54-123, +61-66, 64-67 +65-66, 68-73, +68-173, 174-198, +185-186} +SSRP+BFS +{3-129, 7-110, +40-68, 54-123, +57-180, 57-190, +66-190, 67-190, +68-73, 68-173, +168-187, 172-187, +174-198, 184-185} +77.187 +77.187 +77.187 +{19.76, +6.11, +205.98, +76.9} +{114.76, +6.11, +110.98, +76.9} +1908.2 +IEEE 300 +4 +3 +{57-66, 64-67, +66-190, 68-173, +109-110, 109-129, +122-123, 174-191, +174-198, 184-185, +185-187} +[26] +{7-110, 57-66, +66-190, 67-190, +68-173, 109-129 +122-123, 168-187, +172-187, 174-191, +174-198, 184-185} +77.187 +77.187 +77.187 +{145.98, +79.76, +6.11 +76.9} +{110.98, +114.76, +6.11, +76.9} +1908.2 +TABLE II: Results after applying Algorithms 1 and 2 to the IEEE 300 test cases, considering different initial partitions Π(0). +Power values are reported in MW. Note that bound (17) is computed for 𝑘 = 𝐾. +Now, let us exploit Lemma 1 to prove Theorem III.1. +Proof of Theorem III.1. Consider a triplet (𝑙, 𝑚, 𝑖) fulfilling +(19), and +|𝑃𝑚(𝑘) − 𝑃𝑙(𝑘)| > |𝑝𝑖|; +(26) +we start by showing that, when assuming (19), (26) is equivalent +to (14a), i.e., a migration of node 𝑖 from island M𝑚 to M𝑙 +will occur. +Firstly, we show that (14a) implies (26). When 𝑃𝑚(𝑘) < +𝑃𝑙(𝑘), we have 𝑝𝑖 < 0 from (19b), and from (14a) we have +that +𝑃𝑚(𝑘) < 𝑃𝑙(𝑘 + 1). +(27) +Differently, when 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘), we have 𝑝𝑖 > 0 from (19b), +and from (14a) we have that +𝑃𝑙(𝑘) < 𝑃𝑚(𝑘 + 1). +(28) +From (27) and (28), recalling (15a) and (15b), we have +� +𝑃𝑚(𝑘) − 𝑃𝑙(𝑘) < 𝑝𝑖, +if 𝑝𝑖 < 0, +𝑃𝑚(𝑘) − 𝑃𝑙(𝑘) > 𝑝𝑖, +if 𝑝𝑖 > 0. +(29) +As (29) implies (26), we have proved that (14a) implies (26). +Now, let us prove that (26) implies (14a). To do so, note +that (26) is equivalent to +� +𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) + |𝑝𝑖|, +if 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘), +𝑃𝑚(𝑘) > 𝑃𝑙(𝑘) + |𝑝𝑖|, +if 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘). +(30) +Moreover, exploiting (19b) and recalling (15a) and (15b), (30) +can be recast as +� +𝑃𝑚(𝑘) < 𝑃𝑙(𝑘) + 𝑝𝑖 = 𝑃𝑙(𝑘 + 1), +if 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘), +𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) − 𝑝𝑖 = 𝑃𝑚(𝑘 + 1), +if 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘). +(31) +It is straightforward to see that (31) immediately leads to (14a). +Therefore, we have proved that (when (19) holds) (26) ⇔ (14a). +As (14a) is equivalent to (26) and (19), then if at some step, +say 𝐾, no triplet (𝑙, 𝑚, 𝑖) existed fulfilling (26), the migration +process would stop and, as the network G is connected and so +is the graph GΠ(𝐾) at that step, we would have +|𝑃𝑙(𝐾)−𝑃𝑚(𝐾)| ≤ max +𝑖∈V |𝑝𝑖| +∀𝑙, 𝑚 : V𝑚(𝐾)∩N (V𝑙(𝐾)) ≠ ∅. +(32) +As from Lemma 1, (32) implies that the bound (17) holds, to +prove our thesis we are left with showing that a stopping time +instant 𝐾 exists. Firstly, note that such a step 𝐾 exists if (14a) +fulfills +∥P(𝑘 + 1) − P∗∥2 ≤ 𝛼 ∥P(𝑘) − P∗∥2 ∀𝑘 ∈ {0, ..., 𝐾 −1} (33) +for some positive scalar 𝛼 < 1 as if (33) were satisfied, then +our migration rule would be a contraction mapping. In such +case, from the Banach-Caccioppoli theorem [34], there would +be no limit cycles in the sequence {P(𝑘)} and thus also in +{Π(𝑘)}. Hence, as the number of possible partitions is finite, +so would be the sequence {P(𝑘)} and thus, to complete our +proof, we need to show that (14a) implies (33). As we have +enforced that only one migration occurs at each step 𝑘, then + +7 +0 +2 +4 +6 +8 +10 +-100 +-50 +0 +50 +100 +150 +200 +(a) +(b) +Fig. 2: Partitioning of the IEEE 118 test system into 𝑛𝜇 = 2 +islands, through Algorithms 1 and 2. (a) 𝑃1(𝑘) (red squares), +𝑃2(𝑘) (green circles), 𝐽(𝑘) (black stars), and 𝐽∗ (dashed +line); all are in MW. (b) Final network partition Π(𝐾); red +square denote V1(𝐾), while green circles denote V2(𝐾). Nodes +72, 24, 23, 22, 21, 39, 20, 19, 38 migrated from M1 to M2 in +the given order, while node 43 migrated from M2 to M1 at +𝑘 = 9. Note that the last migration does not change the power +imbalances as the it involves node 38 with nodal power is zero. +P(𝑘 + 1) only differs from P(𝑘) for the 𝑙-th and 𝑚-th entries. +Hence, proving (33) only requires showing that +(𝑃𝑙(𝑘 + 1) − 𝑝∗)2+(𝑃𝑚(𝑘 + 1) − 𝑝∗)2 +< (𝑃𝑙(𝑘) − 𝑝∗)2 + (𝑃𝑚(𝑘) − 𝑝∗)2 +(34) +for all 𝑘 ∈ {0, ..., 𝐾 − 1}. After a few algebraic simplifications, +(34) can be rewritten as +𝑝𝑖(𝑃𝑙(𝑘) − 𝑃𝑚(𝑘) + 𝑝𝑖) < 0 +𝑘 ∈ {0, ..., 𝐾 − 1}, +(35) +which is trivially fulfilled by any triplet (𝑙, 𝑚, 𝑖) fulfilling (19) +and (26), yielding that (19) and (26) imply (33). In turn, as +(19) and (26) imply (14a), the existence of 𝐾 and thus our +thesis remains proved. +□ +0 +2 +4 +6 +8 +10 +12 +-500 +0 +500 +1000 +(a) +(b) +Fig. 3: Partitioning of the IEEE 300 test system into 𝑛𝜇 = 3 +islands, through Algorithms 1 and 2. (a) 𝑃1(𝑘) (red squares), +𝑃2(𝑘) (green circles), 𝑃3(𝑘) (blue triangles), 𝐽(𝑘) (black +stars), and 𝐽∗ (dashed line). (b) Final network partition Π(𝐾); +red squares denote V1(𝐾), green circles denote V2(𝐾), and +blue triangles denote V3(𝐾). The nodes’ migration order is +106, 122, 185, 187, 168, 188, 127, 66, 121, 158, 67 and 40. +VI. Conclusions +We introduced a power network islanding algorithm that +solves the Intentional Controlled Islanding problem in a +distributed manner. Our strategy allows the network nodes +to self-organise so as to minimize the average absolute power +imbalance among islands. To allow the nodes to make informed +decisions, we devised a consensus-based estimator which is +instrumental to the migration process, as it allows nodes to +estimate the power imbalances of neighboring islands in a +distributed manner. We demonstrated analytically that our +algorithm converges in finite time to a partition whose average +absolute power imbalance is in a given neighborhood of +the optimal one. We tested the strategy on two benchmark +power networks, the IEEE 118 and 300 bus systems, after the +disconnection of one of their transmission lines showing the + +72.2423 +39 +2 +20 +9122 +158127185 +187 +1688 +effectiveness of the proposed approach. +References +[1] F. Dörfler, J. W. Simpson-Porco, and F. Bullo, “Breaking the hierarchy: +Distributed control and economic optimality in microgrids,” IEEE +Transactions on Control of Network Systems, vol. 3, no. 3, pp. 241– +253, 2015. +[2] A. Bidram and A. Davoudi, “Hierarchical structure of microgrids control +system,” IEEE Transactions on Smart Grid, vol. 3, no. 4, pp. 1963–1976, +2012. +[3] P. Frasca, H. Ishii, C. Ravazzi, and R. 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Sims, “Handbook of metric fixed point theory,” +Australian Mathematical Society Gazette, vol. 29, no. 2, 2002. + diff --git a/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/load_file.txt b/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1cd4d60155e6044b4758c56a3467ff76ab5e6e3e --- /dev/null +++ b/o9AzT4oBgHgl3EQfOfuK/content/tmp_files/load_file.txt @@ -0,0 +1,664 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf,len=663 +page_content='1 Consensus-based Distributed Intentional Controlled Islanding of Power Grids Francesco Lo Iudice, Ricardo Cardona-Rivera, Antonio Grotta, Marco Coraggio, Mario di Bernardo† Abstract—The problem of partitioning a power grid into a set of islands can be a solution to restore power dispatchment in sections of a grid affected by an extreme failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Current solutions to this problem usually involve finding the partition of the grid into islands that minimizes the sum of their absolute power imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' This combinatorial problem is often solved through heuristic offline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In this paper, we propose instead a distributed online algorithm through which nodes can migrate among islands, self-organizing the network into a suitable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We prove that, under a set of appropriate assumptions, the proposed solution yields a partition whose absolute power imbalance falls within a given bound of the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We validate our analytical results by testing our partitioning strategy on the IEEE 118 and 300 benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Introduction The penetration of renewable and distributed generation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', [1]–[3], and the possible occurrence of cascading failures [4] have made the problem of devising control strategies to govern the operation of power grids of crucial concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Examples of problems of interest include those reported in [5]–[7], [8], [9] and [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' When the control architecture fails to guarantee reliable operation of the transmission grid, last resort strategies have been devised so as to ensure power dispatchment across at least some of its sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Intentional Controlled Islanding (ICI) strategies address this issue [13]–[17] by identifying sections of the grid (or islands) that can isolate and operate independently from the rest of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Recently, intentional islanding has also been proposed in the framework of distribution networks, see [18] and references therein, as the presence of storage devices and renewable energy generation allows these grids to be partitioned into networks of microgrids, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', [19]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The problem of partitioning a grid into a set of islands is usually mathematized as a combinatorial problem—see for example [19], [22]–[24]—and sometimes it is recast as a graph optimization problem [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Often, solving these problems numerically is cumbersome or inefficient, so that heuristic strategies are frequently used to seek a suboptimal solution, while meeting the required computational time that allows the network to stabilize after a contingency [13], [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In this paper, we propose a different distributed approach to solve the ICI problem, where network nodes can migrate from an island to another so as to self-organise into a partition †Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' e-mail: mario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='dibernardo@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Francesco Lo Iudice, Ricardo Cardona-Rivera, and Mario di Bernardo are with the Department of Electrical Engineering and Information Technology, University of Naples Federico II, 80125, Naples, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Antonio Grotta, Marco Coraggio and Mario di Bernardo are (also) with the Scuola Superiore Meridionale, School for Advanced Studies, Naples, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' minimizing the power imbalance between different islands and avoiding large amounts of load shedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Specifically, starting from some initial partition of the grid, we endow the nodes with the ability of locally estimating the power imbalance of their island and of those neighboring it, so as to decide whether to migrate or not to a different island from their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The estimation strategy is completely distributed and de- centralized and relies on nodes running a virtual consensus dynamics parameterized so that the consensus equilibrium the nodes reach is proportional to the power imbalance of the island of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Under suitable assumptions, we analytically show that our migration strategy generates a sequence of partitions that converge in finite time to a configuration whose average absolute power imbalance falls within a certain bound of the minimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We validate our strategy by partitioning the IEEE 118 and IEEE 300 test systems, comparing the viable partitions we obtain to others suggested in previous papers in the Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Preliminaries and problem statement Notation: Given a set Q, we denote by |Q| its cardinality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 1 is the column vector of ones, with appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Power grid: We model a power grid as an undirected connected graph G = (V, E), where V is the set of 𝑛 ∈ N>0 grid nodes (loads or generators) and E is the set of edges representing transmission lines interconnecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Without loss of generality, the 𝑛g ∈ N>0 generators are labeled as nodes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛g, while the 𝑛l ∈ N>0 loads as nodes 𝑛g + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We let 𝑝𝑖 ∈ R be the active power generated or consumed at node 𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 𝑝𝑖 > 0 if 𝑖 is a generator, while 𝑝𝑖 ≤ 0 if 𝑖 is a load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We let 𝐴 be the (symmetric) adjacency matrix associated to the graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' its (𝑖, 𝑗)-th element 𝑎𝑖 𝑗 being 1 if {𝑖, 𝑗} ∈ E or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Islands and neighbours: We define an island as a connected subgraph M𝑙 = (V𝑙, E𝑙) of G, where V𝑙 ⊆ V and E𝑙 = (V𝑙 × V𝑙) ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Given a set of nodes V𝑙, we denote by N (V𝑙) the set of neighbours of the nodes in V𝑙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', N (V𝑙) � {𝑖 ∈ V \\ V𝑙 | ∃𝑗 ∈ V𝑙 : {𝑖, 𝑗} ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We say that island M𝑚 is a neighbor of island M𝑙 if and only if N (V𝑚) ∩ V𝑙 ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Moreover, we denote by N𝑖 the set of neighbours of node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Grid partitions: We say that the grid is partitioned into 𝑛𝜇 ∈ N>0 islands, described by the subgraphs M𝑙, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , M𝑛𝜇, with corresponding node sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , V𝑛𝜇, if Π = {V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , V𝑛𝜇} is a partition of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Additionally, a node, say 𝑖, in an island, say M𝑙, is a boundary node if N𝑖 ∩ (V \\ V𝑙) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Furthermore, we define the condensed graph GΠ = (VΠ, EΠ) induced by the partition Π, where node 𝑙 in VΠ is associated to V𝑙 in Π, and an edge {𝑙, 𝑚} exists in EΠ if and only if V𝑙 ∩ N (V𝑚) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='01167v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='SY] 3 Jan 2023 2 Power imbalance: The power imbalance of an island M𝑙 is 𝑃𝑙 � ∑︁ 𝑖∈V𝑙 𝑝𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (1) the overall grid’s power imbalance is 𝑃tot � 𝑛 ∑︁ 𝑖=1 𝑝𝑖 = 𝑛𝜇 ∑︁ 𝑙=1 𝑃𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (2) The power imbalance in (1) is associated to the synchronous frequency deviation of the island from its nominal value, which in turn is related to the frequency’s stability [9], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Indeed, if the generated power exceeds loads’ demand, the frequency increases, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Excessively large variations in the operating frequency with respect to the nominal one can cause faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Control problem: The problem we study is to find a partition of the power grid G into 𝑛𝜇 ≥ 2 microgrids so as to minimize the average absolute power imbalance, defined as 𝐽 � 1 𝑛𝜇 𝑛𝜇 ∑︁ 𝑙=1 |𝑃𝑙|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (3) Note that, as �𝑛𝜇 𝑙=1 |𝑃𝑙| ≥ ���𝑛𝜇 𝑙=1 𝑃𝑙 �� = |𝑃tot|, then 𝐽 ≥ 𝐽∗ � ���� 𝑃tot 𝑛𝜇 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (4) The cost function in (3) has been used in previous work in the literature on grid partitioning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' [13], [19], [25], as an indicator of the ability of a power system to satisfy the loads’ demand, which is also known as adequacy [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' A consensus based partitioning strategy We propose a strategy that, given an initial partition Π(0) of the power grid into 𝑛𝜇 islands, uses a consensus algorithm to let the nodes self-organise into a new partition that minimizes 𝐽, as defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In particular, at each step 𝑘 of the algorithm, one node can migrate between islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We denote by Π(𝑘) the partition after 𝑘 migrations have occurred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' M𝑙(𝑘) = (V𝑙(𝑘), E𝑙(𝑘)), 𝑙 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛𝜇} being the corresponding islands, 𝑃𝑙(𝑘), 𝑙 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛𝜇} their power imbalances, and 𝐽(𝑘) the corresponding value of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Our strategy is based on two fundamental ingredients: 1) a distributed dynamic estimator based on average consen- sus dynamics that nodes can use to estimate the power imbalance in their island and in those of their neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 2) a migration condition according to which a boundary node can decide whether to migrate from its island to a neighboring one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Next, we describe the two elements above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Distributed power imbalance estimation At any step 𝑘, each node, say 𝑖, can obtain an estimate of the power imbalance, say 𝑃𝑙(𝑘), of the island it belongs to or of an island neighboring it, say M𝑙(𝑘) = (V𝑙(𝑘), E𝑙(𝑘)), by running a consensus based estimation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Specifically, let us define the auxiliary graph � M𝑙(𝑘) � ( � V𝑙(𝑘), �E𝑙(𝑘)) with � V𝑙(𝑘) � � V𝑙(𝑘) \\ 𝑖, if 𝑖 ∈ V𝑙(𝑘), V𝑙(𝑘) ∪ 𝑖, if 𝑖 ∉ V𝑙(𝑘), (5) and �E𝑙(𝑘) � ( � V𝑙(𝑘) × � V𝑙(𝑘)) ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To estimate 𝑃𝑙(𝑘), node 𝑖 must trigger the distributed solution of the two virtual continuous-time consensus dynamics given by �𝑥ℎ(𝑡) = 𝑝ℎ + ∑︁ { 𝑗,ℎ}∈E𝑙 (𝑘) (𝑥 𝑗 (𝑡) − 𝑥ℎ(𝑡)), ∀ℎ ∈ V𝑙(𝑘), (6a) ��𝑥ℎ(𝑡) = 𝑝ℎ + ∑︁ { 𝑗,ℎ}∈�E𝑙 (𝑘) (�𝑥 𝑗 (𝑡) − �𝑥ℎ(𝑡)), ∀ℎ ∈ � V𝑙(𝑘), (6b) starting from null initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Here, 𝑥ℎ(𝑡) and �𝑥ℎ(𝑡) are the virtual states associated to each node ℎ ∈ V𝑙(𝑘) and ℎ ∈ � V𝑙(𝑘), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To run the consensus dynamics (6) in a distributed manner, we assume the virtual states 𝑥ℎ and �𝑥ℎ are broadcast to all neighboring nodes Nℎ ∩ V𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Now, dynamics (6a) can be recast in matrix form as �𝑥𝑥𝑥(𝑡) = p − 𝐿𝑥𝑥𝑥(𝑡), (7) where 𝑥𝑥𝑥 is the stack vector of the virtual states 𝑥ℎ, p is the stack vector of the power values 𝑝ℎ, and 𝐿 is the (symmetric) Laplacian matrix associated to M𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Let us recall that 1T is an eigenvector of the symmetric Laplacian 𝐿, with 0 as an associated eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To obtain the asymptotic behaviour of (7), we premultiply (7) by 1T, obtaining, for all time 𝑡, 1T�𝑥𝑥𝑥(𝑡) = 1Tp = 𝑃𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (8) Moreover, differentiating (7), we obtain the dynamical system �𝑥𝑥𝑥(𝑡) = −𝐿�𝑥𝑥𝑥(𝑡), whose dynamics, determined by the spectral properties of 𝐿, are such that lim 𝑡→∞ �𝑥𝑥𝑥(𝑡) ∈ span(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (9) Altogether, (8) and (9) imply that lim𝑡→∞ �𝑥𝑥𝑥(𝑡) = 1𝜔𝑙, where 𝜔𝑙 � 𝑃𝑙(𝑘) |V𝑙(𝑘)| , (10) Similarly, from (6b), we obtain that lim𝑡→∞ ��𝑥𝑥𝑥(𝑡) = 1�𝜔𝑙, with �𝜔𝑙 � 1 | � V𝑙(𝑘)| ∑︁ 𝑗 ∈ � V𝑙 (𝑘) 𝑝 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (11) Exploting (5), (11) can be recast as �𝜔𝑙 = � 1 |V𝑙 (𝑘) |−1 (𝑃𝑙(𝑘) − 𝑝𝑖) , if 𝑖 ∈ V𝑙(𝑘), 1 |V𝑙 (𝑘) |+1 (𝑃𝑙(𝑘) + 𝑝𝑖) , if 𝑖 ∉ V𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (12) Then, (10) and (12) can be solved together for the unknowns 𝑃𝑙(𝑘) and |V𝑙(𝑘)|, obtaining 𝑃𝑙(𝑘) = 𝑎𝑙𝜔𝑙 𝑝𝑖 − �𝜔𝑙 �𝜔𝑙 − 𝜔𝑙 , (13a) |V𝑙(𝑘)| = 𝑎𝑙 𝑝𝑖 − �𝜔𝑙 �𝜔𝑙 − 𝜔𝑙 , (13b) 3 ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='$ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='% !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='& ℳ" "#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' #" "#" Distributed estimation $ ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='" "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='# $ ℳ" "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='& "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='% "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='$ 1 2 3 4 5 6 ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' ℳ" ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' ℳ" #" = − &$ − \'(" \'(" − (" ## = &$ − \'(# \'(# − (# Migration rule (13) Migration strategy "#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' #" "#" (a) (c) (b) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 1: (a) Initial partition of the power network, with dashed lines representing the communication links among nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' the topology being equal to that of the power network itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (b) Boundary node 3 triggers the distributed simulation of the virtual consensus dynamics in (6) for both islands M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (c) The migration rule (14a) is used to decide whether to migrate from island M1 to M2 and (d) a new partition is eventually generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' with 𝑎𝑙 = � −1, if 𝑖 ∈ V𝑙(𝑘), 1, if 𝑖 ∉ V𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' From (13a), to estimate 𝑃𝑙(𝑘), node 𝑖 needs to compute 𝜔𝑙 and �𝜔𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To do so in a distributed fashion, node 𝑖 starts the distributed computation of the consensus dynamics (6a) and (6b) by broadcasting its virtual states 𝑥𝑖 and �𝑥𝑖 to the nodes in N𝑖 ∩ V𝑙(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In turn, each of these starts sharing its virtual state with its neighbors (within V𝑙(𝑘)), until all nodes in V𝑙(𝑘) join the distributed simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Note that the aforementioned procedure can be conducted through one-hop communication if each node ℎ has knowledge of the index 𝑙 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', 𝑛𝜇} of the island it belongs to, and of its consumed or generated power 𝑝ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Obviously, in a practical implementation, the grid nodes should be equipped with sufficient computational and communication capabilities to run the virtual consensus dynamics on a timescale that is compatible with the grid requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In what follows, we will show how the network nodes can exploit this estimation process to self-organise into a partition of the power network whose power imbalance (3) is rendered minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Migration Condition A boundary node (see § II), say 𝑖, in island M𝑚(𝑘), can decide whether to migrate to a neighboring island M𝑙(𝑘) (see Figure 1) by assessing the power imbalances 𝑃𝑙(𝑘) and 𝑃𝑚(𝑘) (computed through our estimation strategy in § III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Specifically, at step 𝑘, node 𝑖 will migrate from M𝑚(𝑘) to M𝑙(𝑘) if and only if � min(𝑃𝑙(𝑘), 𝑃𝑚(𝑘)) < min(𝑃𝑙(𝑘 + 1), 𝑃𝑚(𝑘 + 1)), (14a) M𝑚(𝑘 + 1) is connected, (14b) with 𝑃𝑙(𝑘 + 1) = 𝑃𝑙(𝑘) + 𝑝𝑖, (15a) 𝑃𝑚(𝑘 + 1) = 𝑃𝑚(𝑘) − 𝑝𝑖, (15b) V𝑙(𝑘 + 1) = V𝑙(𝑘) ∪ 𝑖, (15c) V𝑚(𝑘 + 1) = V𝑚(𝑘) \\ {𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (15d) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Condition (14b) concerning connectivity can be ensured using the estimation strategy in § III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Indeed, given an island M𝑙(𝑘), if there exists a node 𝑖 ∈ V𝑙(𝑘) such that M𝑙(𝑘) \\ {𝑖} is not connected, the virtual derivatives ��𝑥ℎ of its neighbors in � M𝑙 (see (6b)) will in general converge to different values, thus providing a warning signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Migration Algorithm According to our decentralised partitioning strategy, starting from some initial partition at step 𝑘 = 0, each boundary node must trigger the distributed estimation of the power imbalance of the island it belongs to and of its neighboring islands by running the virtual consensus dynamics (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Then, depending on these power imbalances, exploiting the migration condition (14a), the boundary nodes will decide whether to migrate or not to neighboring islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' For the sake of clarity, we illustrate the process by referring to the exemplary situation depicted in Figure 1, where a grid with 𝑛 = 6 nodes is initially partitioned in 𝑛𝜇 = 2 islands, M1(0) and M2(0) (Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Then, each boundary node, as for instance node 3 ∈ V1(0), must decide whether to migrate to the other island (M2) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To this aim, node 3 triggers the distributed estimation of the power imbalances 𝑃1(0) and 𝑃2(0) in both the islands M1(0) and M2(0) (see Figure 1(b)), by running two virtual consensus processes of the form (6) involving all nodes belonging to each of the islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Once a steady state in the distributed simulation of (6) has been reached, node 3 uses the pairs (𝜔1, ˆ𝜔1) and (𝜔2, ˆ𝜔2) to estimate 𝑃1(0) and 𝑃2(0), which it then uses to evaluate the migration condition (14a) (Figure 1(c)) to assess whether to migrate from M1 to M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Once this decision is taken, a new partition is generated (Figure 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In general, our strategy prescribes that the grid nodes get involved in all the (possibly multiple) distributed consensus processes invoked according to (6) by the boundary nodes of their island or of neighboring ones so as to allow the estimation of the power imbalances of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Hence, at any time, each node will have a number of virtual states corresponding to the number of estimation processes it is asked to contribute to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' These steps are summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Additionally, as soon as a node becomes a boundary node (see § II), it must trigger additional virtual dynamics to decide whether to migrate or not from its island to a neighboring one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' This additional procedure is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The following Lemma and Theorem, whose proofs are given in Section V, state that the migration process governed by rule oL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='.4 Algorithm 1: Default routine for any node ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 1 Broadcast all virtual states to neighboring nodes 2 Obtain virtual states from neighboring nodes 3 Integrate (6) for all simulations where ℎ is involved Algorithm 2: Additional steps for a boundary node ℎ ∈ V𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 1 Communicate with the nodes in Nℎ ∩ V𝑚 to trigger a distributed simulation of (6) 2 for 𝑙 : Nℎ ∩ V𝑙 ≠ ∅ do 3 Communicate with the nodes in Nℎ ∩ V𝑙 to trigger a distributed simulation of (6) 4 Wait for steady state in such simulations 5 Estimate 𝑃𝑚 and 𝑃𝑙, ∀𝑙 : Nℎ ∩ V𝑙 ≠ ∅ using (13) 6 Decide whether to migrate from M𝑚 to M𝑙 via (14a) (14a) generates a finite sequence {Π(𝑘)}𝑘 ∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=',𝐾 } of 𝐾 ∈ N migration steps, and give a bound on the difference between the cost 𝐽(𝐾) of the final partition and the optimal cost 𝐽∗ computed in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' If |𝑃𝑙(𝑘) − 𝑃𝑚(𝑘)| ≤ ¯𝑝 ∀𝑙, 𝑚 : N (V𝑚(𝑘)) ∩ V𝑙(𝑘) ≠ ∅, (16) where ¯𝑝 � max𝑖∈V |𝑝𝑖|, then 𝐽(𝑘)−𝐽∗ ≤ 2 𝑛𝜇 � 𝑛𝜇 ∑︁ 𝑙=𝑙∗+1 𝑝∗ + ¯𝑝 � 𝑙 − 𝑛𝜇 + 1 2 �� −(𝑝∗+|𝑝∗|), (17) with 𝑙∗ = � − 𝑝∗ ¯𝑝 + 𝑛𝜇 + 1 2 � , (18) and 𝑝∗ � 𝑃tot/𝑛𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Assume that at each step 𝑘 there exist a node 𝑖 and islands M𝑙(𝑘) and M𝑚(𝑘) (that is a triplet (𝑙, 𝑚, 𝑖)) such that ��� ��� 𝑖 ∈ {V𝑚(𝑘) ∩ N (V𝑙(𝑘))} ∧ M𝑚(𝑘) \\ 𝑖 is connected (19a) and ��� ��� 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) ∧ 𝑝𝑖 < 0 ∨ 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘) ∧ 𝑝𝑖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (19b) Then, the sequence Π(𝑘) obtained under the migration rule (14a) is finite and converges in 𝐾 < +∞ steps to a partition Π(𝐾) such that 𝐽(𝑘) fulfills (17) at 𝑘 = 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In the following section, we validate the strategy numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' A formal proof of convergence is provided later in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Numerical Validation We demonstrate the effectiveness of our algorithm by deploying it to partition the IEEE 118 and 300 testbed cases [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The nodal power values 𝑝𝑖 are computed by solving an Optimal Power Flow (OPF) problem, leveraging Matpower 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='0 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' As the test cases include nodes with null nodal power 𝑝𝑖 = 0, we allow for these nodes to migrate from their island, say M𝑚(𝑘), to a neighboring island, say M𝑙(𝑘), as long as (i) their migration does not render M𝑚(𝑘) disconnected and (ii) 𝑃𝑙(𝑘) ≠ 𝑃𝑙(𝑘′), ∀𝑘′ < 𝑘 : 𝑖 ∈ V𝑙(𝑘′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To apply our partitioning strategy (Algorithms 1, 2), we need some initial partitions Π(0), and to test our algorithm under different conditions, we considered multiple possible Π(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In some cases, we took as Π(0) some selected partitions from [13], [26], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In other cases, we used what we call the SSRP+BFS approach to generete Π(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Namely, we first employ the Search Space Reduction Procedure [26], which generates a spanning tree connecting groups of coherent generators (these are taken from [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Then, the remaining nodes are aggregated to the tree using the Breadth-First Search algorithm [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Throughout our numerical analysis, whenever a node, say 𝑖 ∈ V𝑚(𝑘), can choose to migrate to more than one island, it will select the one maximizing the difference Δ𝑃𝑙 = min{𝑃𝑙(𝑘) + 𝑃𝑖, 𝑃𝑚(𝑘) − 𝑃𝑖} − min{𝑃𝑙(𝑘), 𝑃𝑚(𝑘)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' This choice ensures that the average absolute power imbalance is improved the most after the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' IEEE 118 bus system We used our Algorithm 1-2 to partition the IEEE 118 test system in 𝑛𝜇 = 2 and 𝑛𝜇 = 3 islands, considering only 𝑛g = 19 generators (excluding the reactive compensators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We assume that the migration process is triggered by a three phase solid ground fault at bus 15 forcing line 14-15 to disconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' With 𝑛𝜇 = 2, we considered as initial partition Π(0) the one generated by SSRP+BFS and the final partition reported in [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' with 𝑛𝜇 = 3, we considered as Π(0) the partition generated by SSRP+BFS and the final one reported in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' All relevant information and the results are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We observe that the proposed algorithm is indeed capable of converging in all cases towards partitions that minimize 𝐽, as 𝐽(𝐾) = 𝐽∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' As a representative example, we depict in Figure 2 the case that 𝑛𝜇 = 2 and Π(0) is generated by SSPR+BFS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' namely, Figure 2a portrays the power imbalances 𝑃1(𝑘) and 𝑃2(𝑘) at the various steps, while the final partition Π(𝐾) is reported in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Note that from the OPF results we have max𝑖 |𝑝𝑖| = 542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='78 MW and 𝐽∗ = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25 MW and thus the bound given in Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1 is satisfied as |𝐽(𝐾) − 𝐽∗| = 0 (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' IEEE 300 bus system We used Algorithms 1 and 2 to partition the IEEE 300 test system in 𝑛𝜇 = 3 and 𝑛𝜇 = 4 islands, assuming a failure affects line 194-195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' With 𝑛𝜇 = 3, as Π(0) we consider the SSRP+BFS partition and an arbitrary partition reported in Table II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' with 𝑛𝜇 = 4, as Π(0) we consider the SSRP+BFS partition and that from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In both cases, the groups of coherent generators were selected as in Table II of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' All relevant information and the results are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Again, in all cases, our algorithm is capable of finding an optimal partition, as 𝐽(𝐾) = 𝐽∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' notably, the SSRP+BFS initial 5 Case 𝑛𝜇 𝐾 Cut-set at Π(0) Π(0) Cut-set at Π(𝐾) 𝐽 (0) 𝐽 (𝐾) 𝐽 ∗ 𝑃𝑙 (0) 𝑃𝑙 (𝐾) Bound (17) IEEE 118 2 10 {24-70, 34-43, 37-40, 38-65, 39-40, 71-72} SSRP+BFS {15-19, 18-19, 19-34, 23-25, 23-32, 30-38, 37-38, 37-39, 37-40, 43-44} 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25 {−74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='26, 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='75} {53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='74, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='75} 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='14 IEEE 118 2 9 {1-2, 3-12, 5-8, 6-7, 11-12, 15-17, 15-19, 24-70, 30-38, 34-36, 44-45, 70-71} [13] {4-5, 5-11, 11-12, 15-17, 15-19, 30-38, 34-37, 35-37, 43-44, 69-70, 70-75, 74-75} 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25 {−258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='25, 374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='74} {65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='75, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='74} 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='14 IEEE 118 3 7 {24-70, 34-43, 37-40, 38-65, 39-40, 68-81, 69-77, 71-72, 75-77, 76-118} SSRP+BFS {19-34, 21-22, 23-25, 23-32, 30-38, 34-36, 34-37, 37-38, 37-39, 37-40, 68-81, 69-77 75-77, 76-118} 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='34 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='83 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='83 {−74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='26, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='77} {53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='74, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='77} 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='97 IEEE 118 3 8 {24-70, 24-72, 38-65, 40-42, 41-42, 44-45, 69-77, 75-77, 81-80, 118-76} [26] {24-70, 42-49, 44-45, 61-64, 63-64, 65-66, 65-68, 69-77, 71-72, 75-77, 76-118, 80-81} 147 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='83 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='83 {−199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='26 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98} {83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='66, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='86, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98} 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='97 TABLE I: Results after applying Algorithms 1 and 2 to the IEEE 118 test case, considering different initial partitions Π(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Power values are reported in MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Note that bound (17) is computed for 𝑘 = 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' partitions and that in [26] are already optimal, but our algorithm is able to further decrease the standard deviation between the power imbalances of the three islands (See Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In Figure 3, we report the representative case that 𝑛𝜇 = 3 and Π(0) is the arbitrary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The power imbalances 𝑃1(𝑘), 𝑃2(𝑘), 𝑃3(𝑘) are depicted in Figure 3a, while the final partition Π(𝐾) is portrayed in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Interestingly, across all our numerical experiments, not only does our algorithm ensure fulfillment of the bound given in Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1, but it also always ensures 𝐽(𝐾) = 𝐽∗, and in all cases it succeeds in reducing the standard deviation among the power imbalances of the islands with respect to that of the initial partition (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Finally, we note that, as shown in Table II, for a given test case and a desired number of islands 𝑛𝜇, there are multiple optimal solutions minimizing 𝐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' This opens the possibility of developing a multi-objective partitioning strategy, which might be the subject of future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Proof of Convergence To prove Lemma 1 and Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1 we first need to define the stack vector P(𝑘) � [𝑃1(𝑘) · · · 𝑃𝑛𝜇 (𝑘)]T and P∗ � 𝑝∗1, and then give the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' From (3), we have that 𝐽(𝑘) = 1 𝑛𝜇 �� � ∑︁ 𝑙:𝑃𝑙 (𝑘)>0 𝑃𝑙(𝑘) − ∑︁ 𝑙:𝑃𝑙 (𝑘) ≤0 𝑃𝑙(𝑘)�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (20) Moreover, as ∑︁ 𝑙:𝑃𝑙 (𝑘)>0 𝑃𝑙(𝑘) + ∑︁ 𝑙:𝑃𝑙 (𝑘) ≤0 𝑃𝑙(𝑘) = 𝑃tot = 𝑛𝜇𝑝∗, we can recast (20) as 𝐽(𝑘) = 1 𝑛𝜇 �� � 2 ∑︁ 𝑙:𝑃𝑙 (𝑘)>0 𝑃𝑙(𝑘) − 𝑛𝜇𝑝∗�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Hence, as 𝐽∗ = |𝑝∗| [from (4)], we obtain 𝐽(𝑘) − 𝐽∗ = 2 𝑛𝜇 ∑︁ 𝑙:𝑃𝑙 (𝑘)>0 𝑃𝑙(𝑘) − (𝑝∗ + |𝑝∗|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (21) Without loss of generality, let us relabel the islands so that 𝑃1(𝑘) ≤ 𝑃2(𝑘) ≤ · · · ≤ 𝑃𝑛𝜇 (𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Then, as the graph G (defined in § II) and all the islands remain connected for all 𝑘, at each step also the graph GΠ(𝑘) (defined in § II) will be connected and thus (16) implies that 𝑃𝑙+1(𝑘) ≤ 𝑃𝑙(𝑘) + max 𝑖∈V |𝑝𝑖|, ∀𝑙 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛𝜇 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (22) Note that, from (2), �𝑛𝜇 𝑙=1 𝑃𝑙(𝑘) = 𝑃tot = 𝑛𝜇𝑝∗, and hence from (22) we obtain 𝑃𝑙(𝑘) ≤ 𝑝∗ + ¯𝑝 � 𝑙 − 𝑛𝜇 + 1 2 � , ∀𝑙 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛𝜇}, (23) with ¯𝑝 � max𝑖∈V |𝑝𝑖|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' From (21), 𝐽(𝑘) − 𝐽∗ is maximized (worst case) when (23) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In such a case, to compute 𝐽(𝑘) − 𝐽∗ by leveraging (21), we must first find 𝑙∗ : 𝑃𝑙(𝑘) ≥ 0, ∀𝑙 ∈ {𝑙∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' , 𝑛𝜇}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (24) Hence, to find 𝑙∗ we must find the smallest integer 𝑙 such that 𝑝∗ + ¯𝑝 � 𝑙 − 𝑛𝜇 + 1 2 � ≥ 0, (25) yielding (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Then, from (24), (23), and (21), we obtain (17) and the Lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' □ 6 Case 𝑛𝜇 𝐾 Cut-set at Π(0) Π(0) Cut-set at Π(𝐾) 𝐽 (0) 𝐽 (𝐾) 𝐽 ∗ 𝑃𝑙 (0) 𝑃𝑙 (𝐾) Bound (17) IEEE 300 3 3 {3-129, 7-110, 40-68, 54-123, 57-66, 66-190, 67-190, 68-73, 185-186} SSRP+BFS {3-129, 40-68, 54-123, 57-66, 64-67, 66-190, 68-73, 109-110, 184-185, 185-187} 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='92 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='92 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='92 {6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11, 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='65} {6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11, 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='65} 1254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='95 IEEE 300 3 12 {40-68, 57-66, 66-190, 67-190, 68-73, 106-113, 112-116, 122-123, 185-186} Arbitrary {36-40, 39-40, 61-66, 64-67, 65-66, 68-73, 105-106, 106-107, 106-147, 112-116, 119-121, 121-154, 122-124, 122-128, 127-157, 154-158, 157-158, 168-189, 172-187, 177-188, 184-185} 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='49 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='92 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='92 {−639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='87, 775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='96, 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='65} {129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='21, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='89, 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='65} 1254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='95 IEEE 300 4 5 {3-129, 7-110, 40-68, 54-123, 61-66, 64-67 65-66, 68-73, 68-173, 174-198, 185-186} SSRP+BFS {3-129, 7-110, 40-68, 54-123, 57-180, 57-190, 66-190, 67-190, 68-73, 68-173, 168-187, 172-187, 174-198, 184-185} 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 {19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='76, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11, 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='9} {114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='76, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11, 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='9} 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='2 IEEE 300 4 3 {57-66, 64-67, 66-190, 68-173, 109-110, 109-129, 122-123, 174-191, 174-198, 184-185, 185-187} [26] {7-110, 57-66, 66-190, 67-190, 68-173, 109-129 122-123, 168-187, 172-187, 174-191, 174-198, 184-185} 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='187 {145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='76, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='9} {110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='98, 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='76, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='11, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='9} 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='2 TABLE II: Results after applying Algorithms 1 and 2 to the IEEE 300 test cases, considering different initial partitions Π(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Power values are reported in MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Note that bound (17) is computed for 𝑘 = 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Now, let us exploit Lemma 1 to prove Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Proof of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Consider a triplet (𝑙, 𝑚, 𝑖) fulfilling (19), and |𝑃𝑚(𝑘) − 𝑃𝑙(𝑘)| > |𝑝𝑖|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (26) we start by showing that, when assuming (19), (26) is equivalent to (14a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', a migration of node 𝑖 from island M𝑚 to M𝑙 will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Firstly, we show that (14a) implies (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' When 𝑃𝑚(𝑘) < 𝑃𝑙(𝑘), we have 𝑝𝑖 < 0 from (19b), and from (14a) we have that 𝑃𝑚(𝑘) < 𝑃𝑙(𝑘 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (27) Differently, when 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘), we have 𝑝𝑖 > 0 from (19b), and from (14a) we have that 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (28) From (27) and (28), recalling (15a) and (15b), we have � 𝑃𝑚(𝑘) − 𝑃𝑙(𝑘) < 𝑝𝑖, if 𝑝𝑖 < 0, 𝑃𝑚(𝑘) − 𝑃𝑙(𝑘) > 𝑝𝑖, if 𝑝𝑖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (29) As (29) implies (26), we have proved that (14a) implies (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Now, let us prove that (26) implies (14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To do so, note that (26) is equivalent to � 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) + |𝑝𝑖|, if 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘), 𝑃𝑚(𝑘) > 𝑃𝑙(𝑘) + |𝑝𝑖|, if 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (30) Moreover, exploiting (19b) and recalling (15a) and (15b), (30) can be recast as � 𝑃𝑚(𝑘) < 𝑃𝑙(𝑘) + 𝑝𝑖 = 𝑃𝑙(𝑘 + 1), if 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘), 𝑃𝑙(𝑘) > 𝑃𝑚(𝑘) − 𝑝𝑖 = 𝑃𝑚(𝑘 + 1), if 𝑃𝑙(𝑘) < 𝑃𝑚(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (31) It is straightforward to see that (31) immediately leads to (14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Therefore, we have proved that (when (19) holds) (26) ⇔ (14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' As (14a) is equivalent to (26) and (19), then if at some step, say 𝐾, no triplet (𝑙, 𝑚, 𝑖) existed fulfilling (26), the migration process would stop and, as the network G is connected and so is the graph GΠ(𝐾) at that step, we would have |𝑃𝑙(𝐾)−𝑃𝑚(𝐾)| ≤ max 𝑖∈V |𝑝𝑖| ∀𝑙, 𝑚 : V𝑚(𝐾)∩N (V𝑙(𝐾)) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (32) As from Lemma 1, (32) implies that the bound (17) holds, to prove our thesis we are left with showing that a stopping time instant 𝐾 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Firstly, note that such a step 𝐾 exists if (14a) fulfills ∥P(𝑘 + 1) − P∗∥2 ≤ 𝛼 ∥P(𝑘) − P∗∥2 ∀𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', 𝐾 −1} (33) for some positive scalar 𝛼 < 1 as if (33) were satisfied, then our migration rule would be a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In such case, from the Banach-Caccioppoli theorem [34], there would be no limit cycles in the sequence {P(𝑘)} and thus also in {Π(𝑘)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Hence, as the number of possible partitions is finite, so would be the sequence {P(𝑘)} and thus, to complete our proof, we need to show that (14a) implies (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' As we have enforced that only one migration occurs at each step 𝑘, then 7 0 2 4 6 8 10 100 50 0 50 100 150 200 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 2: Partitioning of the IEEE 118 test system into 𝑛𝜇 = 2 islands, through Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (a) 𝑃1(𝑘) (red squares), 𝑃2(𝑘) (green circles), 𝐽(𝑘) (black stars), and 𝐽∗ (dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' all are in MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (b) Final network partition Π(𝐾);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' red square denote V1(𝐾), while green circles denote V2(𝐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Nodes 72, 24, 23, 22, 21, 39, 20, 19, 38 migrated from M1 to M2 in the given order, while node 43 migrated from M2 to M1 at 𝑘 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Note that the last migration does not change the power imbalances as the it involves node 38 with nodal power is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' P(𝑘 + 1) only differs from P(𝑘) for the 𝑙-th and 𝑚-th entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Hence, proving (33) only requires showing that (𝑃𝑙(𝑘 + 1) − 𝑝∗)2+(𝑃𝑚(𝑘 + 1) − 𝑝∗)2 < (𝑃𝑙(𝑘) − 𝑝∗)2 + (𝑃𝑚(𝑘) − 𝑝∗)2 (34) for all 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', 𝐾 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' After a few algebraic simplifications, (34) can be rewritten as 𝑝𝑖(𝑃𝑙(𝑘) − 𝑃𝑚(𝑘) + 𝑝𝑖) < 0 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=', 𝐾 − 1}, (35) which is trivially fulfilled by any triplet (𝑙, 𝑚, 𝑖) fulfilling (19) and (26), yielding that (19) and (26) imply (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' In turn, as (19) and (26) imply (14a), the existence of 𝐾 and thus our thesis remains proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' □ 0 2 4 6 8 10 12 500 0 500 1000 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' 3: Partitioning of the IEEE 300 test system into 𝑛𝜇 = 3 islands, through Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (a) 𝑃1(𝑘) (red squares), 𝑃2(𝑘) (green circles), 𝑃3(𝑘) (blue triangles), 𝐽(𝑘) (black stars), and 𝐽∗ (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' (b) Final network partition Π(𝐾);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' red squares denote V1(𝐾), green circles denote V2(𝐾), and blue triangles denote V3(𝐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' The nodes’ migration order is 106, 122, 185, 187, 168, 188, 127, 66, 121, 158, 67 and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Conclusions We introduced a power network islanding algorithm that solves the Intentional Controlled Islanding problem in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' Our strategy allows the network nodes to self-organise so as to minimize the average absolute power imbalance among islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' To allow the nodes to make informed decisions, we devised a consensus-based estimator which is instrumental to the migration process, as it allows nodes to estimate the power imbalances of neighboring islands in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We demonstrated analytically that our algorithm converges in finite time to a partition whose average absolute power imbalance is in a given neighborhood of the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' We tested the strategy on two benchmark power networks, the IEEE 118 and 300 bus systems, after the disconnection of one of their transmission lines showing the 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content='2423 39 2 20 9122 158127185 187 1688 effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfOfuK/content/2301.01167v1.pdf'} +page_content=' References [1] F.' 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This work rather relies only on having +knowledge of the current state and intended direction to make +predictions for multiple vehicles at intersections. Furthermore, +message passing of this information between the vehicles +provides each one of them a more holistic overview of the +environment allowing for a more informed prediction. This is +done by training a neural network which takes the state and +intent of the multiple vehicles to predict their future trajectory. +Using the intention as an input allows our approach to be +extended to additionally control the multiple vehicles to drive +towards desired paths. Experimental results demonstrate the +robustness of our approach both in terms of trajectory predic- +tion and vehicle control at intersections. The complete training +and evaluation code for this work is available here: https:// +github.com/Dekai21/Multi_Agent_Intersection. +Index Terms— Trajectory prediction, Multiple vehicles, Neu- +ral network, Deep learning +I. INTRODUCTION +Over the past decade, deep learning has made tremendous +strides towards the ultimate goal of achieving full driving au- +tonomy [1]. Self-driving vehicles deploy a suite of different +sensors such as RADAR, GPS, IMU, LIDAR, cameras or +their combination for various tasks such as object detection, +classification, localization and navigation [2], [3], [4], [5], +[6]. Among them, vision based sensors (Cameras, Lidar etc.) +have been demonstrated to be most promising in achieving +at par human driving performance. This is because these +sensors are closest to emulating the traits of human vision +in perceiving the driving environment. Coupled with other +sensors, they have been successful in various tasks such as +emergency braking [7], [8], lane keeping [9], [10], pedestrian +detection, object tracking [11], [12] etc. However, such +line-of-sight sensors mounted on ego-vehicles are primarily +concerned with tasks involving single vehicles and therefore +have several limitations of their own: +• They can only partially observe an environment due to +limited field of view, occlusions etc. Hence, they may +not be feasible for executing maneuvers at hustling areas +such as traffic intersections. This is important since a +sizable fraction of vehicle collisions occur at traffic +intersections [13] which also tend to be more severe +[14]. +• Each vehicle has an independent sensor and a separate +processing setup. Therefore, the combined computa- +tional power needed for all the vehicles would be high. +1 Computer Vision Group , CIT, Technical University of Munich. +This work was funded by the Munich Center for Machine Learning +* These authors contributed equally +Moreover, these resources occupy space within the ego- +vehicle and may even require cooling. +• Simulated engines have played a crucial role in testing +and evaluating autonomous driving algorithms. How- +ever, sensor data such as images rendered in simulation +may not be a true reflection of reality. Hence, this +domain shift would preclude deployment in the real +world. +To overcome the issue associated with partial observ- +ability at critical areas such as intersections, a camera can +be deployed in a Birds-Eye-View (BEV) manner simul- +taneously observing all agents in the scene as depicted +in Figure 1. Such top-down images are commonplace for +trajectory prediction [15], [16]. The state of the agents can +be captured with a camera permanently mounted on a high +infrastructure [17], [18] or using drone imagery with up +to centimeter accuracy precision [19] using vision based +object detection algorithms. This state information for each +vehicle can then be used to predict the future trajectory or +sequence of control actions. Note that each vehicle can also +aggregate information about other agents before taking the +appropriate action. Using accurate state information rather +than ego-vehicle mounted sensors such as RGB cameras +has 2 additional advantages: 1) The computational burden +on the resources can be relieved since images with many +pixels being processed independently on each vehicle is no +longer necessary. 2) The domain shift problem caused by +the rendering of images in simulation not matching reality +should no longer be a concern. This is because we are using +the state (location, orientation etc.) of the vehicles as an +abstraction to represent information about them. Hence, with +this abstraction it would be possible to train a model on +one domain and test on another as we demonstrate in the +Experiments. +Figure 1 further shows that the state information of all +vehicles along with their desired intention to go straight, turn +left or right is passed to a Multi-vehicle Trajectory Prediction +(MTP) module. The MTP module predicts the future trajec- +tory for each vehicle based on this provided information. +Within the MTP, the future trajectory prediction is in turn +done by the aggregation module which has shared weights +across all the vehicles. This allows the model to handle an +arbitrary number of vehicles in the scene. Note that to make +a prediction for a particular vehicle, the aggregation module +not only takes information about that specific vehicle but +also considers information of other vehicles through message +passing [20]. This provides each vehicle a holistic overview +arXiv:2301.02561v1 [cs.RO] 6 Jan 2023 + +Fig. 1: Multi-vehicle Trajectory Prediction Framework: Step 1: Object detection is done on a top-down image of an +intersection to extract out the state information of each vehicle in the scene. This information is sent to the Multi-vehicle +Trajectory Prediction (MTP) module. Step 2: Meanwhile, each vehicle in the scene also sends its intention to the MTP. +The intention information informs the MTP whether a certain vehicle intends to turn left, right or keep going straight at +the upcoming intersection. Step 3: The MTP then passes this combined state and intention information to the aggregation +sub-module to predict the future trajectory for each vehicle. Note that the trajectory prediction for each vehicle is not only +dependent on its own state and intention information but also considers that of other vehicles too. The focus of this work is +on the MTP module, where we show how the future trajectory of multiple vehicles can be predicted simultaneously using +their state and intent information. +of the environment, thereby making an informed trajectory +prediction. In contrast, Figure 2 shows the implications of not +aggregating information from other vehicles when making +trajectory predictions. To this end, the contributions of this +work are summarized below: +1) We demonstrate that our approach of using only the +state and intention information outperforms the ap- +proach of using past trajectory information. +2) Our model has the ability to predict the future trajec- +tory of an arbitrary number of vehicles. It aggregates +information from other vehicles; thereby giving better +predictions. +3) We show that the model can be trained on one platform +and tested on another. +4) Our approach of predicting the future trajectory can +easily be extended to also control multiple vehicles +simultaneously at intersections. +5) We have also released the entire codebase for train- +ing and testing our method. The code can be found +here: https://github.com/Dekai21/Multi_ +Agent_Intersection. +Note that the primary emphasis of this work is the MTP +module in Step 3 of Figure 1, where we show how the +future trajectory of multiple vehicles can be predicted si- +multaneously using their state and intent information. Step +1, regarding retrieving the BEV information of the vehicles +and Step 2 regarding transmission of intention information +to the MTP is touched upon in the related work Section II. +Based on this, the following assumptions are made: +1) Access to the state information and intention of the +vehicles is available. +2) In case of control, the vehicles are capable of receiving +the control commands to execute the correct maneuvers +at intersections. +II. RELATED WORK +Wireless Vehicle Communication: +Vehicle to Everything/Infrastructure (V2X/V2I) allows for +wireless communication with the vehicles [21]. V2X/V2I +offers various advantages over the usual line of sight sensors +placed on ego-vehicles [22]. It facilitates reliable data +transmission [23] even among non-line-of-sight vehicles in +the immediate vicinity to give prior warnings of impending +traffic jams [24], emergency braking [25], risky overtaking +[26]. In our framework, it would be needed to transmit +state information from and control commands to the +vehicles. However, the emphasis of our work is multi- +vehicle trajectory prediction and not vehicle communication. +Therefore, we assume that the intention information is +known by utilizing different trajectory datasets/platforms in +our experiments. + +ObjectDetection +Step 3:Multi-Vehicle Traiectory +PredictionModule +Step 1 +VehicleState +V +Step 2 +I want to +turn left. +I want to +i go straight. +Intention +Vehicle1 +Vehicle2 +VehicleN +LEGEND +VehicleX +I want to +go straight. +intention +State + Intention +Aggregation +Trajectory Predicted +of VehicleX +Module +Shared Weights +by the Model +Vehicles in the SceneFig. 2: Figure depicts the importance of aggregating information. Left: describes an initial scene comprising of 4 vehicles. +The white vehicle is the first one arriving the intersection from the top and intends to turn to its left. Meanwhile, the +brown and red vehicles arriving from the bottom intend to turn left and go straight respectively. Middle: With information +aggregation, the brown and red vehicles wait until the white vehicle has left the intersection. Right: With no information +aggregation, the brown and red vehicles start moving earlier and crash into the white vehicle. +Trajectory Datasets: +There are multiple real world datasets which provide +trajectory information of various road participants from a +BEV perspective. For e.g. [17], [18] collect data from top +of buildings, while [15], [16] collect from drone imagery. +However, these datasets contain limited proportion of +trajectories comprising of vehicles in the scene. This would +not be enough in terms of quantity and accuracy to train +data driven learning based algorithms. We therefore train +and test on the real world inD dataset [19], which addresses +these limitations. Note that our approach of trajectory +prediction can also be extended to control multiple vehicles. +This falls under the purview of embodied agent evaluation +[27] and is an emerging topic in the area of deep learning. +However, none of the datasets described above provide the +facility to conduct an online evaluation [28]. We therefore +use the Simulation of Urban Mobility (SUMO) platform +[29]. In the context of this work, SUMO allows creation +and control of various scenarios at intersection for e.g. +the number/intention of vehicles, the priority of the roads, +structure of the intersection etc. After training on SUMO, +we then evaluate the online control of the vehicles on +a completely different platform i.e. Car Learning to Act +(CARLA) [30]. CARLA provides the option to pass the +steering and acceleration/throttle commands to maneuver +multiple vehicles in the scene. +Multi-agent trajectory prediction: +Future trajectory prediction of agents using information +about the social interaction between them is being used for +both pedestrians [31], [32], [33], [34] and vehicles [35], [36], +[37], [38]. Many such methods utilize information about +the past trajectory of vehicles to make inferences about the +future [39], [40]. In [41], [42] the output is probabilistic, +while being multimodal in [43], [44] particularly at points +where a road splits into multiple directions. However, +only one of the multiple alternatives would be valid if the +vehicle intends to traverse a certain direction. Our method +in contrast does not require information about the past +trajectory but rather only the current state of the vehicle. +Also, the predictions of the future trajectory is unique as +our model is conditioned on the intention of the vehicle. +[45] showed that being aware of the intention of other +vehicles improves merging at T-intersections. Knowledge +of intention allows our task of trajectory prediction to be +extended to additionally control the vehicles to reach desired +targets. This is done by applying model predictive control +to determine the appropriate throttle and steering angle such +that the vehicle follows the predicted trajectory. We are +not aware of any previous approaches that take only the +current state and intention for future trajectory prediction +and control of multiple vehicles. A recent work by [46], +does use state and intent information but only for the task +of maintaining a longitudinal safety distance between the +front and rear vehicles. Moreover, their approach uses a +rule based approach, whereas our approach is data-driven +by training a neural network. +Multi-agent Control: +Controlling +a +single +vehicle +at +an +intersections +is +a +complicated task [47]. This is further aggravated when +interaction with other agents also needs to be handled +[48]. The work of [49], [50], [51], [52] control the flow of +multiple vehicles to minimize traffic congestion and collision +at intersections. However, this is done by controlling the +traffic lights. Our network on the other hand deals with +controlling the individual vehicles at intersections that are +void of traffic lights. [53], [54] handle multiple agents using +a leader guided formation control. In our work, all vehicles +are independently controlled. Other approaches to control +the individual vehicles involve solving an optimization +problem [55], [56]. Our approach in contrast is learning + +Frame 2 +Frame 2 +Frame 1 +(with information aggregation) +(w/o information aggregation)based. [57], [58], [59], [60] uses reinforcement learning (RL) +for multi agent prediction/control. However, RL methods +tend to be heavily data-inefficient [61]. Our framework, on +the other hand uses imitation learning complemented with +an additional collision cost to prevent vehicle-to-vehicle +collision when controlling multiple vehicles simultaneously. +III. FRAMEWORK +In this section we describe the details of the Multi-Vehicle +Trajectory (MTP) module depicted in Fig.1. It takes the +state and intention information of each of the N vehicles in +the scene as input and predicts their future trajectory for +T timesteps ahead. We summarize the components of our +framework as follows: +Input: +The information input to the MTP about each vehicle is +represented by the vector Xk ∈ R6, k = 1,2,...N. Xk in turn +comprises of 2 components: 1) State and 2) the intention +of the vehicle. State: The state of vehicle k is in turn +represented by a vector ∈ R3, described by its orientation +(θk ∈ R) and location Sk ∈ R2 on the x−y plane. Intention: +of vehicle k represented by Ik ∈ R3 is a one hot encoded +vector describing whether the vehicle intends to go either +left, right or keep going straight at the upcoming intersection. +Input Transformation: +This input vector Xk for each vehicle is then passed through +a series of L Multi-Layer Perceptron (MLP) layers with +trainable parameters. The output of MLP layer l for each +vehicle k is a latent representation given by Xl +k ∈ Rl and is +specified by the following equation: +Xl +k = +� +Xk +l = 0 +σ(WlXl−1 +k ++bl) +0 < l ≤ L +(1) +where Wl ∈ Rl×(l−1) and bl ∈ Rl are the trainable +parameters of the MLP layer l, while σ is the ReLU +non-linear activation function. +Information Aggregation: +Note that the output of the last MLP layer L for vehicle +k is XL +k and is only dependent on the latent representation +of the same vehicle in the previous layer. In order to make +an informed prediction of the future trajectory of a vehicle, +it would be prudent to not only consider latent information +about itself but also the other vehicles too. Therefore, infor- +mation aggregation is done through message passing in the +successive layers l = L+1,L+2,...LF. This produces a new +latent representation of each vehicle given by the following +equation [62]: +Xl +k = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +XL +k +l = L +σ(Wl +sXl−1 +k ++Wl +o +N +∑ +p=1,p̸=k +Xl−1 +p +) +L < l < LF +Wl +sXl−1 +k ++Wl +o +N +∑ +p=1,p̸=k +Xl−1 +p +l = LF +(2) +where Wl +s, Wl +o ∈ Rl×(l−1) are the trainable parameters +of the aggregation layer. The output of each vehicle k in +the last layer is XLF +k +∈ R2T. It is a prediction of the future +trajectory information Sk of the vehicle k for T timesteps +ahead. Note that in the experiments, we demonstrate the +significance of aggregating information from neighbouring +vehicles. Therein, we show that the performance of the +trained model significantly deteriorates when the second term +in Eq. 2 corresponding to aggregation of information from +the neighbouring nodes is removed. +Note that despite having shared weights, each vehicle +predicts a unique trajectory, since the input vector given +by the state and intention information for each vehicle is +different. +Loss Function: +The loss function used to train our model can be decomposed +into the imitation loss (Limitation) and the collision loss +(Lcollision). The imitation loss is the mean of the L2 distance +between the the future trajectory predicted by the model and +the ground truth. +Limitation = 1 +N +N +∑ +k=1 +T +∑ +t=1 +|St +k − ˆSt +k|2 +(3) +where St +k and ˆSt +k are respectively the predicted and ground +truth state information of vehicle k at timestep t. Meanwhile, +if the future trajectory of any 2 vehicles (e.g. vehicle i +and vehicle j) coincide within a certain safety distance +threshold λ at the same time instance t, then a collision cost +proportional to the excess is added as part of the collision +loss: +Lcollision = ∑ +i, j +Lcollisioni,j +(4) +Lcollisioni, j = +� +� +� +0 +if min +t +|St +i −St +j|2 > λ +λ −min +t +|St +i −St +j|2 +otherwise +(5) +where 1 ≤ i < j ≤ N and 1 ≤ t ≤ T. The purpose of +the collision loss is to mitigate the propensity of vehicle- +to-vehicle collision at intersections. We demonstrate the +importance of this component of the loss function in the +experiments. +Vehicle Control: +Note that our Multi-Vehicle Trajectory module can be +extended to also control the individual vehicles. For this, we +model the car with the bicycle model [63] and apply model +predictive control (MPC) to optimize for the acceleration (a) +and steering angle (δ) such as to follow a selected J number + +of points on the predicted trajectory of the vehicle. MPC has +demonstrated to be of better performance compared to other +controllers [64], [65]. The equation of motion considering +the bicycle model are given by: +˙x = v·cosθ; ˙y = v·sinθ; ˙v = a; ˙θ = vtanδ +L +(6) +where L is the wheelbase and v is the velocity of the +vehicle. Meanwhile, the cost function minimized during +optimization is given by: +min +a,δ +J +∑ +i=1 +[(xi − ˆxi)2 +(yi − ˆyi)2 +(θ i − ˆθi)2] +(7) +Data Augmentation: +Note that when it comes to online vehicle control, training +merely on the recorded data may not be enough. This is +because the parameters controlling the car may cause the +ego-vehicle to diverge from the expected trajectory. This +deviation from the norm would cause the ego-vehicle to +reach scenarios not seen by the model during training, such +as the lane of the oncoming traffic or the road boundaries. +Since, such scenarios are not present in the training set, +the prediction of future trajectory by the model would be +incorrect causing the control parameters to further deviate +the car from the normal trajectory such that it eventually +crashes into the side barrier. Therefore, to prevent these +collisions with the barrier, we additionally augment the +original recorded data by adding some noise to the position +of the car. The output future trajectory is then determined +using model predictive control described by Eq. 6 and 7. +However, the only difference is that, the optimization is to +be done only for the final point on the trajectory, rather than +on the J points on the known trajectory. Experiments show +that inclusion of this augmentation reduces collisions with +the barrier during vehicle control. +IV. EXPERIMENTS +To measure the performance of our framework, we con- +duct both an offline and online evaluation. Offline evaluation +is an assessment of future trajectory prediction of a trained +model. For this we use the real world inD dataset [19]. +Note that our approach of trajectory prediction is also +capable of being extended to control the driving of individual +vehicles at intersections. However, offline evaluation may +not necessarily reflect the true driving quality. In fact, [28] +showed that 2 models with similar offline metrics can have +drastically different performance when deployed in a live +setting. For this, online evaluation wherein the agents can ac- +tively interact with the environment is necessary. Therefore, +we use the CARLA [30] platform for online evaluation with +the model trained on a different platform i.e. SUMO [66]. +The SUMO-CARLA co-simulation facilitates this evaluation. +A. Offline Evaluation: +The Bendplatz and Frankenburg intersections from the +inD dataset shown in Figure 3 have been used for offline +evaluation. For each intersection, 3 track files for training +Fig. 3: A snaphot of the Bendplatz and Frankenburg inter- +sections in Germany available in the inD dataset [19] +and 1 for validation are randomly selected. Each track file +contains 20 minutes of track records collected during differ- +ent times. 4 commonly used offline evaluation metrics are +used for comparison, namely: Average Displacement Error +(ADE) , Final Displacement Error (FDE), Miss Rate (MR) +and Collision Rate (CR) [39], [67]. For the interested reader, +mathematical formulation and interpretation of these metrics, +along with further information regarding the inD dataset used +in the experiments is provided in the supplementary file1. +Apart from our model, 3 additional models were trained +for purpose of comparison. Description of which are +described below: +Past Trajectory (VectorNet): +This model is adapted from the approach of [39]. Like our +approach, it retrieves information about the surrounding +vehicles. +It +uses +an +attention +based +mechanism +for +this purpose. However, this approach additionally uses +information of not just the current state of the vehicle +but also the past trajectory in order to better ascertain the +future trajectory. Our model in contrast uses the intention of +where the vehicle desires to go rather than the past trajectory. +No information aggregation: +The architecture of this model is similar to our model, +except that a vehicle does not aggregate information +from other vehicles in the environment. This is done +by preventing message passing among the vehicles for +trajectory prediction. Moreover, only the imitation learning +loss is used for training. +No collision cost: +This is also similar to our approach, except that this model +is trained without the additional collision cost we introduced +into our imitation learning paradigm. +Ours: +This model is trained using the framework described in +Section III. The model takes information about the state and +intention of each vehicle in the scene and predicts the future +trajectory for each based on this available information. Note +that the model is capable of making each vehicle aggregate +information of other vehicles via message passing. This +holistic representation of the environment ought to facilitate +an informed trajectory prediction that minimizes collisions +between the multiple agents. The model is trained with both +1https://github.com/Dekai21/Multi_Agent_ +Intersection/tree/master/supplementary + +BENDPLATZINTERSECTION +ERANKENBERGERINTERSECTIONthe imitation and collision loss functions. However, note that +data augmentation meant for online vehicle control is not +done here. +The result of offline evaluation for all the 4 models are +given in Table.I and Table.II. +B. Online Evaluation/Control +Online evaluation of the driving quality is done on +the CARLA platform. However, the model is trained on +data from SUMO. The intersection is created such that +the vertical road (top-bottom) has higher priority over the +horizontal road (left-right). The metric used for evaluation +of online driving quality is the Distance Collision Ratio +(DCR). It is an online metric describing the distance +covered by the agents before either a vehicle-to-vehicle +(V2V) or vehicle-to-barrier (V2B) collision occurs. It is +mathematically described as the total distance driven by all +the vehicles at an intersection over the total number of V2V +or V2B collisions that occur. A higher value of this metric +is better. +DCR = 1 +C +N +∑ +k=1 +Tk−1 +∑ +t=1 +� +(xt+1,k −xt,k)2 +(yt+1,k −yt,k)2 +(8) +where C is either the number of V2V or V2B collisions. +Meanwhile, Tk is the number of timesteps it takes for a +vehicle k to cross an intersection. Generally, it is larger for +vehicles taking a left turn as opposed to those taking a right +turn due to the difference in the length of the circumference +of the respective curvatures. +Models used for comparison in this online evaluation are the +same as described in Subsection IV-A for offline evaluation. +The only difference is that 2 additional models are trained +with data augmentation to enhance robustness to deviations +caused by imprecise predictions. The first model is trained +with data augmentation but no collision loss and the other +model is trained with both augmentation and collision loss. +DCR metric for V2V and V2B collision for all these +models +are +described +in +Table +III. +For +purpose +of +reproduciblity, +the +inference +code +for +online +control +and +the +details +of +the +SUMO-CARLA +co-simulation +setup +are +provided +in +the +following +repository: +https://github.com/Dekai21/Multi_Agent_ +Intersection#run-the-inference-code. +C. Discussion: +In this subsection we elaborate some findings from the +results. +Significance of aggregation: +As can be seen, the model with no aggregation of information +from other vehicles under-performs our model. This is be- +cause, intersections are locations where plenty of interaction +among multiple vehicles is expected to happen. Therefore, +with no aggregation, an agent only receives information +about itself and is oblivious to the state, intention and +behaviour of the other vehicles. Hence, it cannot holistically +Fig. 4: Shows an example of the implications of not using ag- +gregation in comparison to our model which uses aggregation +at an intersection on the CARLA simulator. The horizontal +lane (left-right) is the non-priority road, while the vertical +lane (top-bottom) is the priority road. +look at the entire scene before taking an informed decision +about its own trajectory prediction. In case of online evalu- +ation on CARLA, we observed something interesting. Most +crashes occurred not within the intersection but rather just +before the vehicle enters the intersection on the non-priority +road. This is because in the training set, these vehicles +yield the right of way to those on the priority road by +slowing down or even stopping completely before entering +intersection. This is to allow the vehicles on the priority +road to pass without hindrance. Only when there is no +hindrance to other vehicles, the vehicle on the non-priority +road moves in to the intersection. However, such situations +are very rare compared to the number of samples where +the vehicle on the non-priority road sits stationary. Hence, +without receiving knowledge about other agents, the model +memorizes to always remain stationary before entering the +intersection from the non-priority road. This blocks the non- +priority road and prevents other vehicles from passing. In an +ideal world, if a vehicle is blocking a road, the other vehicles +approaching this choke point will be expected to slow down +to prevent a crash. However, these vehicles are also oblivious +to the presence of the blocking vehicle and attempt to drive +through it causing a crash. This particularly lowers the DCR +metric especially for V2V collisions. +Figure 4 demonstrates the consequences of not aggregating +information. As described earlier, this leads to collisions +among the vehicles before they enter the intersection due +to the first stationary vehicle. On the right side of the +same figure, an example scenario of our model which uses +aggregation is presented. The pink vehicle desiring to go +straight moves into the intersection as it is aware that there +is no other vehicle at the intersection. +Contribution of Collision Cost: +It can be observed that our model which uses the collision +cost penalty during training performs better than the model +trained without it. The effect is even more pronounced on +the online metric particularly when it comes to preventing +V2V collisions. Note that the DCR metric for V2V drops + +w/o aggregation +with aggregationTABLE I: Results of trajectory prediction at the Bendplatz Intersection in the InD Dataset. (Lower metric values are better) +Model +Past +Traj. +Intention +Message +Passing +Collision +Cost +ADE +FDE +MR +CR +MR+CR +Past +trajectory +✓ +✓ +3.800 +7.515 +0.816 +0.043 +0.859 +No info. +aggregation +✓ +1.341 +2.619 +0.230 +0.127 +0.357 +No +collision cost +✓ +✓ +1.110 +2.193 +0.172 +0.101 +0.273 +Our +model +✓ +✓ +✓ +1.099 +2.126 +0.157 +0.075 +0.232 +TABLE II: Results of trajectory prediction at the Frankenburg Intersection in the InD Dataset. (Lower metric values are +better) +Model +Past +Traj. +Intention +Message +Passing +Collision +Cost +ADE +FDE +MR +CR +MR+CR +Past +trajectory +✓ +✓ +2.192 +4.437 +0.513 +0.092 +0.605 +No info. +aggregation +✓ +1.958 +3.924 +0.411 +0.147 +0.558 +No +collision cost +✓ +✓ +1.752 +3.518 +0.341 +0.112 +0.453 +Our +model +✓ +✓ +✓ +1.850 +3.623 +0.359 +0.072 +0.431 +TABLE III: Results of Online Evaluation on CARLA. (Higher metric values are better) +Model +Past +Traj. +Intention +Message +Passing +Collision +Cost +MPC +Aug. +DCR (V2V) +DCR (V2B) +Past +trajectory +✓ +✓ +99.1 +158.6 +No info. +aggregation +✓ +168.7 +607.4 +No collision cost & +augmentation +✓ +✓ +722.2 +515.9 +No +augmentation +✓ +✓ +✓ +925.2 +341.3 +No +collision cost +✓ +✓ +✓ +753.6 +1256.0 +Our +model +✓ +✓ +✓ +✓ +3915.0 +1957.5 +significantly when this loss component is removed from the +training. The utility of the collision cost is that it has the +ability to make slight modifications to correct the trajectory +of the vehicles if it senses a potential collision thereby +providing it with the ability to evade other vehicles. The +supplementary material contains a video demonstrating the +implications when collision cost is not used as opposed to +our approach. +Importance of Data Augmentation: +Note that we introduced data augmentation to prevent the +vehicle from deviating and crashing into road barriers during +online evaluation. Comparing the performance of the model +trained without data augmentation shows that the DCR met- +ric is significantly reduced particularly for V2B collisions. +Our model in contrast was trained with data samples at +deviated positions from the normal trajectory. Hence, even +if the model were to end up at divergent positions during +online inference, it would know the corrective action to take +to bring the vehicle back on track. This prevents crashes with +the barrier or other vehicles if they are in the way. +Past Trajectory information: +Recall that the model in [39] uses past trajectory information +of a vehicle in order to predict the future trajectory. Hence, +such models have a probabilistic interpretation, wherein +the precise future trajectory tends to be fuzzy and begins +to become more precise by the time the vehicle reaches +well into the intersection. In contrast, since our model is +provided with information about the intention of the vehicle, +the predictions are unique and much more accurate as can +be seen from the results. This intention allows our approach + +Fig. 5: Demonstrates how intention can be used to control the behaviour and interaction among the vehicles. In the first row +of images, the white circled vehicle coming from the bottom desires to go straight. It keeps moving without yielding to any +other vehicle. In the second row, the intention of the same vehicle is modified to turn left. In this case, the white vehicle +slows down to yield to the red circled vehicles which are moving straight. The white vehicle only starts executing the left +turn once the red vehicles have passed. The arrow on the white circle represents the direction of motion. There is no arrow +in case the white circled vehicle is stationary. Note that the vehicle coming from the right intends to turn right, so it is not +a hindrance when the white circled vehicle intends to turn left. +to be extended to vehicle control. Figure 5 shows that +by manipulating the intention, the interaction among the +vehicles is adjusted accordingly. This flexibility in changing +the behaviour is only possible due to the capability derived +from using intention of the vehicle at the input. Not only are +the offline trajectory predictions more accurate (see Table +I and II ) but the online control is also more robust (see +Table III) in comparison to using past trajectory information. +Domain Adaptation: +Note that our model trained only on data from the SUMO +platform to predict the future trajectory can also be used to +control the vehicle on a completely different platform. In +this case, it is the CARLA platform. Note that data from +CARLA was not available to the model during training. The +reason for this successful adaptation of the model to different +domains is because we are using the state information of the +vehicle as the representation. This representation remains +consistent across different platforms/domains. Hence, the +model is immune to the source of origin of this representation +i.e. CARLA or SUMO. Other representations such as im- +ages have difficulty in switching between different domains, +weather/lighting conditions etc. For e.g. a control model +trained on images from a sunny weather condition would +have difficulty controlling the vehicle in a rainy weather +condition even though the domains may be the same [68]. +Note that the entire code for training and conducting both +offline along with online evaluation is contained in the fol- +lowing repository: https://github.com/Dekai21/ +Multi_Agent_Intersection. +V. CONCLUSION +In this paper, we demonstrated how the trajectory +for multiple vehicles can be predicted simultaneously at +intersections. This is done by utilizing their state and +intention information. This allowed extending the approach +to additionally controlling the vehicles to move towards +desired directions. Aggregating information of other vehicles +further facilitated each vehicle to make better informed +decisions. 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Khan, P. Wenzel, D. Cremers, and L. Leal-Taix´e, +“Towards +generalizing sensorimotor control across weather conditions,” +in +Proceedings of the IEEE/RSJ International Conference on Intelligent +Robots and Systems (IROS), 2019. + diff --git a/q9E0T4oBgHgl3EQfrAGu/content/tmp_files/load_file.txt b/q9E0T4oBgHgl3EQfrAGu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..828c13fe8e56bba40ddd9df7b88cfd88e8490d99 --- /dev/null +++ b/q9E0T4oBgHgl3EQfrAGu/content/tmp_files/load_file.txt @@ -0,0 +1,549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf,len=548 +page_content='Multi-Vehicle Trajectory Prediction at Intersections using State and Intention Information Dekai Zhu1∗, Qadeer Khan1∗ and Daniel Cremers1 Abstract— Traditional approaches to prediction of future trajectory of road agents rely on knowing information about their past trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This work rather relies only on having knowledge of the current state and intended direction to make predictions for multiple vehicles at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Furthermore, message passing of this information between the vehicles provides each one of them a more holistic overview of the environment allowing for a more informed prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is done by training a neural network which takes the state and intent of the multiple vehicles to predict their future trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Using the intention as an input allows our approach to be extended to additionally control the multiple vehicles to drive towards desired paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Experimental results demonstrate the robustness of our approach both in terms of trajectory predic- tion and vehicle control at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The complete training and evaluation code for this work is available here: https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='com/Dekai21/Multi_Agent_Intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Index Terms— Trajectory prediction, Multiple vehicles, Neu- ral network, Deep learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' INTRODUCTION Over the past decade, deep learning has made tremendous strides towards the ultimate goal of achieving full driving au- tonomy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Self-driving vehicles deploy a suite of different sensors such as RADAR, GPS, IMU, LIDAR, cameras or their combination for various tasks such as object detection, classification, localization and navigation [2], [3], [4], [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Among them, vision based sensors (Cameras, Lidar etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=') have been demonstrated to be most promising in achieving at par human driving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is because these sensors are closest to emulating the traits of human vision in perceiving the driving environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Coupled with other sensors, they have been successful in various tasks such as emergency braking [7], [8], lane keeping [9], [10], pedestrian detection, object tracking [11], [12] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, such line-of-sight sensors mounted on ego-vehicles are primarily concerned with tasks involving single vehicles and therefore have several limitations of their own: They can only partially observe an environment due to limited field of view, occlusions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, they may not be feasible for executing maneuvers at hustling areas such as traffic intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is important since a sizable fraction of vehicle collisions occur at traffic intersections [13] which also tend to be more severe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Each vehicle has an independent sensor and a separate processing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, the combined computa- tional power needed for all the vehicles would be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 1 Computer Vision Group , CIT, Technical University of Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This work was funded by the Munich Center for Machine Learning These authors contributed equally Moreover, these resources occupy space within the ego- vehicle and may even require cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Simulated engines have played a crucial role in testing and evaluating autonomous driving algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' How- ever, sensor data such as images rendered in simulation may not be a true reflection of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, this domain shift would preclude deployment in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' To overcome the issue associated with partial observ- ability at critical areas such as intersections, a camera can be deployed in a Birds-Eye-View (BEV) manner simul- taneously observing all agents in the scene as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Such top-down images are commonplace for trajectory prediction [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The state of the agents can be captured with a camera permanently mounted on a high infrastructure [17], [18] or using drone imagery with up to centimeter accuracy precision [19] using vision based object detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This state information for each vehicle can then be used to predict the future trajectory or sequence of control actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that each vehicle can also aggregate information about other agents before taking the appropriate action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Using accurate state information rather than ego-vehicle mounted sensors such as RGB cameras has 2 additional advantages: 1) The computational burden on the resources can be relieved since images with many pixels being processed independently on each vehicle is no longer necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 2) The domain shift problem caused by the rendering of images in simulation not matching reality should no longer be a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is because we are using the state (location, orientation etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=') of the vehicles as an abstraction to represent information about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, with this abstraction it would be possible to train a model on one domain and test on another as we demonstrate in the Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Figure 1 further shows that the state information of all vehicles along with their desired intention to go straight, turn left or right is passed to a Multi-vehicle Trajectory Prediction (MTP) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The MTP module predicts the future trajec- tory for each vehicle based on this provided information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Within the MTP, the future trajectory prediction is in turn done by the aggregation module which has shared weights across all the vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This allows the model to handle an arbitrary number of vehicles in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that to make a prediction for a particular vehicle, the aggregation module not only takes information about that specific vehicle but also considers information of other vehicles through message passing [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This provides each vehicle a holistic overview arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='02561v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='RO] 6 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 1: Multi-vehicle Trajectory Prediction Framework: Step 1: Object detection is done on a top-down image of an intersection to extract out the state information of each vehicle in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This information is sent to the Multi-vehicle Trajectory Prediction (MTP) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Step 2: Meanwhile, each vehicle in the scene also sends its intention to the MTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The intention information informs the MTP whether a certain vehicle intends to turn left, right or keep going straight at the upcoming intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Step 3: The MTP then passes this combined state and intention information to the aggregation sub-module to predict the future trajectory for each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the trajectory prediction for each vehicle is not only dependent on its own state and intention information but also considers that of other vehicles too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The focus of this work is on the MTP module, where we show how the future trajectory of multiple vehicles can be predicted simultaneously using their state and intent information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' of the environment, thereby making an informed trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In contrast, Figure 2 shows the implications of not aggregating information from other vehicles when making trajectory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' To this end, the contributions of this work are summarized below: 1) We demonstrate that our approach of using only the state and intention information outperforms the ap- proach of using past trajectory information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 2) Our model has the ability to predict the future trajec- tory of an arbitrary number of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It aggregates information from other vehicles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' thereby giving better predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 3) We show that the model can be trained on one platform and tested on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 4) Our approach of predicting the future trajectory can easily be extended to also control multiple vehicles simultaneously at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 5) We have also released the entire codebase for train- ing and testing our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The code can be found here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='com/Dekai21/Multi_ Agent_Intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the primary emphasis of this work is the MTP module in Step 3 of Figure 1, where we show how the future trajectory of multiple vehicles can be predicted si- multaneously using their state and intent information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Step 1, regarding retrieving the BEV information of the vehicles and Step 2 regarding transmission of intention information to the MTP is touched upon in the related work Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Based on this, the following assumptions are made: 1) Access to the state information and intention of the vehicles is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 2) In case of control, the vehicles are capable of receiving the control commands to execute the correct maneuvers at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' RELATED WORK Wireless Vehicle Communication: Vehicle to Everything/Infrastructure (V2X/V2I) allows for wireless communication with the vehicles [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' V2X/V2I offers various advantages over the usual line of sight sensors placed on ego-vehicles [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It facilitates reliable data transmission [23] even among non-line-of-sight vehicles in the immediate vicinity to give prior warnings of impending traffic jams [24], emergency braking [25], risky overtaking [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In our framework, it would be needed to transmit state information from and control commands to the vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, the emphasis of our work is multi- vehicle trajectory prediction and not vehicle communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, we assume that the intention information is known by utilizing different trajectory datasets/platforms in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' ObjectDetection Step 3:Multi-Vehicle Traiectory PredictionModule Step 1 VehicleState V Step 2 I want to turn left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' I want to i go straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Intention Vehicle1 Vehicle2 VehicleN LEGEND VehicleX I want to go straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' intention State + Intention Aggregation Trajectory Predicted of VehicleX Module Shared Weights by the Model Vehicles in the SceneFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 2: Figure depicts the importance of aggregating information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Left: describes an initial scene comprising of 4 vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The white vehicle is the first one arriving the intersection from the top and intends to turn to its left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Meanwhile, the brown and red vehicles arriving from the bottom intend to turn left and go straight respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Middle: With information aggregation, the brown and red vehicles wait until the white vehicle has left the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Right: With no information aggregation, the brown and red vehicles start moving earlier and crash into the white vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Trajectory Datasets: There are multiple real world datasets which provide trajectory information of various road participants from a BEV perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' [17], [18] collect data from top of buildings, while [15], [16] collect from drone imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, these datasets contain limited proportion of trajectories comprising of vehicles in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This would not be enough in terms of quantity and accuracy to train data driven learning based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' We therefore train and test on the real world inD dataset [19], which addresses these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that our approach of trajectory prediction can also be extended to control multiple vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This falls under the purview of embodied agent evaluation [27] and is an emerging topic in the area of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, none of the datasets described above provide the facility to conduct an online evaluation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' We therefore use the Simulation of Urban Mobility (SUMO) platform [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In the context of this work, SUMO allows creation and control of various scenarios at intersection for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' the number/intention of vehicles, the priority of the roads, structure of the intersection etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' After training on SUMO, we then evaluate the online control of the vehicles on a completely different platform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Car Learning to Act (CARLA) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' CARLA provides the option to pass the steering and acceleration/throttle commands to maneuver multiple vehicles in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Multi-agent trajectory prediction: Future trajectory prediction of agents using information about the social interaction between them is being used for both pedestrians [31], [32], [33], [34] and vehicles [35], [36], [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Many such methods utilize information about the past trajectory of vehicles to make inferences about the future [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In [41], [42] the output is probabilistic, while being multimodal in [43], [44] particularly at points where a road splits into multiple directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, only one of the multiple alternatives would be valid if the vehicle intends to traverse a certain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our method in contrast does not require information about the past trajectory but rather only the current state of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Also, the predictions of the future trajectory is unique as our model is conditioned on the intention of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' [45] showed that being aware of the intention of other vehicles improves merging at T-intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Knowledge of intention allows our task of trajectory prediction to be extended to additionally control the vehicles to reach desired targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is done by applying model predictive control to determine the appropriate throttle and steering angle such that the vehicle follows the predicted trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' We are not aware of any previous approaches that take only the current state and intention for future trajectory prediction and control of multiple vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' A recent work by [46], does use state and intent information but only for the task of maintaining a longitudinal safety distance between the front and rear vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Moreover, their approach uses a rule based approach, whereas our approach is data-driven by training a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Multi-agent Control: Controlling a single vehicle at an intersections is a complicated task [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is further aggravated when interaction with other agents also needs to be handled [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The work of [49], [50], [51], [52] control the flow of multiple vehicles to minimize traffic congestion and collision at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, this is done by controlling the traffic lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our network on the other hand deals with controlling the individual vehicles at intersections that are void of traffic lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' [53], [54] handle multiple agents using a leader guided formation control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In our work, all vehicles are independently controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Other approaches to control the individual vehicles involve solving an optimization problem [55], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our approach in contrast is learning Frame 2 Frame 2 Frame 1 (with information aggregation) (w/o information aggregation)based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' [57], [58], [59], [60] uses reinforcement learning (RL) for multi agent prediction/control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, RL methods tend to be heavily data-inefficient [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our framework, on the other hand uses imitation learning complemented with an additional collision cost to prevent vehicle-to-vehicle collision when controlling multiple vehicles simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' FRAMEWORK In this section we describe the details of the Multi-Vehicle Trajectory (MTP) module depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It takes the state and intention information of each of the N vehicles in the scene as input and predicts their future trajectory for T timesteps ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' We summarize the components of our framework as follows: Input: The information input to the MTP about each vehicle is represented by the vector Xk ∈ R6, k = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Xk in turn comprises of 2 components: 1) State and 2) the intention of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' State: The state of vehicle k is in turn represented by a vector ∈ R3, described by its orientation (θk ∈ R) and location Sk ∈ R2 on the x−y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Intention: of vehicle k represented by Ik ∈ R3 is a one hot encoded vector describing whether the vehicle intends to go either left, right or keep going straight at the upcoming intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Input Transformation: This input vector Xk for each vehicle is then passed through a series of L Multi-Layer Perceptron (MLP) layers with trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The output of MLP layer l for each vehicle k is a latent representation given by Xl k ∈ Rl and is specified by the following equation: Xl k = � Xk l = 0 σ(WlXl−1 k +bl) 0 < l ≤ L (1) where Wl ∈ Rl×(l−1) and bl ∈ Rl are the trainable parameters of the MLP layer l, while σ is the ReLU non-linear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Information Aggregation: Note that the output of the last MLP layer L for vehicle k is XL k and is only dependent on the latent representation of the same vehicle in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In order to make an informed prediction of the future trajectory of a vehicle, it would be prudent to not only consider latent information about itself but also the other vehicles too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, infor- mation aggregation is done through message passing in the successive layers l = L+1,L+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This produces a new latent representation of each vehicle given by the following equation [62]: Xl k = � � � � � � � � � � � � � � � � � XL k l = L σ(Wl sXl−1 k +Wl o N ∑ p=1,p̸=k Xl−1 p ) L < l < LF Wl sXl−1 k +Wl o N ∑ p=1,p̸=k Xl−1 p l = LF (2) where Wl s, Wl o ∈ Rl×(l−1) are the trainable parameters of the aggregation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The output of each vehicle k in the last layer is XLF k ∈ R2T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It is a prediction of the future trajectory information Sk of the vehicle k for T timesteps ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that in the experiments, we demonstrate the significance of aggregating information from neighbouring vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therein, we show that the performance of the trained model significantly deteriorates when the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 2 corresponding to aggregation of information from the neighbouring nodes is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that despite having shared weights, each vehicle predicts a unique trajectory, since the input vector given by the state and intention information for each vehicle is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Loss Function: The loss function used to train our model can be decomposed into the imitation loss (Limitation) and the collision loss (Lcollision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The imitation loss is the mean of the L2 distance between the the future trajectory predicted by the model and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Limitation = 1 N N ∑ k=1 T ∑ t=1 |St k − ˆSt k|2 (3) where St k and ˆSt k are respectively the predicted and ground truth state information of vehicle k at timestep t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Meanwhile, if the future trajectory of any 2 vehicles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' vehicle i and vehicle j) coincide within a certain safety distance threshold λ at the same time instance t, then a collision cost proportional to the excess is added as part of the collision loss: Lcollision = ∑ i, j Lcollisioni,j (4) Lcollisioni, j = � � � 0 if min t |St i −St j|2 > λ λ −min t |St i −St j|2 otherwise (5) where 1 ≤ i < j ≤ N and 1 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The purpose of the collision loss is to mitigate the propensity of vehicle- to-vehicle collision at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' We demonstrate the importance of this component of the loss function in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Vehicle Control: Note that our Multi-Vehicle Trajectory module can be extended to also control the individual vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For this, we model the car with the bicycle model [63] and apply model predictive control (MPC) to optimize for the acceleration (a) and steering angle (δ) such as to follow a selected J number of points on the predicted trajectory of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' MPC has demonstrated to be of better performance compared to other controllers [64], [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The equation of motion considering the bicycle model are given by: ˙x = v·cosθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' ˙y = v·sinθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' ˙v = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' ˙θ = vtanδ L (6) where L is the wheelbase and v is the velocity of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Meanwhile, the cost function minimized during optimization is given by: min a,δ J ∑ i=1 [(xi − ˆxi)2 +(yi − ˆyi)2 +(θ i − ˆθi)2] (7) Data Augmentation: Note that when it comes to online vehicle control, training merely on the recorded data may not be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is because the parameters controlling the car may cause the ego-vehicle to diverge from the expected trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This deviation from the norm would cause the ego-vehicle to reach scenarios not seen by the model during training, such as the lane of the oncoming traffic or the road boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Since, such scenarios are not present in the training set, the prediction of future trajectory by the model would be incorrect causing the control parameters to further deviate the car from the normal trajectory such that it eventually crashes into the side barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, to prevent these collisions with the barrier, we additionally augment the original recorded data by adding some noise to the position of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The output future trajectory is then determined using model predictive control described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, the only difference is that, the optimization is to be done only for the final point on the trajectory, rather than on the J points on the known trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Experiments show that inclusion of this augmentation reduces collisions with the barrier during vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' EXPERIMENTS To measure the performance of our framework, we con- duct both an offline and online evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Offline evaluation is an assessment of future trajectory prediction of a trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For this we use the real world inD dataset [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that our approach of trajectory prediction is also capable of being extended to control the driving of individual vehicles at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, offline evaluation may not necessarily reflect the true driving quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In fact, [28] showed that 2 models with similar offline metrics can have drastically different performance when deployed in a live setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For this, online evaluation wherein the agents can ac- tively interact with the environment is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, we use the CARLA [30] platform for online evaluation with the model trained on a different platform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' SUMO [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The SUMO-CARLA co-simulation facilitates this evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Offline Evaluation: The Bendplatz and Frankenburg intersections from the inD dataset shown in Figure 3 have been used for offline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For each intersection, 3 track files for training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 3: A snaphot of the Bendplatz and Frankenburg inter- sections in Germany available in the inD dataset [19] and 1 for validation are randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Each track file contains 20 minutes of track records collected during differ- ent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 4 commonly used offline evaluation metrics are used for comparison, namely: Average Displacement Error (ADE) , Final Displacement Error (FDE), Miss Rate (MR) and Collision Rate (CR) [39], [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For the interested reader, mathematical formulation and interpretation of these metrics, along with further information regarding the inD dataset used in the experiments is provided in the supplementary file1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Apart from our model, 3 additional models were trained for purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Description of which are described below: Past Trajectory (VectorNet): This model is adapted from the approach of [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Like our approach, it retrieves information about the surrounding vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It uses an attention based mechanism for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, this approach additionally uses information of not just the current state of the vehicle but also the past trajectory in order to better ascertain the future trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our model in contrast uses the intention of where the vehicle desires to go rather than the past trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' No information aggregation: The architecture of this model is similar to our model, except that a vehicle does not aggregate information from other vehicles in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is done by preventing message passing among the vehicles for trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Moreover, only the imitation learning loss is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' No collision cost: This is also similar to our approach, except that this model is trained without the additional collision cost we introduced into our imitation learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Ours: This model is trained using the framework described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The model takes information about the state and intention of each vehicle in the scene and predicts the future trajectory for each based on this available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the model is capable of making each vehicle aggregate information of other vehicles via message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This holistic representation of the environment ought to facilitate an informed trajectory prediction that minimizes collisions between the multiple agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The model is trained with both 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='com/Dekai21/Multi_Agent_ Intersection/tree/master/supplementary BENDPLATZINTERSECTION ERANKENBERGERINTERSECTIONthe imitation and collision loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, note that data augmentation meant for online vehicle control is not done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The result of offline evaluation for all the 4 models are given in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='I and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Online Evaluation/Control Online evaluation of the driving quality is done on the CARLA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, the model is trained on data from SUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The intersection is created such that the vertical road (top-bottom) has higher priority over the horizontal road (left-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The metric used for evaluation of online driving quality is the Distance Collision Ratio (DCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It is an online metric describing the distance covered by the agents before either a vehicle-to-vehicle (V2V) or vehicle-to-barrier (V2B) collision occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It is mathematically described as the total distance driven by all the vehicles at an intersection over the total number of V2V or V2B collisions that occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' A higher value of this metric is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' DCR = 1 C N ∑ k=1 Tk−1 ∑ t=1 � (xt+1,k −xt,k)2 +(yt+1,k −yt,k)2 (8) where C is either the number of V2V or V2B collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Meanwhile, Tk is the number of timesteps it takes for a vehicle k to cross an intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Generally, it is larger for vehicles taking a left turn as opposed to those taking a right turn due to the difference in the length of the circumference of the respective curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Models used for comparison in this online evaluation are the same as described in Subsection IV-A for offline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The only difference is that 2 additional models are trained with data augmentation to enhance robustness to deviations caused by imprecise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The first model is trained with data augmentation but no collision loss and the other model is trained with both augmentation and collision loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' DCR metric for V2V and V2B collision for all these models are described in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For purpose of reproduciblity, the inference code for online control and the details of the SUMO-CARLA co-simulation setup are provided in the following repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='com/Dekai21/Multi_Agent_ Intersection#run-the-inference-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Discussion: In this subsection we elaborate some findings from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Significance of aggregation: As can be seen, the model with no aggregation of information from other vehicles under-performs our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is be- cause, intersections are locations where plenty of interaction among multiple vehicles is expected to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Therefore, with no aggregation, an agent only receives information about itself and is oblivious to the state, intention and behaviour of the other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, it cannot holistically Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 4: Shows an example of the implications of not using ag- gregation in comparison to our model which uses aggregation at an intersection on the CARLA simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The horizontal lane (left-right) is the non-priority road, while the vertical lane (top-bottom) is the priority road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' look at the entire scene before taking an informed decision about its own trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In case of online evalu- ation on CARLA, we observed something interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Most crashes occurred not within the intersection but rather just before the vehicle enters the intersection on the non-priority road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is because in the training set, these vehicles yield the right of way to those on the priority road by slowing down or even stopping completely before entering intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is to allow the vehicles on the priority road to pass without hindrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Only when there is no hindrance to other vehicles, the vehicle on the non-priority road moves in to the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, such situations are very rare compared to the number of samples where the vehicle on the non-priority road sits stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, without receiving knowledge about other agents, the model memorizes to always remain stationary before entering the intersection from the non-priority road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This blocks the non- priority road and prevents other vehicles from passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In an ideal world, if a vehicle is blocking a road, the other vehicles approaching this choke point will be expected to slow down to prevent a crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' However, these vehicles are also oblivious to the presence of the blocking vehicle and attempt to drive through it causing a crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This particularly lowers the DCR metric especially for V2V collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Figure 4 demonstrates the consequences of not aggregating information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' As described earlier, this leads to collisions among the vehicles before they enter the intersection due to the first stationary vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' On the right side of the same figure, an example scenario of our model which uses aggregation is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The pink vehicle desiring to go straight moves into the intersection as it is aware that there is no other vehicle at the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Contribution of Collision Cost: It can be observed that our model which uses the collision cost penalty during training performs better than the model trained without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The effect is even more pronounced on the online metric particularly when it comes to preventing V2V collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the DCR metric for V2V drops w/o aggregation with aggregationTABLE I: Results of trajectory prediction at the Bendplatz Intersection in the InD Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' (Lower metric values are better) Model Past Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Intention Message Passing Collision Cost ADE FDE MR CR MR+CR Past trajectory ✓ ✓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='800 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='859 No info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' aggregation ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='341 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='357 No collision cost ✓ ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='110 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='273 Our model ✓ ✓ ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='099 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='232 TABLE II: Results of trajectory prediction at the Frankenburg Intersection in the InD Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' (Lower metric values are better) Model Past Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Intention Message Passing Collision Cost ADE FDE MR CR MR+CR Past trajectory ✓ ✓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='192 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='605 No info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' aggregation ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='958 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='558 No collision cost ✓ ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='752 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='453 Our model ✓ ✓ ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='850 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='623 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='431 TABLE III: Results of Online Evaluation on CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' (Higher metric values are better) Model Past Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Intention Message Passing Collision Cost MPC Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' DCR (V2V) DCR (V2B) Past trajectory ✓ ✓ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='1 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='6 No info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' aggregation ✓ 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='7 607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='4 No collision cost & augmentation ✓ ✓ 722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='2 515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='9 No augmentation ✓ ✓ ✓ 925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='2 341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='3 No collision cost ✓ ✓ ✓ 753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='6 1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='0 Our model ✓ ✓ ✓ ✓ 3915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='0 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='5 significantly when this loss component is removed from the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The utility of the collision cost is that it has the ability to make slight modifications to correct the trajectory of the vehicles if it senses a potential collision thereby providing it with the ability to evade other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The supplementary material contains a video demonstrating the implications when collision cost is not used as opposed to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Importance of Data Augmentation: Note that we introduced data augmentation to prevent the vehicle from deviating and crashing into road barriers during online evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Comparing the performance of the model trained without data augmentation shows that the DCR met- ric is significantly reduced particularly for V2B collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our model in contrast was trained with data samples at deviated positions from the normal trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, even if the model were to end up at divergent positions during online inference, it would know the corrective action to take to bring the vehicle back on track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This prevents crashes with the barrier or other vehicles if they are in the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Past Trajectory information: Recall that the model in [39] uses past trajectory information of a vehicle in order to predict the future trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, such models have a probabilistic interpretation, wherein the precise future trajectory tends to be fuzzy and begins to become more precise by the time the vehicle reaches well into the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In contrast, since our model is provided with information about the intention of the vehicle, the predictions are unique and much more accurate as can be seen from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This intention allows our approach Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' 5: Demonstrates how intention can be used to control the behaviour and interaction among the vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In the first row of images, the white circled vehicle coming from the bottom desires to go straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' It keeps moving without yielding to any other vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In the second row, the intention of the same vehicle is modified to turn left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In this case, the white vehicle slows down to yield to the red circled vehicles which are moving straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The white vehicle only starts executing the left turn once the red vehicles have passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The arrow on the white circle represents the direction of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' There is no arrow in case the white circled vehicle is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the vehicle coming from the right intends to turn right, so it is not a hindrance when the white circled vehicle intends to turn left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' to be extended to vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Figure 5 shows that by manipulating the intention, the interaction among the vehicles is adjusted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This flexibility in changing the behaviour is only possible due to the capability derived from using intention of the vehicle at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Not only are the offline trajectory predictions more accurate (see Table I and II ) but the online control is also more robust (see Table III) in comparison to using past trajectory information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Domain Adaptation: Note that our model trained only on data from the SUMO platform to predict the future trajectory can also be used to control the vehicle on a completely different platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' In this case, it is the CARLA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that data from CARLA was not available to the model during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' The reason for this successful adaptation of the model to different domains is because we are using the state information of the vehicle as the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This representation remains consistent across different platforms/domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Hence, the model is immune to the source of origin of this representation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' CARLA or SUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Other representations such as im- ages have difficulty in switching between different domains, weather/lighting conditions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' a control model trained on images from a sunny weather condition would have difficulty controlling the vehicle in a rainy weather condition even though the domains may be the same [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Note that the entire code for training and conducting both offline along with online evaluation is contained in the fol- lowing repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content='com/Dekai21/ Multi_Agent_Intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' CONCLUSION In this paper, we demonstrated how the trajectory for multiple vehicles can be predicted simultaneously at intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This is done by utilizing their state and intention information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' This allowed extending the approach to additionally controlling the vehicles to move towards desired directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Aggregating information of other vehicles further facilitated each vehicle to make better informed decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' Our framework is also capable of being trained on one domain while being tested on another domain, data of which was not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' REFERENCES [1] SAE-International, “Taxonomy & definitions for terms related to driving automation systems for on-road motor vehicles,” SAE Inter- national, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E0T4oBgHgl3EQfrAGu/content/2301.02561v1.pdf'} +page_content=' [2] Kadir Korkmaz, “Producing the location information with the kalman 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Structural and +Commonsense Knowledge +HAODI MA, University of Florida, USA +Knowledge graphs (KG) have served as the key component of various natural language processing applications. Commonsense +knowledge graphs (CKG) are a special type of KG, where entities and relations are composed of free-form text. However, previous +works in KG completion and CKG completion suffer from long-tail relations and newly-added relations which do not have many know +triples for training. In light of this, few-shot KG completion (FKGC), which requires the strengths of graph representation learning and +few-shot learning, has been proposed to challenge the problem of limited annotated data. In this paper, we comprehensively survey +previous attempts on such tasks in the form of a series of methods and applications. Specifically, we first introduce FKGC challenges, +commonly used KGs, and CKGs. Then we systematically categorize and summarize existing works in terms of the type of KGs and the +methods. Finally, we present applications of FKGC models on prediction tasks in different areas and share our thoughts on future +research directions of FKGC. +CCS Concepts: • Computing methodologies → Knowledge representation and reasoning; Reasoning about belief and +knowledge. +Additional Key Words and Phrases: Knowledge graph embeddings, link prediction, knowledge distillation, knowledge graph +ACM Reference Format: +Haodi Ma. 2022. A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge. 1, 1 (Janu- +ary 2022), 27 pages. https://doi.org/XXXXXXX.XXXXXXX +1 +INTRODUCTION +Knowledge graphs (KGs) are a collection of triples, where each triple represents a relation r between the head entity h +and tail entity t. Examples of real-world KGs include Freebase [5], Yago [58] and NELL [10]. These KGs contain millions +of facts and are the fundamental basis for applications like question-answering, recommender systems, and natural +language processing. +Although an immense amount of information is stored in today’s large-scale KGs, they are highly incomplete, which +makes Knowledge Graph Completion (KGC) a challenge for its downstream applications. Recent trends target learning +low-dimension representations of entities and relations for missing link predictions [(Bordes et al., 2013; Trouillon et al., +2016; Dettmers et al., 2017)]. The general idea of these methods is to model and inference various relation patterns +between entities based on known facts in the KG. For example, TransE models relations as translation, aiming at +inversion and composition patterns. Rotate, as one representative, can infer symmetric, asymmetric, inversion, and +composition patterns. +However, such methods usually require sufficient training triples for all relations to learn embeddings. Previous +works [78] show that a large portion of KG relations is long-tail. In other words, they only have a few instances in the +Author’s address: Haodi Ma, ma.haodi@ufl.edu, University of Florida, Gainesville, Florida, USA. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2022 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.01172v1 [cs.CL] 3 Jan 2023 + +2 +Haodi Ma +KG. For example, about 10% of relations in Wikidata have no more than 10 triples. Besides, real-world KGs are often +dynamic, which means new relations and entities will be added whenever new knowledge is acquired. To tackle these +challenges, the model should be capable of predicting new triples given only a small number of examples. +To address the above challenges, [78] proposed two benchmarks, NELL-One and Wiki-One, for few-shot knowledge +graph completion (FKGC) and a baseline model called GMatching. The model introduces a local neighbor encoder +to learn expensive entity representations with only a few samples for each query relation. One branch of recent +works [54, 86] follows a similar approach and achieves considerable performance by improving the quality of embeddings +by considering local graph neighbors. They further argue that entity neighbors should have varied impacts associated +with different task relations. Since relations can be polysemous, reference triples should also make different contributions +to a particular query. For example, if the task relation is isPartOf, as shown in Figure 1, such relation has different +meanings, e.g., organization-related as in (Liverpool, isPartOf, Premier League) or location-related as in +(Gainesville, isPartOf, Florida). Apparently, for a query (Dallas, isPartOf, Taxes), the location-related +references should be more influential than others. These models [43, 54] propose to use attention networks to capture +the dynamic properties of both entities and references. +Fig. 1. Example of (a) an entity with diverse roles, credits to [54] +(b)references showing different impact to a particular query +Bill +Gates +Jennifer +Gates +Melinda +Gates +Microsoft +Chairman +Paul +Allen +WorkWith +ProxyFor +HasJobPosition +MarryTo +HasChild +HasChild +Rory +Gates +CeoOf +? +ParentOfPerson +? +Reference +Query +(Petersburg, SubPartOf, Virginia) +(Vacaville, SubPartOf, California) +(Prague, SubPartOf, Czech) +(Cavaliers, SubPartOf, NBA) +(Los Angeles Lakers, SubPartOf, NBA) +(Chicago Bulls, SubPartOf, NBA) +(b) +(a) +(a) +Reference +(Denver Nugget, isPartOf, NBA) +(Liverpool, isPartOf, Premier League) +(Gainesville, isPartOf, Florida) +(Venice, isPartOf, Italy) +(Montreal, isPartOf, Quebec) +Query +(Dallas, isPartOf, Taxes) +(b) +Fig. 3. A subset of ATOMIC20 +20, credits to [28] +X gets X’s car repaired +a mechanic +money +X wants to call +Uber for a ride +X wants to pay +their bill +The car costs +too much to repair +Fix leaky radiator +garage +repair shop +earned by working +fold into origami +pay repairs +paper +to maintain +the car +Person drives +an old car +Person spent +a fortune +social-interaction +event-centered +physical-entity +capable of +as a result, +X wants +is located at +used for +used for +has property +is made of +desires +because X +wanted +can be +hindered by +happens before +happens after +before, +X needs +Another track of FKGC models [12, 24, 37, 95] is developed based on model-agnostic meta-learning (MAML) [19]. +These models leverage meta-learning to learn the learning process of expressive embeddings of entities and relations +with only a few instances. In particular, they use the high-frequency relations in the training set to capture meta- +information, which includes common features across different task relations. With a good parameter initialization +provided by the meta-information, these models can rapidly adapt to the test tasks where only a few instances are +provided for each task relation. +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +3 +On the other hand, as a particular type of knowledge graphs, commonsense knowledge graphs (CKGs) like +ATOMIC [50] and ConceptNet [57], where entities and relations are composed of free-form text, gain less atten- +tion from embedding-based models. CKGs are dynamic since entities with unseen text are constantly introduced, which +makes them natural benchmarks for FKGC. Besides, entities and attributes in CKGs are usually free-form texts. As +shown in Figure 3, unlike general KGs that have structured entity and relation names, entity descriptions in CKGs have +rich semantic meaning, and implicit semantic relations can be used to infer commonsense knowledge directly, but the +such feature also makes CKGs more sparse comparing with general KGs since entities referring to the same concept can +be distinct nodes. As shown in [67], the average in-degree of ConceptNet and ATOMIC is only 1/15 and 1/8 compared +with FB15K-237. Since CKGs do not cleanly fit into a schema comparing two entities with a relation, embedding-based +methods are limited to capturing implicit commonsense knowledge. +Meanwhile, recent progress in training transformer-based contextual language models [16, 48] has inspired the +interest in using the language models (LMs) as knowledge bases. For example, recent works have focused on querying +the LMs with prompts (e.g., "Beatles was formed in __"). COMET [8] is a transformer-based KG completion model +trained to predict the unseen tail entity conditioning on the head entity and relation on ATOMIC [50]. BertNet [26] +takes a step further to directly extract triples for unseen entities from pre-trained language models by automatically +paraphrasing an initial prompt for FKGC/KGC tasks. +Finally, in this survey, we cover typical applications of FKGC models in data science, visual extraction, and medical +communities. We further discuss future research directions for FKGC on general and commonsense knowledge graphs +based on the observed weakness of current models. +2 +PRELIMINARIES +In this section, we first review different KGs. Then we formally define knowledge graph completion and few-shot +knowledge graph completion. In the final part of this section, we briefly introduce few-shot learning and meta-learning, +which are widely used in FKGC tasks. +2.1 +Knowledge Graph +Let E and R denote the set of entities and relations, a knowledge graph G = {(𝑒𝑖,𝑟𝑘,𝑒𝑗)} ⊂ E × R × E is a collection of +factual triples, where E represents the set of entities, R is the set of relations; 𝑒𝑖 and 𝑟𝑘 are the 𝑖-th entity and 𝑘-th +relation, respectively. We usually refer 𝑒𝑖 and 𝑒𝑗 as the head and tail entity. A knowledge graph can also be represented +as X ∈ {0, 1}|E |×|R |×|E |, which is called the adjacancy tensor of G. The (𝑖, 𝑗,𝑘) entry X𝑖,𝑘,𝑗 = 1 when triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) +is true, otherwise X𝑖,𝑘,𝑗 = 0. A list of commonly-used KGs with their source, size and examples are shown in Table 1 +2.1.1 +structural Knowledge Graph. +As introduced earlier, previous works tend to extract semi-structured text to construct knowledge graphs [5, 39, 58]. +These KGs are usually constructed by crowdsourcing or extracted from crowdsources. +FreeBase. Freebase is a crowdsourced curated KG first introduced in 2008 [5] and has been used as a standard baseline +KG for many tasks, including KG completion. The most up-to-date and complete version of Freebase contains about 3 +billion total triples and about 50 million entities 1. A wide-used subset of Freebase, FB15K-237, excludes inverse relations +from Freebase and includes 14541 entities, 237 relations, and 272,155 training triples. Relations contained in Freebase +1https://developers.google.com/freebase/ +Manuscript submitted to ACM + +4 +Haodi Ma +are hierarchical, which form a well-defined space of entities and relations that motivates the thread of embedding models. +Wikidata. Wikidata is also a crowdsourced KG, containing approximately 78 million data items, with about 23,000 +types and 1,600 relations [45]. At its inception, it was designed to be an alternative method to manage the information +found in Wikipedia. As well as providing factual information, Wikidata gives the context around a fact by storing its +source. As of 2014, Wikidata supported 287 languages [66]. In 2014 Google transferred the data stored in Freebase into +Wikidata [46]. Entities and relations in Wikidata are described through property-value pairs; +YAGO. YAGO is a large knowledge base that is built automatically from Wikipedia. The knowledge graph com- +bines information from Wikipedia in 10 different languages into a whole to provide a multilingual dimension of the +knowledge. It also attaches spatial and temporal information to many facts and thus allows the user to query the data +over space and time. As constructed from Wikipedia, YAGO inherits the hierarchy from Wikipedia and uses structural +text for entities and relations as well. There exists multiple iterative version of YAGO, including YAGO2 and YAGO3. +YAGO3 contains 87 million facts, 10.8 million entities, and 76 million keywords. +2.1.2 +Commonsense Knowledge Graph. +Commonsense knowledge graphs mean to organize commonsense or domain-specific knowledge for downstream appli- +cations. Though existing CKGs [50, 57] are also commonly constructed by human crowdsourcing, they use free-form +text for entities. +ATOMIC. The ATOMIC dataset2, released by [50], contains 877K tuples covering a variety of commonsense so- +cial knowledge around specific event prompts (e.g., "X goes to the store"). ATOMIC contains everyday commonsense +knowledge entities organized as if-then relations. It contains over 300K entities in total, and entities are composed of +text descriptions with an average of 4.4 words. Specifically, ATOMIC distills its commonsense in 9 dimensions, covering +the event’s causes (e.g., "X needs to drive there"), its effects on the agent (e.g., "to get food") and its effect on other direct +(orimplied) participants (e.g., "Others will be fed"). +ConceptNet. ConceptNet [57] is a multilingual knowledge graph that connects words and phrases of natural language +with labeled edges. Its knowledge is collected from many sources that include expert-created resources, crowdsourcing, +and games with a purpose. It represents the general knowledge involved in understanding language using words +and phrases of different languages. Such "concepts" can help natural language applications to understand better the +meanings behind the words people use. ConceptNet contains over 13 million links between these concepts. +Visual Genome. Instead of just using natural language sources, Visual Genome [31] collects commonsense knowledge +from images as well. It collects dense annotations of objects, attributes, and relations with each image to construct the +knowledge. Specifically, Visual Genome contains over 100K images in total, with each image having, on average, 21 +objects, 18 attributes, and 18 relations between objects. Since the objects, attributes, and relations are extracted from +images, the dataset categorizes them with WordNet [41] synsets. In this section, we first review different KGs. Then +we formally define knowledge graph completion and few-shot knowledge graph completion. In the final part of this +section, we briefly introduce few-shot learning and meta-learning, which are widely used in FKGC tasks. +2https://homes.cs.washington.edu/ msap/atomic/ +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +5 +Table 1. Survey of existing knowledge graphs and examples. +source +size +examples +Freebase +https://developers.google.com/freebase +4 relation groups, +2M nodes, 18M +edges +/m/070xg, /sports/sports_team/colors, /m/01g5v +(Seattle seahawks) +(blueish) +/m/06kxk2, /people/person/place_of_birth, /m/01_d4 +(Carl Foreman) +(America/Chicago) +Wikidata +https://www.wikidata.org +1.2k +relations, +75M +objects, +900M edges +Austria:Q40, part_of:P361, European Union:Q458 +The Beatles:Q1299, location_of_formation:P740, +Liverpool:Q24826 +YAGO +https://yago-knowledge.org/ +87 million facts, +10.8 million enti- +ties, and 76 mil- +lion keywords +pl/Henryk_Pietras, wasBornIn, de/Debiensko +fr/Chateau_de_Montcony, isLocatedIn, Burgundy +Concept Net +https://conceptnet.io/ +36 relations, 8M +nodes, 21M edges +/c/en/go_to_bed, /r/HasPrerequisite, +/c/en/get_ready_for_bed +/c/en/section_of_children’s_books, /r/AtLocation, +/c/en/bookstore +/c/pt/atordoaremos/v, /r/FormOf, /c/pt/atordoar +ATOMIC +https://allenai.org/data/atomic-2020 +9 relations, 300k +nodes, 877k edges +money, is used for, pay repairs +money, is made of, paper +PersonX accepts the job, xEffect, joyful +Visual +Genome +https://visualgenome.org/ +42k +relations, +3.8M +nodes, +2.3M edges, 2.8M +attributes +men.n.01, wears.v.01, backpack.n.01 +chair.n.01, has.v.01, padding.n.01 +juice bottle.n.01, on.v.01, desk.n.01 +2.2 +Few-shot Knowledge Graph Completion +2.2.1 +Knowledge Graph Completion. +The objective of knowledge graph completion (KGC) is to predict valid but unobserved triples in G. Formally, given a +head entity 𝑒𝑖 (tail entity 𝑒𝑗) with a relation 𝑟𝑘, models are expected to find the tail entity 𝑒𝑗 (head entity 𝑒𝑖) to form the +most plausible triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) in G. KGC models usually define a scoring function 𝑓 : E × R × E → R to assign a +score 𝑠(𝑒𝑖,𝑟𝑘,𝑒𝑗) to each triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) ∈ E × R × E which indicates the plausibility of the triple. +2.2.2 +Knowledge Graph Embedding. +Knowledge graph embedding (KGE) proposes to project entities and relations into a well-defined space that can be +modeled with high-dimensional vectors. Knowledge embedding (KGE) models usually associate each entity 𝑒𝑖 and +relation 𝑟𝑗 with vector representations e𝑖, r𝑗 in the embedding space. Then they define a scoring function to model the +interactions among entities and relations. +Manuscript submitted to ACM + +6 +Haodi Ma +KGE models can be generally classified into translation and bilinear models. The representative of translation models +is TransE [6], which models the relations between entities as the difference between their embeddings. This method is +effective in inferencing composition, anti-symmetry, and inversion patterns but cannot handle the 1-to-N, N-to-1, and +N-N relations. RotatE [59] models relations as rotations in complex space so that symmetric relations can be captured, +but is as limited as TransE otherwise. ComplEx [64], as a representative of bilinear models, introduces a diagonal matrix +with complex numbers to capture anti-symmetry. Other models, such as BoxE [1], and HAKE [91], can express multiple +types of relationship patterns with complex KG embeddings. +2.2.3 +Graph Neural Network Models. +The Graph Neural Network (GNN) has gained wide attention on KGC tasks in recent years [71, 84, 92]. With the high +expressiveness of GNNs, these methods have shown promising performance. However, SOTA GNN-based models do +not show great advantages compared with KGE models while introducing additional computational complexity [92]. +For example, NBFNet [97] and RED-GNN [90] achieve competitive performance on KGC benchmarks, but the leverage +of the Bellman-Ford algorithm which needs to propagate through the whole knowledge graph, which restrict their +application on large graphs. +2.2.4 +Few-shot Learning. +Few-shot Learning (FSL) [74] focus on learning transferable general prior knowledge from existing tasks for new +tasks with limited labeled data. It usually adopts a meta-learning framework [19] that treats entire tasks as training +examples so that the model can adapt fast to new tasks. Specifically, given a set of tasks T and their training data, in the +meta-training phase, the objective of the model is to learn global parameters 𝜃 ′ that are effective across all tasks in T: +𝜃 ′ = argmax +𝜃 +∑︁ +T𝑖∼𝑝 (T) +L(DT𝑖,𝜃) +where 𝑝(T) is distribution of tasks; DT𝑖 is the training data of task T⟩; L is the loss function of the downstream task. +Then in the meta-testing phase, 𝜃∗ is taken as the initialized parameters (prior knowledge) that are quickly adapted on +a new task T𝑗: +𝜃∗ = L(DT𝑗,𝜃) +where T𝑗 only has limited labeled data. Previous FSL methods can be generally categorized into (1) metric-based +methods [56, 65] that exploit task-specific similarity metrics to generalize from support set data to query data; (2) +optimization-based methods [19, 20, 49] that aim to find model parameters that are sensitive to changes in the task so +that the base learner can quickly adapt to new few-shot tasks with a small number of gradient updates. +2.2.5 +Few-shot Knowledge Graph Completion. +Following the definition of KGC and FSL, we now formally define few-shot knowledge graph completion (FKGC). +Consider a knowledge graph G = {(ℎ,𝑟,𝑡)} ⊂ E × R × E is a collection of factual triples, where E represents the set of +entities, R is the set of relations, respectively. Given a relation𝑟 ∈ R and its supporting set S𝑟 = {(ℎ𝑘,𝑡𝑘)|(ℎ𝑘,𝑟,𝑡𝑘) ∈ T }, +the task is to complete triple (ℎ,𝑟,𝑡) with the tail entity 𝑡 ∈ E missing. In other words, the model needs to predict 𝑡 from +a candidate entity set C given (ℎ,𝑟). When |S𝑟 | = 𝐾 and 𝐾 is very small, the task is called 𝐾-shot KG completion. An +extreme scenario is when 𝑘 = 0, which means there are no supporting triples. Such a task is also referred to as inductive +KGC, zero-shot KGC, or out-of-graph KGC, where models are expected to predict correct relations for unseen entities. +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +7 +A few-shot KGC model aims to rank the true entity higher than the the false candidate entities. In FKGC, each training +task corresponds to a relation 𝑟 ∈ R with its own supporting/query entity pairs, i.e., T𝑟 = {S𝑟, Q𝑟 }. As previously +defined, S contains 𝐾-shot supporting entity pairs. Q𝑟 = {(ℎ𝑚,𝑡𝑚)/Cℎ𝑚,𝑟 } consists of all queries and the corresponding +candidates Cℎ𝑚,𝑟 which are selected based on the entity type constraint [78]. We further denote all the tasks in training +as the meta-training set T𝑚𝑒𝑡𝑎−𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔. +After training on the meta-training set, the few-shot learning model will be tested by predicting facts of new relations +𝑟 ′ ∈ R′. The relations for testing are unseen from the meta-training set, i.e., R ∪ R′ = ∅. Each relation in the testing +phase also has its few-shot supporting and query set: T𝑟′ = {S𝑟′, Q𝑟′}, defined similarly as in meta-training. We denote +all tasks in testing as the meta-testing set T𝑚𝑒𝑡𝑎−𝑡𝑒𝑠𝑡𝑖𝑛𝑔. The model also has access to a background KG G′ which is a +subset of G with all the relations instead of those in T𝑚𝑒𝑡𝑎−𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 and T𝑚𝑒𝑡𝑎−𝑡𝑒𝑠𝑡𝑖𝑛𝑔. +3 +FKGC MODELS +Generally, FKGC models with Structural Knowledge combine KGC models with few-shot learning for various applica- +tions. Besides KGE models, GNN-based methods have also shown competitive performance in FKGC since only limited +labeled data are provided in the supporting set for each few-shot task. The models that leverage semantic features, on +the other hand, utilize prompts to combine structural and semantic information. +Three main challenges exist for FKGC task [30]: +• (1) How to learn the most representative information of triples in the few-shot setting? General machine +learning algorithms require a large number of data for model training, while there are only a few references in +the few-shot scenario. Learning representative patterns of different relations from limited triples becomes the +key to solving the FKGC problem. +• (2) How to decrease the over-reliance on background KGs? Most prior few-shot methods rely on a back- +ground KG to access information from neighborhoods of entities or pre-train the entity embeddings. Some +recent models argue that a thorough background KG is not always accessible, and storing it in memory is also +space-consuming. +• (3) How to utilize the negative samples to enhance the model efficacy? The most intuitive matching ap- +proaches generally compare the similarity between queries and positive references while neglecting the similarity +between queries and negative references, which can improve the accuracy of triplet validity measurement. +In this section, we systematically categorize recent FKGC models with structural knowledge into metric-based +methods and optimization-based methods depending on how they adopt FSL techniques and how they tackle the +three questions above. Then we go from prompt-based structural models to the ones that take advantage of pretrained +language models. A list of representative FKGC works with their open-source dataset/codes is provided in Table2 +3.1 +Metric-based Method +Existing metric-based FKGC models share the framework of either Matching Network [65] or Translation Network [6]. +For models that are built based on Matching Network, they first implement a GNN-based entity encoder to generate +entity embeddings. Then an aggregation module is applied on entity pairs in the supporting set to compute the +embedding of each relation. Finally, the model computes the probability of acceptance of each query triple based on its +similarity to supporting triples. KGE models like TransE [6] and ConvE [15] are also widely used in entity encoder as +the intermediate representation to be further enhanced with other information. +Manuscript submitted to ACM + +8 +Haodi Ma +Method +Learning Task +FSL technique +Venue +Code/Data Link +Structural FKGC +GMatching [78] +Relation prediction +Matching-based +EMNLP’18 +https://github.com/xwhan/One-shot-Relational-Learning +FSRL [86] +Relation prediction +Matching-based +AAAI’20 +https://github.com/chuxuzhang/AAAI2020_FSRL +FAAN [54] +Relation prediction +Matching-based +EMNLP’20 +https://github.com/JiaweiSheng/FAAN +GEN [3] +Relation prediction +Matching-based +NeurIPS’20 +https://github.com/JinheonBaek/GEN +REFORM [70] +Relation prediction +Matching-based +CIKM’21 +https://github.com/SongW-SW/REFORM +MetaR [12] +Relation prediction +Matching-based +EMNLP’19 +https://github.com/AnselCmy/MetaR +GANA [43] +Relation prediction +Matching-based +SIGIR’21 +https://github.com/ngl567/GANA-FewShotKGC +MetaP [30] +Multi-hop relation prediction +Metric-based +SIGIR’21 +https://github.com/jzystc/metap +Meta-KGR [37] +Multi-hop relation prediction +Optimization-based +EMNLP’19 +https://github.com/THU-KEG/MetaKGR +ADK-KG [89] +Multi-hop relation prediction +Optimization-based +SDM’22 +https://github.com/ADK-KG/ADK-KG +ZS-GAN [47] +Multi-hop relation prediction +Optimization-based +NeurIPS’21 +https://github.com/Panda0406/Zero-shot-knowledge-graph-relational-learning +Commonsense FKGC +ConMask [55] +Relation prediction +Text fusion-based +AAAI’18 +https://github.com/bxshi/ConMask +MIA [44] +Relation prediction +Text fusion-based +WWW’21 +-- +InductiveE [54] +Entity prediction +LM-based +IJCNN’21 +https://github.com/BinWang28/InductivE +BERTRL [85] +Relation prediction +LM-based +AAAI’22 +https://github.com/zhw12/BERTRL +OntoPrompt [83] +Entity prediction +Prompt-based +WWW’22 +https://github.com/zjunlp/KnowPrompt +ZS-SKA [12] +Relation prediction +Prompt-based +arxiv +-- +COMET [8] +Relation prediction +LM-based +AAAI’21 +https://github.com/atcbosselut/comet-commonsense +BERTNet [26] +Triple prediction +LM-based +– +https://github.com/tanyuqian/knowledge-harvest-from-lms +Table 2. Representative FKGC methods with open-source code/data. +Following this framework, GMatching [78] is the first work to solve the one-shot KGC problem. It first proposes a +neighbor encoder, which utilizes the local graph structure to generate better entity embedding. The motivation here is +that, although the entity embedding of previous KGE models can encode relational information, previous work [77] +shows that explicitly modeling structure patterns like path can still benefit relational prediction. The neighbor encoder +in GMatching encodes only the one-hop neighbors of each given entity, i.e., a set of (relation, entity) tuples to guarantee +it is general to large-scale KGs. Specifically, starting with pre-trained KGE embeddings for every tuple in the one-hop +neighbor set, GMatching applies a feed-forward layer to encode the interaction between the relation and the entity +in each tuple. A neighbor encoder is then applied on both supporting entity pairs and query entity pairs to generate +each representation. Then, the model exploits an LSTM-based recurrent processing block [65] to perform multi-step +matching between the reference pair and each query pair. The matching scores are finally used to rank every entity in +the candidate set of each query. Besides proposing the first baseline model on FKGC task, the work also proposes two +widely-used benchmarks: NELL-One and Wiki-One [78]. Both are built following the FKGC task setting. More statistics +and details are provided in Table 3 and Sec 4.2. +Sharing the same idea, FSRL [86] extends GMatching to the few-shot setting. It further proposes a relation-aware +heterogeneous neighbor encoder to enhance entity embeddings based on the heterogeneous graph structure and +attention mechanism so that the model can encode the different impacts of different neighbors on the task relation. The +main argument here is that different neighbors should impact the task relation differently, which models like GMatching +neglect. For example, taking ParentOfPerson as the task relation, the neighbor (MarryTo, Melinda Gates) should +have a higher weight compared with (CeoOf, Microsoft). To tackle such an issue, FSRL introduces an attention +module to generate entity embeddings by assigning different attention weights when encoding all neighbors. +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +9 +By applying the attentive neighbor encoder, FSRL acquires the representation of each entity pair in the supporting +set. It then implements an RNN-based aggregator to model interactions between supporting entity pairs for each task +related to generate an informative representation of the entire supporting set. Inspired by aggregating node embeddings +with recurrent neural network [25], FSRL applies a recurrent autoencoder aggregator on all entity pairs. In order to +formulate the embedding of the reference set, it aggregates all hidden states of the encoder and extends them by adding +residual connection [27]and attention weight. +With the aggregated representation of the reference set, FSRL applies a matching network to discover similar entity +pairs of the reference set. Instead of comparing each reference entity pair with the query pair, a similar recurrent +matching processor with LSTM cells is used to directly compute the similarity between the reference set and query +entity pair for the final answer ranking. During the training session, each time model samples a task relation and +optimizes the model for that task. The model will sample few-shot entity pairs as the supporting set and a batch of query +entity pairs. Negative training sets are constructed by polluting the tail entities in query entity pairs. Meta-learning is +exploited in the gradient descent step for parameter optimization so that FSRL can transform well onto test few-shot +relations. +Although FSRL [86] proposes to treat neighbors differently based on their relevance to the central entity, it still +assigns fixed weights to all neighbors throughout all task relations. Such a solution leads to static entity embeddings +in different tasks, hurting the system’s effectiveness. FAAN [54], taking a step further, argues that entity neighbors +should have varied impacts with different task relations. For example, SteveJobs is associated with task relations +HasJobPosition and HasChild. Intuitively, if the task relation is CeoOf, the model should pay more attention to the +job position role of entity SteveJobs than the family role. +Besides, task relations can have different meanings under different contexts. For example, if the task relation is +isPartOf, as shown in Figure 1, such relation has different meanings, e.g. organization-related as in (Liverpool, +isPartOf, Premier League) or location-related as in (Gainesville, isPartOf, Florida). Apparently, for a query +(Dallas, isPartOf, Taxes), the location-related references should be more influential than others. Therefore, the +reference (supporting) triples should also contribute variously to different queries. +To address the above challenges, FAAN proposes an adaptive attentional neighbor encoder to model entity embeddings +with one-hop entity neighbors. They also follow TransE [6] to model the task relation embedding as a translation +between the head and tail entity embeddings, i.e., r ≈ h − t. Then, to further model various roles of the reference +entities, FAAN train an attention metric based on the relevance of entity neighbor relations and the task relation to +further obtain a role-aware neighbor embedding for each entity in the reference set. The encoder allows dynamic +attention scores adaptive to different task relations. The adaptive mechanism helps to capture the diverse roles of +entities based on the different contributions of neighbors. The final representation of each entity encodes both the +pre-trained embedding and its role-aware neighbor embedding. +With the enhanced entity representation provided by the encoder, FAAN further applies a stack of Transformer +blocks for supporting and query triples to capture various meanings of the task relation. It borrows the idea of learning +dynamic KG embeddings from [69]. For each element, it passes the element embedding and position embedding through +several Transformer blocks to acquire meaningful entity pair embeddings. +Then, instead of using static representations when predicting different queries, FAAN obtains a general adaptively +representation of the supporting set by aggregating all the references with their attention score to the task relation. +FAAN also uses meta-training in the same fashion as FSRL, i.e., the model is trained on different task relations in +the meta-training set to generate a set of parameters that performs well across all the tasks and can quickly adapt to +Manuscript submitted to ACM + +10 +Haodi Ma +few-shot tasks in the test set. With all the above, FAAN improves the quality of entities and reference representations +by capturing their fine-grained meanings. Sharing a similar matching score as in FSRL, FAAN outperforms previous +models on FKGC task. +HARV, on the other hand, focuses on capturing the differences between neighbor relations and entities and interaction +between relations, which are previously neglected. It introduces a hierarchical neighbor aggregator for central entity +representation by separating the information between the head entity and relation (relation-level) and between the +relation and tail entity (entity-level). The relation-level attention weights are computed based on the head entity and +relation embeddings. The relation-level embeddings are generated by aggregating neighbor relations of head entity +ℎ with such attention. Concatenations of relation-level embedding and each tail entity are then used to generate the +entity-level attention weights. The two-level weights finally generate the triple-level weight, which is used to compute +the enhanced entity representation. Interactions between relations are taken into account by the relation encoder. The +encoder is an extension of the LSTM aggregator in FSRL with Bi-LSTM, which updates representations of all support +entity pairs. The concatenation of the embedding of support entity pairs and the embedding from the Bi-LSTM encoder +is used as the final representation of each entity pair, and the supporting set is represented by an attention-based +aggregation of all support entity pairs. +In addition, GEN [3] investigates an out-of-graph FKGC scenario for relation prediction between unseen entities +or between seen and unseen entities. It is meta-learned to extrapolate the knowledge from seen to unseen entities, +and transfer knowledge from entities with many to few links. GEN further develops a stochastic embedding layer +for transductive inference to model uncertainty in the link prediction between unseen entities. Gen is compatible +with any GNNs. Specifically, two GENs are employed at the meta-training stage for both inductive and transductive +link prediction. The first GEN is inductive GEN. It learns to encode the unseen entities that are not observed and +predicts the links between seen and unseen entities. The second GNN, respectively, is transductive GEN. It takes a +step further to learn to predict the links between unseen entities themselves. To enable transductive inference, the +meta-learning framework in GEN can simulate the unseen entities during meta-training while they are not observed +in conventional learning schemes. Also, since link prediction for unseen entities is inherently unreliable, which gets +worse in the few-shot scenario where only a few triplets are available for each entity, GEN learns the distribution of +unseen representations for stochastic embedding to account for the uncertainty. Further, we apply a transfer learning +strategy to model the long-tail distribution. These lead GEN to represent the unseen entities that are well aligned with +the seen entities. As mentioned, a naive GEN may be affected by the intrinsic unreliability of few-shot out-of-graph +link prediction due to the uncertainty of unseen entities’ representations caused by lacking supporting triples. The +stochastic layer tackling this issue embeds an unseen entity by learning the distribution over that entity embedding. +GEN also models the source of uncertainty on the output embedding from the transductive GEN with Monte Carlo +dropout. +More recently, REFORM [70] proposes an error-aware module to control the negative impact of errors affecting +FKGC. It slightly varies from the original FKGC to predict the missing relation category of the query entity pair from the +few-shot relation categories. Since most real-world KGs are automatically constructed, many errors are incorporated +into KGs without manual validation. Such errors significantly alleviate the performance of previous methods on FKGC, +especially when there are only a few supporting triples to rely on. The neighbor encoder of REFORM focus on selecting +the most reliable neighbors with an attention mechanism to enhance entity representations. The attention weight matrix +is trained with pre-trained embeddings (in REFORM, TransE) to ensure those correct neighbors have higher weights. The +matrix is then normalized using a softmax function to acquire a robust embedding for each entity. Reference entity pairs +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +11 +are represented by the concatenation of their head and tail entity embeddings. Then, to generate robust embedding for +relations in the supporting set, REFORM contains a cross-relation aggregation module based on a transformer encoder +to capture the relation correlations and support instances. The transformer encoder makes each input embedding +participate in the encoding of all other input embeddings based on a multi-head attention mechanism. Then in the +error mitigation module, REFORM exploits the graph convolution network (GCN) to generate confidence weights +of various relations for each query task. The confidence weights can be considered attention weights to limit errors’ +impact. Specifically, REFORM builds a query-oriented graph to measure the effect of different supporting instances +on a specific query relation. The GCN is trained to minimize the loss that a query relation is grouped into the wrong +category. +A representative that uses Translation Network is MetaR. The idea is that instead of encoding neighbor information, +MetaR focus on transferring the common and shared information within one task from reference instances to query +triples. Such information is referred to as relation meta in MetaR. The relation-meta learner generates representations +of entity pairs from head and tail entity embeddings in the supporting set. Given the head and tail entity pairs in the +supporting set, the learner first extracted entity-pair specific relation meta through a fully connected neural network +which uses LeakyReLU [38] as the activation function. The final relation meta of a task is the average of all entity-pair +specific relation meta in the current supporting set. +MetaR also exploits Meta-learning to accelerate the learning process, which is referred to as gradient meta. As +mentioned in Section 2.2.5, the model should be able to update a new few-shot task rapidly. Inheriting the idea of +TransE [6], MetaR applies a similar score function ||h𝑖 + RT𝑟 − t𝑖 || to calculate the score of each entity pair with the +relation meta. Then, by minimizing the loss over the supporting set with the score of all positive and negative triples, +the gradients of parameters can indicate how they should be updated. Following this gradient update rule, MetaR can +make quick updates on relation meta and use the updated one to score the query set with the same scoring function. +The model is trained to minimize the sum of query loss over all the tasks in one batch. Compared with GMatching [78], +which relies on a background knowledge graph, our MetaR is independent of them, thus, it is more robust as background +knowledge graphs might not be available for few-shot link prediction in real scenarios. +GANA [43], taking a step further, extends MetaR by refining embedding and relation meta computation with attention +mechanism and an LSTM aggregator. The motivation here is that noise neighbor information may hurt the model when +the neighbors are spare or even if no proper neighbor is available to represent the few-shot relation. GANA proposes a +global-local framework. At the global stage, a gated and attentive neighbor aggregator is built to accurately integrate the +semantics of a few-shot relation’s neighborhood, which helps filter the noise neighbors even if a KG contains extremely +sparse neighborhoods. The head and tail entities associated with the few-shot relation and their neighborhoods are +combined to eliminate noise neighbor information due to the sparse neighborhood. A gating mechanism could determine +the importance of the neighborhood representation to represent a few-shot relation. Specifically, a graph attention +network (GAT)-based neighbor encoder is developed at the global stage to capture different impacts of neighbors +to improve the quality of entity embedding. The encoder generates the attention weight for each neighbor based +on a trainable linear transformation matrix. GANA employs a gate value with linear transformation to eliminate +noise neighbors due to the sparse neighborhood to automatically determine the impact of the neighbor of an entity +for the few-shot task relation. An entity is then represented by combining its entity embedding with its neighbor +representations. The final triple neighbor representation of the supporting set is the concatenation of the head and tail +representations. With the supporting set encoded, GANA employs an attentive Bi-LSTM encoder to integrate multiple +neighborhood representations of a query relation in the support set. The query relation representation is a weighted +Manuscript submitted to ACM + +12 +Haodi Ma +sum of the final hidden states of the Bi-LSTM by combining all the neighbor embeddings in the supporting set. For the +local stage, a meta-learning-based TransH(MTransH) method is designed to model complex relations and train our +model in a few-shot learning fashion. The reason to use TransH is its ability to model complex relations. A similar loss +function is applied with MAML approach to learning well-initialized parameters over all few-shot (query) relations in +the meta-training set. +Another similar FKGC model HiRe [76], can be seen as an extension of GANA. It proposes to jointly capture three +levels of relational information: entity-level, triple-level, and context-level. Contrastive learning is used to encode the +union of the neighbors of the head and tail entities together in a triple to encode a wider context. HiRe proposes a +context encoder for the target triplet to learn the embeddings of its true/false contexts based on the self-attention +mechanism so that important neighbors within the context would be given higher weights. Furthermore, a contrastive +loss is employed to pull close the triplet towards its actual context and push it apart from its false context. Then at +the triple-level relational learning stage, instead of LSTM, HiRe develops a transformer-based meta-relation learner to +capture interactions among reference triples and generates meta relational representation of target relations. Finally, +HiRe employs a TransD-based [29] meta score function to capture the diversity of entities and relations. MAML-based +training strategy is also applied similarly to GANA. With the three-level relational information, HiRe performs better +on NELL-One and Wiki-One compared with state-of-art models. The ablation study further proved that all three levels +of relational information are crucial to the performance of HiRe, which future models can further leverage. +Meta-iKG, another recent work on this track, proposes to utilize local subgraphs to transfer subgraph-specific +information and rapidly learn transferable patterns through meta-gradients with meta-learning. Graph neural network +is recently incorporated into inductive relation reasoning to capture multi-hop information around the target triplet. For +example, GraIL [63] proposes a subgraph-based relation reasoning framework to process unseen entities. CoMPILE [40] +extends the idea by introducing a node-edge communicative message-passing mechanism to model the directed +subgraphs. Meta-iKG can be interpreted as an extension of CoMPILE method to FKGC. Instead of being limited to +transductive settings and unable to process unseen entities, Meta-iKG targets a few-shot inductive KGC task, including +new entities in the test set. The model splits relations into few-shot and large-shot relations with a threshold K on +relation instances number and meta-train with large-shot relations to find well0initialized parameters and adapt the +model on triples with few-shot relations following the framework of MAML. Inheriting the structure from MetaR, +Meta-iKG first extracts direct enclosing subgraphs between target and tail entities at the relation-specific learning +stage. Then an inductive node labeling function is applied to identify the different roles of entities in the subgraph. +The node embedding is initialized by the distances to the target entities to embed the relative position of each node in +the subgraph. Meta-iKG then follows the idea of CoMPILE to use communicative message-passing neural network to +score each subgraph to encode its plausibility of the target triple as the task loss. Regular meta-learning steps promise +performance on few-shot relations. However, they may introduce bias to the updated parameters since the task relation +query set only updates the final parameters. To guarantee the performance of Meta-iKG on large-shot relations as well, +it introduces the large-shot relation update procedure, which further updates the final parameters using the support set +with a lower learning rate. This operation enables Meta-iKG to generalize well on the whole inductive dataset. +To tackle the KG-dependent problem and further exploit negative samples in the training stage, a meta pattern +learning framework, MetaP [30], is proposed. Patterns in data are representative regularities to classify data. Triples +in KGs also follow relation-specific patterns, which can be used to measure the validity of triples. The pattern of a +relation refers to the regularity of feature co-occurrence of the head entity, relation, and tail entity. MetaP designs a +pattern learner based on convolutional filters to extract patterns of triples directly. It can learn latent representations of +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +13 +relation-specific patterns from limited references and thus is independent of the background KG. Besides, by leveraging +negative references, MetaP can measure the validity of query triples more accurately. A pattern matcher with a validity +balance mechanism (VBM) is proposed to predict the probabilities of whether patterns of query triples are positive or +negative. +3.2 +Optimization-based Method +Optimization-based FKGC models rely on MAML for model optimization. In other words, such models tackle the +challenge of the few-shot relation prediction problem by optimizing GNN with MAML. +Since methods like GMatching and FSRL only focus on fact prediction and exploit only one-hop neighbors, they +miss more structure information provided in KGs, and results lack interpretation. Accordingly, multi-hop relation +reasoning was proposed to infer facts using multi-hop reasoning paths. e.g., (The Beatles, FoundIn, Liverpool) ∧ +(Liverpool, PartOf, British) → (The Beatles, BaseIn, British). Recent models [14, 34] propose multi-hop +reasoning methods, which leverage the symbolic composition information of relations in KGs to achieve explainable +reasoning results. These works also state multi-hop reasoning as a sequential decision process and exploit reinforcement +learning to tackle such tasks. A meta-based algorithm for multi-hop reasoning (Meta-KGR) sharing similar ideas is +then proposed to provide explainable and effective few-shot relations reasoning. Specifically, Meta-KGR introduces +a reinforcement learning framework to model the multi-hop reasoning process, where a recurrent neural network +encodes the search path. It then adopts MAML to learn effective meta-parameters from high-frequency relations that +could quickly adapt to few-shot relations. +Specifically, deriving the on-policy reinforcement learning (RL) from [34], the multi-hop reasoning is treated as a +Markov Decision Process (MDP): give the query relation 𝑟𝑞, the model starts from the source entity 𝑒𝑠, sequentially +step through several relations and entities until it arrives at the target entity 𝑒𝑜. The MDP module includes (1) state, +which is the entity at the current step. All the states share the source entity and task relation as the global context. +(2) action: the action space at state 𝑠𝑡 includes all the current entity’s outgoing relation and entity tuples. (3) reward: +the model will receive a terminal reward equal to 1 if it reaches the correct target entity. Otherwise, a reward will be +given based on the similarity between the target entity and the current entity using pre-trained KG embeddings. The +policy network is then used to determine action at each state. Then, Meta-KGR applies the policy network considering +the search history over background KG. The search history before the current step is encoded with LSTM. The action +space is represented by stacking all the action embeddings in the action space at the current step. The policy network is +trained to maximize the expected reward over all triples. In meta-learning, Meta-KGR employs a meta-policy network +similar to MAML so that Meta-KGR can quickly adapt to a relation-aware policy network for every query relation with +well-initialized parameters learned in this stage. +FIRE [87] extends Meta-KGR with a heterogeneous neighbor aggregator and a search space pruning strategy. +Specifically, FIRE leverages on-policy reinforcement learning to model the path of multi-hop reasoning and encodes +entity embedding using multi-hop heterogeneous structural information. It then prunes the reasoning search space +using knowledge graph embedding to improve the reasoning efficiency. Meta-learning is also applied in the optimization +procedure so that the learned parameters can be fast adapted for few-shot task relations. +Specifically, since the original RL module in Meta-KGR does not encode the heterogeneous structure information into +the entity embedding, FIRE keeps the structure encoding module as in FSRL to enhance entity embeddings. Note that +some entities in KGs have a large number of neighbors, making the action space redundant at specific steps. Different +from [14, 34] that cut edges based on centrality score, FIRE takes structural correlation between states as an important +Manuscript submitted to ACM + +14 +Haodi Ma +feature to guide action search and applies a knowledge-aware search space pruning strategy. The model keeps only +top-𝑚 most correlated entities at each step based on the structural correlation between entities at step 𝑡 and possible +candidates at step 𝑡 + 1 with pre-trained KGE like TransE. Fast adaptation with meta-learning is then utilized to learn +well-initialized parameters that can quickly generalize to few-shot relations. +More recently, ADK-KG [89]further improves FIRE by developing a text-enhanced heterogeneous graph neural +network to encode node embeddings, where entity and relation embeddings are pre-trained using content information +and augmenting MAML with task weight. It is the first work to leverage a pre-trained language model to capture +content information in the FKGC task. The reinforcement learning module is similar to the ones in FIRE and Meta-KGR +and generates the encoding of each entity. The problem is that RL only encodes the reasoning process but ignores the +content and structural information in KG. ADK-KG thus develops a text-enhanced heterogeneous graph neural network +to enrich entity embeddings. Firstly, for each entity and relation in KG, ADK-KG extracts their text information as +their content features. It then merges all text features of entities and relations and feeds them into the pre-trained +BERT language model [16]to obtain the corresponding content feature vector. For enumerated content (e.g., entity +and relation ids in Wikidata), ADK-KG applies one-hot encoding to convert it to a binary feature vector. After that, a +neural network is utilized for encoding and aggregating content embeddings of selected neighbors for each relation +type. The selected neighbors include first-order neighbors and relations and also high-order neighbors sharing the +same relation and the first-order ones from a random walk. Finally, because different relation types of neighbors will +make different contributions to the final entity representation, the model employs the attention mechanism to utilize +these relation-type-based neighborhood embeddings to generate the final embedding of each entity. Such embeddings +are then used in RL-based reasoning to replace pre-trained entity embeddings. In the meta-learning step, since relations +in KG usually have different meanings, AKD-KG assign different weights to them with self-attention mechanism. +Another recent work extending Meta-KGR and FIRE is THML [94]. THML argues that RL-based models’ generalization +is usually limited by low reasoning performance on hard relations (relations with high training loss). THML challenges +this problem in FKGC by identifying the hard relations at each training batch and then further training the model on +those effectively generated new hardness-aware training batches from both relation and relation cluster levels. THML +also formulates the reasoning process as an MDP as in Meta-KGR and FIRE at the hardness-aware meta-reinforcement +learning module. The main difference is that to solve the sparse reward caused by false reasoning paths and efficiency +concerns, THML splits the reward into three parts. The terminal reward is the same as in FIRE, except that THML uses +ConvE [15] for pre-trained embeddings. The path reward encodes the reasoning chain length to encourage the model +to find the target entity with a relation chain that is as short as possible since shorter paths often provide more reliable +reasoning than longer paths [14]. A path may be declined if the length exceeds 3. Another problem with KG reasoning +is that models tend to infer paths with similar semantic meaning in the training stage, which may lead the model into a +local-optimal path. THML thus proposes that the diversity reward encourages the model to find different paths. Then, +instead of random sampling triple queries at the training stage, THML applies a two-level hardness-aware sampling +strategy. Relation level hardness-aware sampling ranks the reasoning accuracy of all relations in a batch online to select +hard relations for the next batch. At relational-cluster level hardness-aware sampling, THML obtains pre-trained TransE +embeddings for all relations, then performs the k-means algorithm to form relation clusters. It then selects the cluster +with the hard relation with the lowest accuracy at the current batch as the hard cluster and adds it to the next batch. +Apart from the methods discussed above, there are also other types of optimization-based solutions on FKGC. For +example, ZSGAN [47] studies zero-shot KGC by establishing the connection between text and knowledge graph with +generative adversarial networks. The motivation is that the semantic features of new classes can be reflected by their +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +15 +textual descriptions. Moreover, textual descriptions contain rich and unambiguous information, which is critical for +large-scale recognition tasks. The core of ZAGAN is the design of a conditional generative model to learn the qualified +relation embeddings from raw text descriptions. +ZSGAN leverages a feature encoder for real data representations to generate reasonable real data distribution from +KG embeddings. The feature encoder is trained in advance from the training set and fixed during the adversarial training +process. The neighbor encoder only considers one-hop neighbors of each entity and used GCN [51] to generate the +structure-based representation of each entity. Then a feed-forward layer is used as the entity encoder to extract the +information from each head, tail entity pair. The relation fact representation is finally formulated as the concatenation +of head and tail neighbor embeddings and the entity pair embedding. +Given text representations, the generator is to generate reasonable relation embeddings that capture the corresponding +relational semantic information in the knowledge graph feature space. Based on this, the prediction of query relations +is converted to a supervised classification task. Specifically, text embedding is the vector sum of word embeddings +weighted by TF-IDF values after removing stop-words and punctuations. The text embedding is passed to the generative +adversarial model (GAN) to generate the relation embedding. On the contrary, the discriminator seeks to separate the +fake data from the real data distribution and identifies the relation type as well. The input features are first transformed +via a fully-connected (FC) layer with LeakyReLU [38]. Two network branches follow after. The first branch is a FC +layer that acts as a binary classifier to separate real data from fake data. The other branch is classification performance. +In order to stabilize training behavior and eliminate mode collapse, ZSGAN also adopts the gradient penalty, which +penalizes the model if the gradient norm moves away from the target norm value 1. Similarly, RAN [88] worked out a +general feature generation-based framework for addressing unseen relations in both few-shot KG completion with +unseen relations and few-shot relation extraction from text. +Unlike previous works that leverage entity pair matching, P-INT [79] utilizes the paths from the head to the tail +entities to represent an entity pair and computes the interactions between paths for the FKGC problem. The motivation +is still to involve more structural and expressive information and exclude noise neighbors in few-shot reasoning. To +extract the support subgraph for each support entity pair, P-INT employs a two-side BFS algorithm [77] to prune the +search space. The intersected neighbors of the left and right paths are used to generate paths from head to tail entities +with different lengths. The relations in these paths are then combined as a set of supporting relations. Pre-trained +TransE embeddings are used to compute the similarity between each pair of relations in the KG. Then to reason the +query subgraph, P-INT extends the limited number of neighbors with a fixed number of steps and, for each of them, gets +the maximum similarity with the supporting relations and returns top-𝐿 ones. After 𝑇 hops, the model extends a query +subgraph with at most 𝑇 × 𝐿 entities. Simultaneous to reasoning, P-INT can trace all paths from the head entity of the +query to every extended entity in the subgraph. In the matching component, P-INT calculates the similarity between +every two paths. Then the RBF aggregation function in [77] is used to extract similarity features of the similarity matrix +as the interaction between paths. Inspired by FAAN [54], P-INT computes relation-aware attentions for different paths +to model their impact on the matching result based on the relevance of a path to the query relation. +3.3 +Ontology-based Methods +Apart from the FSL methods that are mentioned above, there is an emerging interest in extracting knowledge from +large language models as pre-training/transforming fine-tuning models have become a default paradigm for natural +language processing. Thus, how to effectively transfer between structured relational knowledge and natural language +knowledge has become a challenging task. Recent works attempt to integrate structural knowledge like ontology to +Manuscript submitted to ACM + +16 +Haodi Ma +enhance language understanding. One of the representatives on this track is OntoZSL [21], which proposes a novel +zero-shot learning framework that not only enhances the class semantics with an ontological schema. It also employs +an ontology-based generative adversarial network (GAN), as in ZSGAN, to synthesize training samples for unseen +classes. OntoZSL first designs an ontology encoder for learning relation representations from the ontological schema +by considering the structural relation between concepts and their correlations in the textual description. Pre-trained +TransE is used to generate the default embedding of all concepts in the ontological schema. Then a text-aware semantic +embedding model is employed by projecting the structural and textual representations into the same embedding space +and learning them simultaneously with the same scoring function as in TransE. With two types of representation +learned with the ontology encoder, OntoZSL then follows GAN [47] to learn and train the real relation embeddings +in bags containing all the one-hop neighbor triples of the task relation. The embeddings of all entity pairs in the bag +consist of the real embedding of the task relation, which contains semantic and structural features of the task relation. +OntoZSL finally generates plausible relation embedding for each unseen relation with its text description using the +well-trained generator. For prediction, the model calculates the similarity between the relation embedding and the +candidate’s head, tail entity pair. +3.4 +FKGC with Commonsense Knowledge +Previous models primarily focus on structural information in KGs like one-hop neighbors and paths but relatively +ignore semantic information. Models like ADK-KG [89] consider content information but still rely on neighbors to +generate embeddings for entities and relations. As mentioned, due to the recent broad investigation of pre-trained +language models such as BERT [16] and GPT [9], some methods are developed by fine-tuning these models to exploit +textual information for few/zero-shot KG completion. This section covers several representatives in this category to +discuss how to leverage commonsense/textual information in KGs effectively. +ConMask [55] is one of the first models that is proposed to tackle the zero-shot KGC problem by encoding unseen +entities with their names and text descriptions. In general, it feeds the text embeddings of the entities and the relation of +a triple into a model composed of an attention-based relation-dependent text masking module and a CNN-based target +fusion module. To capture text information that is relevant to task relation, ConMask uses a relation-dependent content +masking module to reduce noise in the given descriptions. The component first pre-processes the input description to +select small relevant snips based on the task relation. ConMask utilizes the attention mechanism to mask the irrelevant +task, which assigns a relation-dependent similarity score to words. A common problem here is that the words with the +highest scores are not the target entity but words with similar semantic meaning to the task relation. Since actual target +words are always around these indicator words, ConMask adjusts the similarity score of each word based on its context. +Then the model extracts relation-based entity embeddings via target fusion. Three fully convolutional neural networks +(FCN) layers without de-convolution operations are developed for this task. Since directly generating entity embeddings +with the target fusion module may be costly, ConMask employs a semantic averaging function that aggregates word +embeddings to represent entity names and generates representations of other textual features for each entity. +Although ConMask successfully learns embeddings of the entity’s name and parts of its text description to connect +unseen entities to the KGs, it does not take full advantage of the rich feature information in the text descriptions. +Besides, the proposed relationship-dependent content masking method in ConMask may quickly fail to find the target +words. To challenge such problems, a Multiple Interaction Attention (MIA) model [44] is proposed to acquire the +interactions between the head entity description, head entity name, the relationship name, and the candidate tail +entity descriptions for more graphic representations. Besides, MIA similarly uses the additional textual features of head +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +17 +entity descriptions to enhance the head entity representation and apply the attention mechanism between candidate +tail entities to enhance their representation of them. Specifically, MIA first transforms each word in the head entity +description, question, and candidate tail entity description into several continuous representations, including GloVe, +POS, NER, and BERT embeddings, and concatenates them to form the input representations of each word. Then, to +enhance entity representations with the interaction with all the relevant textual information, MIA leverages the same +word-level sequence alignment attention mechanism for each interaction since words in the same description are not +equally important, and relevant descriptions usually mention each other. The third component of the model is an RNN +layer which uses Bi-LSTM to encode text context. The exact attention mechanism is applied between multiple candidate +tail entity descriptions to enhance their representations. MIA also explores different scoring functions to enhance the +convergence of the model, which achieves significant improvements against other state-of-the-art models. However, the +problem with this approach is that it relies heavily on entity descriptions and only works when necessary information +is available. +InductiveE [67] proposes a commonsense KG link prediction method that can deal with unseen entities by utilizing +textual entity descriptions. It enables inductive learning by directly building representations from entity descriptions +instead of leveraging textual entity representations as training initialization. Specifically, It first represents an entity +using the concatenation of its text embeddings by the fastText word embedding model [4] and the last layer for +[𝐶𝐿𝑆] token of the pre-trained BERT [16]. To further enhance semantic entity representations with entity neighbor +information, InductiveE adds similarity links for unseen entities to initialize their neighbor information. It then feeds +the entity representations of the densified graph into a model composed of a gated-relational GCN encoder and a +simplified ConvE [15] decoder to predict each triple’s score. The gated encoder is employed to guarantee adaptively +control over the amount of information fused to the center node from their neighboring connections. For the decoder, +Conv-TransE [53] has been proven effective and efficient in scoring triples in KGC. The decoder of InducctiveE further +improved it by adding a shuffling operation before convolution to allow more interaction between embeddings and +improve the convolutional model’s expressive ability. +InductiveE only explores inductive learning on unseen entities, while inductive learning on unseen/new relations is +also valuable for real-world commonsense FKGC tasks. A KGC model, KG-BERT [82], is developed to target such a +challenge. It transforms a triple head entity, relation, and tail entity into a text sequence. It then makes triple prediction +as a downstream text classification task, where BERT is fine-tuned with given training triples. For unseen entities and +relations with name information, the candidate triples associated with them can be directly predicted by transforming +them into text sequences. +Following this approach, BERTRL [85] also proposes to predict triples as a downstream text classification task of +BERT, utilizing the text information of entities and relations. However, it fine-tunes BERT using single triples and +possible paths connecting two entities where reasoning is conducted explicitly. Given a query triple (ℎ, ?,𝑡) to exploit +the neighborhood knowledge of head and tail entities and select proper neighbors for efficiency concern, BERTRL +entirely relies on BERT to encode such information. To formalize structural knowledge in KGs to fit into BERT models, +BERTRL collects all the length-𝑘 reasoning paths between the head and tail entity in the query triple. It then takes each +path as a separate input to BERT. Each path individually induces the query triple with a confidence score. The problem +is then transformed into a binary classification problem where the score of the linear layer on top of [𝑐𝑙𝑠] indicates the +correctness of the query triple given a reasoning path. The maximum aggregation of bag scoring is used at inference +time to generalize the score of all reasoning paths. The path with the highest score can be used to explain the reasoning +process of the prediction. +Manuscript submitted to ACM + +18 +Haodi Ma +As all these works manage to consider textual information simultaneously with structural information, they still treat +them as two types of knowledge. On the other hand, prompt-tuning proposed in GPT3 [9] as an arising methodology +has been used for tasks like relation extraction names, entity recognition, etc. Recent works have tried to integrate +external knowledge into prompt designing. OntoPrompt [83] is one of the representatives of this approach. It utilizes +prompts to bridge commonsense knowledge from pre-trained language models (LMs) and structural knowledge from +knowledge graphs for the FKGC task. OntoPrompt first employs ontology transformation to enrich and convert structure +knowledge to text format, where it utilizes pre-defined templates to convert knowledge to text as prompts. Specifically +for KGC, the model leverages head entity types and tail entity types from the ontology representation as constraints. +It uses corresponding items obtained from the external Wikidata query as the source of ontology and extracts the +textual description. It follows KG-BERT to consider KGC as a triple classification task and concatenate entities and +relations as an input sequence. It also uses the learnable virtual token to enhance the prompt representation. Next, +OntoPrompt proposes span-sensitive knowledge injection to select informative knowledge. Considering that irrelevant +and noisy knowledge may lead to changes in the meaning of the original sentence, OntoPrompt leverages a visible +matrix based on spans to limit the impact of corresponding knowledge on the knowledge injection. In this way, not all +tokens in the input sentences will be affected by external knowledge. Third, OntoPrompt develops a collective training +algorithm to optimize representations jointly. Note that the injected external knowledge should be associated with the +surrounding context; learnable tokens are added with random initialization and optimized along with injected ontology +tokens with a fixed language model. Inspired by the previous study [23] that prompt-tuning in the low-data regime is +unstable and may obtain poor performance, the model further optimizes all parameters to train the ontology text and +input text representations collectively. With the components above, OntoPrompt can enrich task-relevant knowledge +using pre-trained LMs, prevent negative knowledge fusion, and integrate commonsense knowledge into structural +which solve challenging problems in knowledge missing, knowledge noise, and knowledge heterogeneity and achieve +promising performance in FKGC task. +ZS-SKA [22] also utilizes prompt to tackle the FKGC task. Instead of leveraging ontology knowledge, they directly +work on semantic knowledge augmentation for zero-shot relation classification. +ZS-SKA first generates augmented instances with unseen relations from instances with seen relations following a +word-level sentence translation rule. To encode every instance, ZS-SKA tokenizes all the words in a sentence and feeds +them to BERT to generate a contextual representation for each token. A CNN layer is used after obtaining the tokenized +input sentence. Then, they design prompts based on ConceptNet [57] to integrate semantic knowledge information +learned from seen relations and exploit such knowledge to infer the features of unseen relations. They consider multiple +types of semantic knowledge, including relation descriptions and name entities, to learn unseen relations effectively. +The input sequence is wrapped with a natural language snip template. Instead of using the actual label sets in the +prompt template, they automatically construct weighted virtual label words based on the knowledge graph of each +label. Prompts are represented by embedding the super-class of the input words and the virtual label embedding for +unseen relation. By generating the representations of both seen and unseen relations with augmented instances and +prompts through prototypical networks [56], distance is calculated to predict unseen relations. +The works above have already shown how pre-trained language models, as external sources, can help with FKGC +tasks for both entity and relation learning. Taking one step further, instead of completing a zero/few-shot query tuple, +embraced by the power of pre-trained LMs, another trending topic is to directly complete KGs by constructing triples +for unseen entities. Representative models on this task include COMET [8] and BERTNet [26]. +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +19 +COMET [8] is the first comprehensive study on automatic commonsense knowledge graph completion. It is a +generative transformer model over commonsense knowledge which learns to generate detailed and diverse commonsense +descriptions in natural language. As mentioned in Sec 2.1.2, one main challenge on commonsense knowledge graph +reasoning is that commonsense knowledge, represented by open-text, usually does not fit into a fixed schema. Generally, +COMET tackles this problem by constructing commonsense KG/KB by training a transformer over existing tuples as a +seed set of knowledge to learn an adaptive representation of commonsense knowledge with a pre-trained LM. Then +the LM can be used to produce novel tuples with unseen entities. The relations are identified by generating phrases +that can semantically complete an existing seed phrase and relation type. In detail, the transformer language model in +COMET follows the structure of GPT [48], which consists of multiple transformer blocks of multi-headed attention and +fully connected layers to encode input text. The input of the model is concatenated sequence of words for knowledge +tuples. To encode the order information between tokens that is ignored by the transformer, COMET adds a position +embedding for each position in the sequence. The position embedding and word embedding of each word is added for +the final representation, which is the input to the first transformer layer. COMET is trained to produce the tail entity +given the tuple’s head entity and relation, which follows the setting of the KGC task but is expected to generate novel +tuples that do not exist in the training set. +The limitation of COMET is that it can only generate triples for new entities with seen relations. The ideal zero-shot +KGC model should be able to construct tuples with unseen entities and relations. BERTNet [26] is then proposed to +harvest KGs with implicit knowledge from pre-trained LMs. BERTNet only requires a few-shot seed set, including an +initial prompt and seed entities for each relation as input, and can extract knowledge for unseen relations. In general, +the model automatically generates different prompts and searches within a given LM for novel knowledge. +BERTNet tackles two challenges in KGC/KG construction. First, LMs have shown to be inconsistent even with a +slightly different prompt, making it difficult to extract knowledge reliably from LMs. An intuitive solution is to learn +the optimal prompts automatically. Such methods require extensive training data, which is unavailable in few-shot +or zero-shot settings. To this end, BERTNet employs unsupervised paraphrasing on an initial prompt to generate a +set of various prompts with their confidence. Then entity pairs that consistently satisfy these prompts are extracted +to generate novel triples. Another challenge then comes into space when searching for proper entity pairs due to the +ample candidate space in LMs. BERTNet devises an efficient search-and-rescoring strategy that strikes the balance +between knowledge accuracy and coverage. a prompt/entity pair compatibility function is designed to dynamically +reassign weights for both candidate prompts and entities at each knowledge searching step. Specifically for entity +searching, BERTNet first uses individual compatible score, which is more accessible to threshold and prune, to weighted +average across all prompts to generate a large set of candidate entity pairs. These candidates are then re-ranked by the +total compatible score to select the output entity pairs. +Since BERTNet only requires a set of seed prompts and few-shot entities for relations other than a pre-trained LM, +the model guarantees the flexibility to extract novel knowledge even for relations that have complex structures or +include multiple entities. Besides, the resulting triples can be considered as an interpretation of the respective black-box +LMs. Another novelty of BERTNet is that instead of only looking at matrices like hits@10 and BLUE score, it directly +integrates the generated tuples into background KGs and applies the new KGs for downstream tasks. The performance +on those tasks indicates that BERTNet can generate novel high-quality tuples. +Manuscript submitted to ACM + +20 +Haodi Ma +4 +APPLICATIONS AND RESOURCES +4.1 +Applications +FKGC models have been applied to problems other than prediction tasks on knowledge graphs. One main application is +to leverage the few-shot link prediction technique for other graph-related tasks. For example, Meta-Graph [7]investigates +few-shot link prediction on different networks, including protein-protein interaction (PPI) networks [98], 3D point cloud +data [42] and academic social networks [62]. MetaTNE [32] also leverages a meta transformer commonly used in FKGC +tasks to predict protein-protein interaction. Similarly, molecular property prediction has always been a demanding +problem since manual prediction can be costly and inefficient. Meta-MGNN [24]and Pre-PAR [73] have been proposed +to solve such tasks with few-shot link prediction methods. Meta-MGNN takes each molecule as a graph and uses a +graph-level-GNN to encode each molecule. It further introduces an attention mechanism to make MAML aware of +molecular property differences to improve model encoding further. Moreover, Pre-PAR improves Meta-MGNN by +capturing relational structure among different molecular properties to effectively and efficiently propagate limited +labels among similar molecules. Similarly, GEN [3] has also been used on drug-drug interaction prediction. +Another challenging real-world task related to link prediction is recommendation problems. For example, as proposed +in the Yelp challenge [2], user reviews have become a significant part of web services like Yelp. Since users can post their +opinions about businesses, products, and services through reviews consisting of free-form text and a numeric star rating, +the interaction between users, services/products, and reviews intuitively form structural and textual knowledge. Recent +techniques in collecting user-related information, like GPS-enabled devices, help form location-based social networks +that provide the location information that is valuable for the recommendation system. One challenge for such systems is +that when a new user joins, there is little existing knowledge in the platform besides basic information and location. Thus, +few-shot link prediction models like SEATLE [33] and heterogeneous information network-based models [36, 93] aim +to tackle cold-start recommendation problems over graphs with meta-learning. Other models [17, 18, 35, 61, 68, 80, 81] +tackles recommendation problems with methods like attribute matching with previous users, transforming knowledge +from other platforms with cross-network meta-learning, modeling user with updated information with dynamic +meta-graph reasoning, etc. +As mentioned in BERTNet [26], since FKGC models help to improve the quality and coverage of original KGs, the +output can be leveraged by downstream tasks. For example, affordance reasoning and extraction [96] can used few-shot +KGC models to generate affordance for unseen entities with pre-trained LMs or similarity-matching with seen entities +in the background KG. More recently, in large-scale video extraction/recognition datasets like EPIC [13] and STAR [75] +or action planning tasks, procedural reasoning is required due to the rapid changing of reasoning scenarios and dynamic +environment knowledge. FKGC models can be powerful on such tasks since they can easily capture new few-shot +information and swiftly adapt and optimize themselves to fit each reasoning step. +4.2 +Resources +With the bursting growth of zero-shot KG completion methods, various benchmarks have been proposed for evaluation. +They are usually constructed based on existing commonly used typical KG completion datasets, including FB15k [5], +FB15k-237 [46], NELL-995 [77], and some other sub-KGs extracted from popular KGs such as Wikidata [66]. Such +benchmarks usually obtain entity information from the original KGs. For example, DBpedia50k, FB20k, and Wiki- +data5M collect correspondingly text descriptions of entities from DBpedia and Wikipedia. Here we introduce several +representatives for zero/few-shot KGC: +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +21 +Table 3. Statistics of FKGC Benchmarks +(a) Relation prediction Benchmarks +Ent # +Rel # +Triples # +Task rel # (train/valid/test) +Source +NELL-One [78] +68,545 +358 +181,109 +51/5/11 +NELL +Wiki-One [78] +4,838,244 +822 +5,859,240 +133/16/34 +Wikidata +NELL-ZS [47] +65,567 +181 +181,109 +139/10/32 +NELL +Wiki-ZS [47] +605,812 +537 +724,967 +469/20/48 +Wikidata +(b) Entity prediction Benchmarks +Ent # +Rel # +Triples(train/valid/test) # +Unseen Ent # (train/valid/test) # +Source +WN18RR-sub [3] +4,478 +11 +93,003 (-) +3000 +WN188RR +FB15K-237-sub [3] +10,938 +237 +72,065/6,246/9,867 +2,500/1,000/1,500 +FB15K-237 +NELL-995-sub [3] +5,694 +200 +22,345/3,676/5,852 +1,500/600/900 +NELL-995 +FB15K-237-OWE [52] +14,405 +235 +242,489/10,963/36,250 +2081 (-) +FB15K-237 +Wikidata5M [72] +4,594,485 +1222 +20,496,514v/6,699/6,894 +4,579,609/7,374/7,475 +Wikidata +• FB15k-237-OWE [52] is built on FB15k-237 for zero-shot KGC. They first sample a set of tail entities and +randomly pick associated head entities from FB15K-237. Then all triples with their head in the associated entity +set are moved to the testing set, which forms the testing set for tail entity prediction. At the same time, the ones +with their tail in the associated entity set are removed. The testing set for head entity prediction is similarly +generated. These two sets form the final testing set by further removing triples whose relations are not included in +the training set. This testing set is further split into a validation set and the final testing set. The dataset contains +2,081 unseen entities, 12,324 seen entities, and 235 relations. The numbers of triples for training/validation/testing +are 242,489/10,963/36,250. +• Wikidata5M [72] was originally constructed for evaluating text-aware KGE models but is now widely used for +zero-shot KGC. It is developed based on Wikidata and English Wikipedia dump. Each entity uses the first section +of its Wikipedia page as its description. The dataset excludes entities without Wikipedia pages or descriptions +shorter than 5 words. Next, they extract all the triples from Wikidatadump. The dataset keeps triples with +qualified head and tail entities, leaving it with 4,594,485 entities, 822 relations, and 20,624,575 triplets. To support +the zero-shot setting, they randomly extract two sub-KGs as the validation and testing sets and use the remaining +as the training sets. They ensure that the entities and triples are mutually disjoint across the training, validation, +and testing sets. Detailed information on each set can be found in Table 3. +• Subsets of WN18RR, FB15k-237, and NELL-995 are constructed by [3] for out-of-graph completion between +seen, unseen entities, and unseen entities themselves. These subsets are systematically extracted from their +original benchmarks. They randomly sample a group of entities with less than 100 associated triples as unseen +entities. These triples are further separated to compose three meta sets (meta-training, meta-validation, and +Manuscript submitted to ACM + +22 +Haodi Ma +meta-testing sets). The rest of the original benchmarks are considered as the background KGs/In-Graph, and +entities inside are respectively seen entities. The meta sets are then cleaned to guarantee that each of their triples +has at least one unseen entity and that all the triples are out of In-Graph. More statistics about seen/unseen +entities are presented in Table 3. +• NELL/Wiki-ZS are two zero-shot KG completion benchmarks. Each benchmark contains three relation-disjoint +sets: the training set holds seen relations, validation/testing set holds unseen relations. Associated triples are +separated accordingly, while entities included in the testing/validation set are all involved in the training set. +NELL-ZS has 139/10/32 relations in the training/validation/testing set , while Wiki-ZS involves 469/20/48 relations +for training/validation/testing. GEN [47] uses relation textual descriptions as the textual information, while +OntoZSL [21] constructs ontological schemas, which contain not only textual information but also other relation +knowledge, including relation hierarchies and relation domains for both benchmarks. +• NELL/Wiki-One are originally developed in GMatching [78] for evaluating few-shot KG completion with +unseen relations only one supporting instance. To construct the testing set, they extract relations with less than +500 but more than 50 associated triples from the original benchmarks and use those as task relations. With this +preprocessing, 67 relations are extracted in NELL-One, and the benchmark further partitions them into 51/5/11 to +further extracted associated triples and compose the training/validation/testing set. Similarly, in Wiki-One, 183 +relations are extracted and partitioned into 133/16/34 for constructing triples in the training/validation/testing +set. On the entity side, 68,545 entities are extracted for NELL-One, and 4,838,244 for Wiki-One. Additionally, +another 291 and 639 relations are extracted as background relations to construct more triples for the entities. +The relations in the training/validation/testing set are guaranteed to be disjoint so that the two benchmarks can +be used for both zero-shot KGC and few-shot KGC tasks. +5 +ANALYSIS +Starting with GMatching [78], since the problem setting comes from KGC problem, FKGC models intuitively enhance +KGE or GNN models which originally capture the structural information with meta-learning frameworks. Early metric- +based models like MetaR, FSRL [86] and FAAN [54], focusing on how to effectively leverage neighbor information by +assigining static or task-aware attentions to different neighbors. Optimization models like Meta-KGR [37], ADK-GK [89], +ZS-GAN [47], explore to improve entity and relation embeddings with multi-hop path information and. These paths +can also be used to explain reasoning logic which improve the interpretablility of the models. Meta-learning provides +the opportunity for these models to quickly adapt to few-shot tasks at testing stage. +However, these models ignore the semantic information in background KGs like entity/relation names, text description. +To combine the two types of information, prompt-based models like OntoPrompt [83] and ZS-SKA [22] are proposed. +These models employs prompts to translate triples in KGs into sentence-like knowledge. Pre-trained language models +like BERT are then used to generate entity and relation embeddings. Semantic information like text description of +entities and relations helps to enrich the embeddings. +All these models are dependent on large-scale KGs since they provide structural information for entity and relation +embeddings. On the other hand, GPT models [9, 48] show that pre-trained language models originally contain structural +knowledge as well, some models take a step further to totally depend on pre-trained LMs. COMET [8] and BERTNet [26] +are the representatives of this category. COMET follows the framework of GPT models and BERTNet uses prompts to +Manuscript submitted to ACM + +A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge +23 +GMatching [78] +FSRL [86] +MetaR [12] +OntoPrompt [83] +InductiveE [54] +COMET![8] +BERTNet [26] +NELL-One [78] +✓ +✓ +✓ +✗ +✗ +✗ +✗ +FB15K-237-sub [3] +✓ +✓ +✓ +✓ +✗ +✗ +✗ +WN18RR-sub [3] +✓ +✓ +✓ +✓ +✗ +✗ +✗ +NELL-ZS [47] +✗ +✗ +✗ +✗ +✗ +✗ +✗ +ConceptNet [57] +✗ +✗ +✗ +✗ +✓ +✓ +✓ +ATOMIC [50] +✗ +✗ +✗ +✗ +✓ +✓ +✗ +Table 4. Evalution of Representative FKGC Models +generate novel triples. Comparing with models with structural knowledge, these models can usually work on both +structural and commonsense KGs. This approach free FKGC models from background KGs and extend the usage to +downstream tasks in more area. +6 +FUTURE DIRECTIONS +As discussed, earlier FKGC models are primarily extensions of meta-learning and transfer learning methods to FKGC. +Another type of knowledge in meta-learning is the learning process. In the future, representing knowledge on learning +(e.g., previous reasoning process [60]) as KGs and integrating them with meta-learning or transfer learning algorithms +could lead to more general neural-symbolic paradigms that apply to different FSL tasks. Propagation-based methods +like GEN [3] and REFORM [70] solve few-shot KG completion by utilizing the few-shot samples, i.e., triples model the +correlation of unseen entities with the distribution and correlation of seen ones. It would be a promising solution to +utilize these few-shot links and the unseen entities’ correlations auxiliary information such as textual descriptions, +attributes, and schemas. +Another track of FKGC models like OntoPrompt [83] has proved that exploiting ontology/rule structured knowledge +is a promising approach to infer symbolic knowledge like triples for unseen entities/relations. On the other hand, +generation-based models like OntoZSL [21] are not biased to seen or unseen classes in prediction compared with the +widely explored mapping-based methods [11]. Therefore, generation-based ZSL methods conditioned on the embeddings +of KGs could be a future direction for FKGC task. +It is till recently that semantic information has gained attention in FKGC models. GANA [43]first proposes to integrate +the semantics of a few-shot relation’s neighborhood, and ZSGAN [47] generates reasonable relation embeddings with +text representations of task relations. Works like COMET [8], KG-BERT [82], and BERTNet [26] further present the +effectiveness of learning for representation of unseen/few-shot entities and relation. The performance of these models +indicates that multi-modal knowledge can also be beneficial for FKGC tasks. Challenging the problem of efficiently +integrating data from more modalities into current FKGC models can be a promising direction to tackle not only +knowledge graph completion tasks but also zero/few-shot reasoning in other areas. +According to Table 4, currently there is not enough evaluation to compare the performance between FKGC models +with large-scale KGs and the ones leveraging pre-trained LMs. It is still at theoretic level that models like BERTNet and +COMET are effective for strucural KGs like NELL and Freebase. Such evaluations will also be valuable for few-shot +knowledge graph completion with multi-modal information. +Manuscript submitted to ACM + +24 +Haodi Ma +7 +CONCLUSION +As knowledge graphs are a popular source for tasks in various domains, few-shot knowledge graph completion models +provide a chance to efficiently integrate new knowledge into existing KGs to improve the quality and coverage of KGs. +In this survey, we first introduce the major challenges and bases of FKGC. Then we comprehensively reviewed previous +studies on FKGC. We categorize these methods into two groups: FKGC with structural knowledge and FKGC with +semantic knowledge. This categorization shows the trending of FKGC methods from meta-learning and attention-based +models to leveraging semantic and textual information or even directly extracting structural triples from pre-trained +language models. Then we discuss how FKGC models can be transferred or applied in various fields. 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Bioinformatics 33, 14 (2017), +i190–i198. +Manuscript submitted to ACM + diff --git a/qNAzT4oBgHgl3EQfOvuh/content/tmp_files/load_file.txt b/qNAzT4oBgHgl3EQfOvuh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00a0e9e423c93d4395912d61f4afbfcdb95d381b --- /dev/null +++ b/qNAzT4oBgHgl3EQfOvuh/content/tmp_files/load_file.txt @@ -0,0 +1,1281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf,len=1280 +page_content='A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge HAODI MA, University of Florida, USA Knowledge graphs (KG) have served as the key component of various natural language processing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Commonsense knowledge graphs (CKG) are a special type of KG, where entities and relations are composed of free-form text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, previous works in KG completion and CKG completion suffer from long-tail relations and newly-added relations which do not have many know triples for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In light of this, few-shot KG completion (FKGC), which requires the strengths of graph representation learning and few-shot learning, has been proposed to challenge the problem of limited annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In this paper, we comprehensively survey previous attempts on such tasks in the form of a series of methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, we first introduce FKGC challenges, commonly used KGs, and CKGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we systematically categorize and summarize existing works in terms of the type of KGs and the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, we present applications of FKGC models on prediction tasks in different areas and share our thoughts on future research directions of FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' CCS Concepts: • Computing methodologies → Knowledge representation and reasoning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Reasoning about belief and knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Additional Key Words and Phrases: Knowledge graph embeddings, link prediction, knowledge distillation, knowledge graph ACM Reference Format: Haodi Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 1, 1 (Janu- ary 2022), 27 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION Knowledge graphs (KGs) are a collection of triples, where each triple represents a relation r between the head entity h and tail entity t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Examples of real-world KGs include Freebase [5], Yago [58] and NELL [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These KGs contain millions of facts and are the fundamental basis for applications like question-answering, recommender systems, and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Although an immense amount of information is stored in today’s large-scale KGs, they are highly incomplete, which makes Knowledge Graph Completion (KGC) a challenge for its downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Recent trends target learning low-dimension representations of entities and relations for missing link predictions [(Bordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Trouillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Dettmers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', 2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The general idea of these methods is to model and inference various relation patterns between entities based on known facts in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, TransE models relations as translation, aiming at inversion and composition patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Rotate, as one representative, can infer symmetric, asymmetric, inversion, and composition patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, such methods usually require sufficient training triples for all relations to learn embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Previous works [78] show that a large portion of KG relations is long-tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In other words, they only have a few instances in the Author’s address: Haodi Ma, ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='haodi@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='edu, University of Florida, Gainesville, Florida, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' © 2022 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01172v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='CL] 3 Jan 2023 2 Haodi Ma KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, about 10% of relations in Wikidata have no more than 10 triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, real-world KGs are often dynamic, which means new relations and entities will be added whenever new knowledge is acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To tackle these challenges, the model should be capable of predicting new triples given only a small number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To address the above challenges, [78] proposed two benchmarks, NELL-One and Wiki-One, for few-shot knowledge graph completion (FKGC) and a baseline model called GMatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model introduces a local neighbor encoder to learn expensive entity representations with only a few samples for each query relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' One branch of recent works [54, 86] follows a similar approach and achieves considerable performance by improving the quality of embeddings by considering local graph neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They further argue that entity neighbors should have varied impacts associated with different task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since relations can be polysemous, reference triples should also make different contributions to a particular query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, if the task relation is isPartOf, as shown in Figure 1, such relation has different meanings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', organization-related as in (Liverpool, isPartOf, Premier League) or location-related as in (Gainesville, isPartOf, Florida).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Apparently, for a query (Dallas, isPartOf, Taxes), the location-related references should be more influential than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These models [43, 54] propose to use attention networks to capture the dynamic properties of both entities and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Example of (a) an entity with diverse roles, credits to [54] (b)references showing different impact to a particular query Bill Gates Jennifer Gates Melinda Gates Microsoft Chairman Paul Allen WorkWith ProxyFor HasJobPosition MarryTo HasChild HasChild Rory Gates CeoOf ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ParentOfPerson ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Reference Query (Petersburg, SubPartOf, Virginia) (Vacaville, SubPartOf, California) (Prague, SubPartOf, Czech) (Cavaliers, SubPartOf, NBA) (Los Angeles Lakers, SubPartOf, NBA) (Chicago Bulls, SubPartOf, NBA) (b) (a) (a) Reference (Denver Nugget, isPartOf, NBA) (Liverpool, isPartOf, Premier League) (Gainesville, isPartOf, Florida) (Venice, isPartOf, Italy) (Montreal, isPartOf, Quebec) Query (Dallas, isPartOf, Taxes) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A subset of ATOMIC20 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' credits to [28] X gets X’s car repaired a mechanic money X wants to call Uber for a ride X wants to pay their bill The car costs too much to repair Fix leaky radiator garage repair shop earned by working fold into origami pay repairs paper to maintain the car Person drives an old car Person spent a fortune social-interaction event-centered physical-entity capable of as a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' X wants is located at used for used for has property is made of desires because X wanted can be hindered by happens before happens after before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' X needs Another track of FKGC models [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 95] is developed based on model-agnostic meta-learning (MAML) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These models leverage meta-learning to learn the learning process of expressive embeddings of entities and relations with only a few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In particular, they use the high-frequency relations in the training set to capture meta- information, which includes common features across different task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With a good parameter initialization provided by the meta-information, these models can rapidly adapt to the test tasks where only a few instances are provided for each task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 3 On the other hand, as a particular type of knowledge graphs, commonsense knowledge graphs (CKGs) like ATOMIC [50] and ConceptNet [57], where entities and relations are composed of free-form text, gain less atten- tion from embedding-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' CKGs are dynamic since entities with unseen text are constantly introduced, which makes them natural benchmarks for FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, entities and attributes in CKGs are usually free-form texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As shown in Figure 3, unlike general KGs that have structured entity and relation names, entity descriptions in CKGs have rich semantic meaning, and implicit semantic relations can be used to infer commonsense knowledge directly, but the such feature also makes CKGs more sparse comparing with general KGs since entities referring to the same concept can be distinct nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As shown in [67], the average in-degree of ConceptNet and ATOMIC is only 1/15 and 1/8 compared with FB15K-237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since CKGs do not cleanly fit into a schema comparing two entities with a relation, embedding-based methods are limited to capturing implicit commonsense knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meanwhile, recent progress in training transformer-based contextual language models [16, 48] has inspired the interest in using the language models (LMs) as knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, recent works have focused on querying the LMs with prompts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', "Beatles was formed in __").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' COMET [8] is a transformer-based KG completion model trained to predict the unseen tail entity conditioning on the head entity and relation on ATOMIC [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' BertNet [26] takes a step further to directly extract triples for unseen entities from pre-trained language models by automatically paraphrasing an initial prompt for FKGC/KGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, in this survey, we cover typical applications of FKGC models in data science, visual extraction, and medical communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We further discuss future research directions for FKGC on general and commonsense knowledge graphs based on the observed weakness of current models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2 PRELIMINARIES In this section, we first review different KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we formally define knowledge graph completion and few-shot knowledge graph completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In the final part of this section, we briefly introduce few-shot learning and meta-learning, which are widely used in FKGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1 Knowledge Graph Let E and R denote the set of entities and relations, a knowledge graph G = {(𝑒𝑖,𝑟𝑘,𝑒𝑗)} ⊂ E × R × E is a collection of factual triples, where E represents the set of entities, R is the set of relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 𝑒𝑖 and 𝑟𝑘 are the 𝑖-th entity and 𝑘-th relation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We usually refer 𝑒𝑖 and 𝑒𝑗 as the head and tail entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A knowledge graph can also be represented as X ∈ {0, 1}|E |×|R |×|E |, which is called the adjacancy tensor of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The (𝑖, 𝑗,𝑘) entry X𝑖,𝑘,𝑗 = 1 when triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) is true, otherwise X𝑖,𝑘,𝑗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A list of commonly-used KGs with their source, size and examples are shown in Table 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1 structural Knowledge Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As introduced earlier, previous works tend to extract semi-structured text to construct knowledge graphs [5, 39, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These KGs are usually constructed by crowdsourcing or extracted from crowdsources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' FreeBase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Freebase is a crowdsourced curated KG first introduced in 2008 [5] and has been used as a standard baseline KG for many tasks, including KG completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The most up-to-date and complete version of Freebase contains about 3 billion total triples and about 50 million entities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A wide-used subset of Freebase, FB15K-237, excludes inverse relations from Freebase and includes 14541 entities, 237 relations, and 272,155 training triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Relations contained in Freebase 1https://developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/freebase/ Manuscript submitted to ACM 4 Haodi Ma are hierarchical, which form a well-defined space of entities and relations that motivates the thread of embedding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Wikidata is also a crowdsourced KG, containing approximately 78 million data items, with about 23,000 types and 1,600 relations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' At its inception, it was designed to be an alternative method to manage the information found in Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As well as providing factual information, Wikidata gives the context around a fact by storing its source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As of 2014, Wikidata supported 287 languages [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In 2014 Google transferred the data stored in Freebase into Wikidata [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Entities and relations in Wikidata are described through property-value pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' YAGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' YAGO is a large knowledge base that is built automatically from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The knowledge graph com- bines information from Wikipedia in 10 different languages into a whole to provide a multilingual dimension of the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It also attaches spatial and temporal information to many facts and thus allows the user to query the data over space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As constructed from Wikipedia, YAGO inherits the hierarchy from Wikipedia and uses structural text for entities and relations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' There exists multiple iterative version of YAGO, including YAGO2 and YAGO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' YAGO3 contains 87 million facts, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='8 million entities, and 76 million keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2 Commonsense Knowledge Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Commonsense knowledge graphs mean to organize commonsense or domain-specific knowledge for downstream appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Though existing CKGs [50, 57] are also commonly constructed by human crowdsourcing, they use free-form text for entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ATOMIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The ATOMIC dataset2, released by [50], contains 877K tuples covering a variety of commonsense so- cial knowledge around specific event prompts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', "X goes to the store").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ATOMIC contains everyday commonsense knowledge entities organized as if-then relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It contains over 300K entities in total, and entities are composed of text descriptions with an average of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='4 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, ATOMIC distills its commonsense in 9 dimensions, covering the event’s causes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', "X needs to drive there"), its effects on the agent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', "to get food") and its effect on other direct (orimplied) participants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', "Others will be fed").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ConceptNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ConceptNet [57] is a multilingual knowledge graph that connects words and phrases of natural language with labeled edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Its knowledge is collected from many sources that include expert-created resources, crowdsourcing, and games with a purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It represents the general knowledge involved in understanding language using words and phrases of different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such "concepts" can help natural language applications to understand better the meanings behind the words people use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ConceptNet contains over 13 million links between these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Visual Genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Instead of just using natural language sources, Visual Genome [31] collects commonsense knowledge from images as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It collects dense annotations of objects, attributes, and relations with each image to construct the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, Visual Genome contains over 100K images in total, with each image having, on average, 21 objects, 18 attributes, and 18 relations between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since the objects, attributes, and relations are extracted from images, the dataset categorizes them with WordNet [41] synsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In this section, we first review different KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we formally define knowledge graph completion and few-shot knowledge graph completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In the final part of this section, we briefly introduce few-shot learning and meta-learning, which are widely used in FKGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2https://homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='edu/ msap/atomic/ Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Survey of existing knowledge graphs and examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' source size examples Freebase https://developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/freebase 4 relation groups, 2M nodes, 18M edges /m/070xg, /sports/sports_team/colors, /m/01g5v (Seattle seahawks) (blueish) /m/06kxk2, /people/person/place_of_birth, /m/01_d4 (Carl Foreman) (America/Chicago) Wikidata https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2k relations, 75M objects, 900M edges Austria:Q40, part_of:P361, European Union:Q458 The Beatles:Q1299, location_of_formation:P740, Liverpool:Q24826 YAGO https://yago-knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org/ 87 million facts, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='8 million enti- ties, and 76 mil- lion keywords pl/Henryk_Pietras, wasBornIn, de/Debiensko fr/Chateau_de_Montcony, isLocatedIn, Burgundy Concept Net https://conceptnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='io/ 36 relations, 8M nodes, 21M edges /c/en/go_to_bed, /r/HasPrerequisite, /c/en/get_ready_for_bed /c/en/section_of_children’s_books, /r/AtLocation, /c/en/bookstore /c/pt/atordoaremos/v, /r/FormOf, /c/pt/atordoar ATOMIC https://allenai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org/data/atomic-2020 9 relations, 300k nodes, 877k edges money, is used for, pay repairs money, is made of, paper PersonX accepts the job, xEffect, joyful Visual Genome https://visualgenome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='org/ 42k relations, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='8M nodes, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='3M edges, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='8M attributes men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, wears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, backpack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01 chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01 juice bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01, desk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2 Few-shot Knowledge Graph Completion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1 Knowledge Graph Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The objective of knowledge graph completion (KGC) is to predict valid but unobserved triples in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Formally, given a head entity 𝑒𝑖 (tail entity 𝑒𝑗) with a relation 𝑟𝑘, models are expected to find the tail entity 𝑒𝑗 (head entity 𝑒𝑖) to form the most plausible triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' KGC models usually define a scoring function 𝑓 : E × R × E → R to assign a score 𝑠(𝑒𝑖,𝑟𝑘,𝑒𝑗) to each triple (𝑒𝑖,𝑟𝑘,𝑒𝑗) ∈ E × R × E which indicates the plausibility of the triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2 Knowledge Graph Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Knowledge graph embedding (KGE) proposes to project entities and relations into a well-defined space that can be modeled with high-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Knowledge embedding (KGE) models usually associate each entity 𝑒𝑖 and relation 𝑟𝑗 with vector representations e𝑖, r𝑗 in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then they define a scoring function to model the interactions among entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM 6 Haodi Ma KGE models can be generally classified into translation and bilinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The representative of translation models is TransE [6], which models the relations between entities as the difference between their embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This method is effective in inferencing composition, anti-symmetry, and inversion patterns but cannot handle the 1-to-N, N-to-1, and N-N relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' RotatE [59] models relations as rotations in complex space so that symmetric relations can be captured, but is as limited as TransE otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ComplEx [64], as a representative of bilinear models, introduces a diagonal matrix with complex numbers to capture anti-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Other models, such as BoxE [1], and HAKE [91], can express multiple types of relationship patterns with complex KG embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='3 Graph Neural Network Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The Graph Neural Network (GNN) has gained wide attention on KGC tasks in recent years [71, 84, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the high expressiveness of GNNs, these methods have shown promising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, SOTA GNN-based models do not show great advantages compared with KGE models while introducing additional computational complexity [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, NBFNet [97] and RED-GNN [90] achieve competitive performance on KGC benchmarks, but the leverage of the Bellman-Ford algorithm which needs to propagate through the whole knowledge graph, which restrict their application on large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='4 Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Few-shot Learning (FSL) [74] focus on learning transferable general prior knowledge from existing tasks for new tasks with limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It usually adopts a meta-learning framework [19] that treats entire tasks as training examples so that the model can adapt fast to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, given a set of tasks T and their training data, in the meta-training phase, the objective of the model is to learn global parameters 𝜃 ′ that are effective across all tasks in T: 𝜃 ′ = argmax 𝜃 ∑︁ T𝑖∼𝑝 (T) L(DT𝑖,𝜃) where 𝑝(T) is distribution of tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' DT𝑖 is the training data of task T⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' L is the loss function of the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then in the meta-testing phase, 𝜃∗ is taken as the initialized parameters (prior knowledge) that are quickly adapted on a new task T𝑗: 𝜃∗ = L(DT𝑗,𝜃) where T𝑗 only has limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Previous FSL methods can be generally categorized into (1) metric-based methods [56, 65] that exploit task-specific similarity metrics to generalize from support set data to query data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' (2) optimization-based methods [19, 20, 49] that aim to find model parameters that are sensitive to changes in the task so that the base learner can quickly adapt to new few-shot tasks with a small number of gradient updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='5 Few-shot Knowledge Graph Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Following the definition of KGC and FSL, we now formally define few-shot knowledge graph completion (FKGC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Consider a knowledge graph G = {(ℎ,𝑟,𝑡)} ⊂ E × R × E is a collection of factual triples, where E represents the set of entities, R is the set of relations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Given a relation𝑟 ∈ R and its supporting set S𝑟 = {(ℎ𝑘,𝑡𝑘)|(ℎ𝑘,𝑟,𝑡𝑘) ∈ T }, the task is to complete triple (ℎ,𝑟,𝑡) with the tail entity 𝑡 ∈ E missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In other words, the model needs to predict 𝑡 from a candidate entity set C given (ℎ,𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' When |S𝑟 | = 𝐾 and 𝐾 is very small, the task is called 𝐾-shot KG completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' An extreme scenario is when 𝑘 = 0, which means there are no supporting triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such a task is also referred to as inductive KGC, zero-shot KGC, or out-of-graph KGC, where models are expected to predict correct relations for unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 7 A few-shot KGC model aims to rank the true entity higher than the the false candidate entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In FKGC, each training task corresponds to a relation 𝑟 ∈ R with its own supporting/query entity pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', T𝑟 = {S𝑟, Q𝑟 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As previously defined, S contains 𝐾-shot supporting entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Q𝑟 = {(ℎ𝑚,𝑡𝑚)/Cℎ𝑚,𝑟 } consists of all queries and the corresponding candidates Cℎ𝑚,𝑟 which are selected based on the entity type constraint [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We further denote all the tasks in training as the meta-training set T𝑚𝑒𝑡𝑎−𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' After training on the meta-training set, the few-shot learning model will be tested by predicting facts of new relations 𝑟 ′ ∈ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relations for testing are unseen from the meta-training set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', R ∪ R′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Each relation in the testing phase also has its few-shot supporting and query set: T𝑟′ = {S𝑟′, Q𝑟′}, defined similarly as in meta-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We denote all tasks in testing as the meta-testing set T𝑚𝑒𝑡𝑎−𝑡𝑒𝑠𝑡𝑖𝑛𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model also has access to a background KG G′ which is a subset of G with all the relations instead of those in T𝑚𝑒𝑡𝑎−𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 and T𝑚𝑒𝑡𝑎−𝑡𝑒𝑠𝑡𝑖𝑛𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 3 FKGC MODELS Generally, FKGC models with Structural Knowledge combine KGC models with few-shot learning for various applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides KGE models, GNN-based methods have also shown competitive performance in FKGC since only limited labeled data are provided in the supporting set for each few-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The models that leverage semantic features, on the other hand, utilize prompts to combine structural and semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Three main challenges exist for FKGC task [30]: (1) How to learn the most representative information of triples in the few-shot setting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' General machine learning algorithms require a large number of data for model training, while there are only a few references in the few-shot scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Learning representative patterns of different relations from limited triples becomes the key to solving the FKGC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' (2) How to decrease the over-reliance on background KGs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Most prior few-shot methods rely on a back- ground KG to access information from neighborhoods of entities or pre-train the entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Some recent models argue that a thorough background KG is not always accessible, and storing it in memory is also space-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' (3) How to utilize the negative samples to enhance the model efficacy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The most intuitive matching ap- proaches generally compare the similarity between queries and positive references while neglecting the similarity between queries and negative references, which can improve the accuracy of triplet validity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In this section, we systematically categorize recent FKGC models with structural knowledge into metric-based methods and optimization-based methods depending on how they adopt FSL techniques and how they tackle the three questions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we go from prompt-based structural models to the ones that take advantage of pretrained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A list of representative FKGC works with their open-source dataset/codes is provided in Table2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1 Metric-based Method Existing metric-based FKGC models share the framework of either Matching Network [65] or Translation Network [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For models that are built based on Matching Network, they first implement a GNN-based entity encoder to generate entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then an aggregation module is applied on entity pairs in the supporting set to compute the embedding of each relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, the model computes the probability of acceptance of each query triple based on its similarity to supporting triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' KGE models like TransE [6] and ConvE [15] are also widely used in entity encoder as the intermediate representation to be further enhanced with other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM 8 Haodi Ma Method Learning Task FSL technique Venue Code/Data Link Structural FKGC GMatching [78] Relation prediction Matching-based EMNLP’18 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/xwhan/One-shot-Relational-Learning FSRL [86] Relation prediction Matching-based AAAI’20 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/chuxuzhang/AAAI2020_FSRL FAAN [54] Relation prediction Matching-based EMNLP’20 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/JiaweiSheng/FAAN GEN [3] Relation prediction Matching-based NeurIPS’20 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/JinheonBaek/GEN REFORM [70] Relation prediction Matching-based CIKM’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/SongW-SW/REFORM MetaR [12] Relation prediction Matching-based EMNLP’19 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/AnselCmy/MetaR GANA [43] Relation prediction Matching-based SIGIR’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/ngl567/GANA-FewShotKGC MetaP [30] Multi-hop relation prediction Metric-based SIGIR’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/jzystc/metap Meta-KGR [37] Multi-hop relation prediction Optimization-based EMNLP’19 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/THU-KEG/MetaKGR ADK-KG [89] Multi-hop relation prediction Optimization-based SDM’22 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/ADK-KG/ADK-KG ZS-GAN [47] Multi-hop relation prediction Optimization-based NeurIPS’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/Panda0406/Zero-shot-knowledge-graph-relational-learning Commonsense FKGC ConMask [55] Relation prediction Text fusion-based AAAI’18 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/bxshi/ConMask MIA [44] Relation prediction Text fusion-based WWW’21 -- InductiveE [54] Entity prediction LM-based IJCNN’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/BinWang28/InductivE BERTRL [85] Relation prediction LM-based AAAI’22 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/zhw12/BERTRL OntoPrompt [83] Entity prediction Prompt-based WWW’22 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/zjunlp/KnowPrompt ZS-SKA [12] Relation prediction Prompt-based arxiv -- COMET [8] Relation prediction LM-based AAAI’21 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/atcbosselut/comet-commonsense BERTNet [26] Triple prediction LM-based – https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='com/tanyuqian/knowledge-harvest-from-lms Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Representative FKGC methods with open-source code/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Following this framework, GMatching [78] is the first work to solve the one-shot KGC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It first proposes a neighbor encoder, which utilizes the local graph structure to generate better entity embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The motivation here is that, although the entity embedding of previous KGE models can encode relational information, previous work [77] shows that explicitly modeling structure patterns like path can still benefit relational prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The neighbor encoder in GMatching encodes only the one-hop neighbors of each given entity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', a set of (relation, entity) tuples to guarantee it is general to large-scale KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, starting with pre-trained KGE embeddings for every tuple in the one-hop neighbor set, GMatching applies a feed-forward layer to encode the interaction between the relation and the entity in each tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A neighbor encoder is then applied on both supporting entity pairs and query entity pairs to generate each representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, the model exploits an LSTM-based recurrent processing block [65] to perform multi-step matching between the reference pair and each query pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The matching scores are finally used to rank every entity in the candidate set of each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides proposing the first baseline model on FKGC task, the work also proposes two widely-used benchmarks: NELL-One and Wiki-One [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Both are built following the FKGC task setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' More statistics and details are provided in Table 3 and Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Sharing the same idea, FSRL [86] extends GMatching to the few-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It further proposes a relation-aware heterogeneous neighbor encoder to enhance entity embeddings based on the heterogeneous graph structure and attention mechanism so that the model can encode the different impacts of different neighbors on the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The main argument here is that different neighbors should impact the task relation differently, which models like GMatching neglect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, taking ParentOfPerson as the task relation, the neighbor (MarryTo, Melinda Gates) should have a higher weight compared with (CeoOf, Microsoft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To tackle such an issue, FSRL introduces an attention module to generate entity embeddings by assigning different attention weights when encoding all neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 9 By applying the attentive neighbor encoder, FSRL acquires the representation of each entity pair in the supporting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then implements an RNN-based aggregator to model interactions between supporting entity pairs for each task related to generate an informative representation of the entire supporting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Inspired by aggregating node embeddings with recurrent neural network [25], FSRL applies a recurrent autoencoder aggregator on all entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In order to formulate the embedding of the reference set, it aggregates all hidden states of the encoder and extends them by adding residual connection [27]and attention weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the aggregated representation of the reference set, FSRL applies a matching network to discover similar entity pairs of the reference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Instead of comparing each reference entity pair with the query pair, a similar recurrent matching processor with LSTM cells is used to directly compute the similarity between the reference set and query entity pair for the final answer ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' During the training session, each time model samples a task relation and optimizes the model for that task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model will sample few-shot entity pairs as the supporting set and a batch of query entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Negative training sets are constructed by polluting the tail entities in query entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-learning is exploited in the gradient descent step for parameter optimization so that FSRL can transform well onto test few-shot relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Although FSRL [86] proposes to treat neighbors differently based on their relevance to the central entity, it still assigns fixed weights to all neighbors throughout all task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such a solution leads to static entity embeddings in different tasks, hurting the system’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' FAAN [54], taking a step further, argues that entity neighbors should have varied impacts with different task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, SteveJobs is associated with task relations HasJobPosition and HasChild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Intuitively, if the task relation is CeoOf, the model should pay more attention to the job position role of entity SteveJobs than the family role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, task relations can have different meanings under different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, if the task relation is isPartOf, as shown in Figure 1, such relation has different meanings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' organization-related as in (Liverpool, isPartOf, Premier League) or location-related as in (Gainesville, isPartOf, Florida).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Apparently, for a query (Dallas, isPartOf, Taxes), the location-related references should be more influential than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Therefore, the reference (supporting) triples should also contribute variously to different queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To address the above challenges, FAAN proposes an adaptive attentional neighbor encoder to model entity embeddings with one-hop entity neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They also follow TransE [6] to model the task relation embedding as a translation between the head and tail entity embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', r ≈ h − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, to further model various roles of the reference entities, FAAN train an attention metric based on the relevance of entity neighbor relations and the task relation to further obtain a role-aware neighbor embedding for each entity in the reference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The encoder allows dynamic attention scores adaptive to different task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The adaptive mechanism helps to capture the diverse roles of entities based on the different contributions of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The final representation of each entity encodes both the pre-trained embedding and its role-aware neighbor embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the enhanced entity representation provided by the encoder, FAAN further applies a stack of Transformer blocks for supporting and query triples to capture various meanings of the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It borrows the idea of learning dynamic KG embeddings from [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For each element, it passes the element embedding and position embedding through several Transformer blocks to acquire meaningful entity pair embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, instead of using static representations when predicting different queries, FAAN obtains a general adaptively representation of the supporting set by aggregating all the references with their attention score to the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' FAAN also uses meta-training in the same fashion as FSRL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', the model is trained on different task relations in the meta-training set to generate a set of parameters that performs well across all the tasks and can quickly adapt to Manuscript submitted to ACM 10 Haodi Ma few-shot tasks in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With all the above, FAAN improves the quality of entities and reference representations by capturing their fine-grained meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Sharing a similar matching score as in FSRL, FAAN outperforms previous models on FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' HARV, on the other hand, focuses on capturing the differences between neighbor relations and entities and interaction between relations, which are previously neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It introduces a hierarchical neighbor aggregator for central entity representation by separating the information between the head entity and relation (relation-level) and between the relation and tail entity (entity-level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relation-level attention weights are computed based on the head entity and relation embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relation-level embeddings are generated by aggregating neighbor relations of head entity ℎ with such attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Concatenations of relation-level embedding and each tail entity are then used to generate the entity-level attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The two-level weights finally generate the triple-level weight, which is used to compute the enhanced entity representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Interactions between relations are taken into account by the relation encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The encoder is an extension of the LSTM aggregator in FSRL with Bi-LSTM, which updates representations of all support entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The concatenation of the embedding of support entity pairs and the embedding from the Bi-LSTM encoder is used as the final representation of each entity pair, and the supporting set is represented by an attention-based aggregation of all support entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In addition, GEN [3] investigates an out-of-graph FKGC scenario for relation prediction between unseen entities or between seen and unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is meta-learned to extrapolate the knowledge from seen to unseen entities, and transfer knowledge from entities with many to few links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GEN further develops a stochastic embedding layer for transductive inference to model uncertainty in the link prediction between unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Gen is compatible with any GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, two GENs are employed at the meta-training stage for both inductive and transductive link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The first GEN is inductive GEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It learns to encode the unseen entities that are not observed and predicts the links between seen and unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The second GNN, respectively, is transductive GEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It takes a step further to learn to predict the links between unseen entities themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To enable transductive inference, the meta-learning framework in GEN can simulate the unseen entities during meta-training while they are not observed in conventional learning schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Also, since link prediction for unseen entities is inherently unreliable, which gets worse in the few-shot scenario where only a few triplets are available for each entity, GEN learns the distribution of unseen representations for stochastic embedding to account for the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Further, we apply a transfer learning strategy to model the long-tail distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These lead GEN to represent the unseen entities that are well aligned with the seen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As mentioned, a naive GEN may be affected by the intrinsic unreliability of few-shot out-of-graph link prediction due to the uncertainty of unseen entities’ representations caused by lacking supporting triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The stochastic layer tackling this issue embeds an unseen entity by learning the distribution over that entity embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GEN also models the source of uncertainty on the output embedding from the transductive GEN with Monte Carlo dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' More recently, REFORM [70] proposes an error-aware module to control the negative impact of errors affecting FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It slightly varies from the original FKGC to predict the missing relation category of the query entity pair from the few-shot relation categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since most real-world KGs are automatically constructed, many errors are incorporated into KGs without manual validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such errors significantly alleviate the performance of previous methods on FKGC, especially when there are only a few supporting triples to rely on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The neighbor encoder of REFORM focus on selecting the most reliable neighbors with an attention mechanism to enhance entity representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The attention weight matrix is trained with pre-trained embeddings (in REFORM, TransE) to ensure those correct neighbors have higher weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The matrix is then normalized using a softmax function to acquire a robust embedding for each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Reference entity pairs Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 11 are represented by the concatenation of their head and tail entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, to generate robust embedding for relations in the supporting set, REFORM contains a cross-relation aggregation module based on a transformer encoder to capture the relation correlations and support instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The transformer encoder makes each input embedding participate in the encoding of all other input embeddings based on a multi-head attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then in the error mitigation module, REFORM exploits the graph convolution network (GCN) to generate confidence weights of various relations for each query task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The confidence weights can be considered attention weights to limit errors’ impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, REFORM builds a query-oriented graph to measure the effect of different supporting instances on a specific query relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The GCN is trained to minimize the loss that a query relation is grouped into the wrong category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A representative that uses Translation Network is MetaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The idea is that instead of encoding neighbor information, MetaR focus on transferring the common and shared information within one task from reference instances to query triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such information is referred to as relation meta in MetaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relation-meta learner generates representations of entity pairs from head and tail entity embeddings in the supporting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Given the head and tail entity pairs in the supporting set, the learner first extracted entity-pair specific relation meta through a fully connected neural network which uses LeakyReLU [38] as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The final relation meta of a task is the average of all entity-pair specific relation meta in the current supporting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' MetaR also exploits Meta-learning to accelerate the learning process, which is referred to as gradient meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='5, the model should be able to update a new few-shot task rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Inheriting the idea of TransE [6], MetaR applies a similar score function ||h𝑖 + RT𝑟 − t𝑖 || to calculate the score of each entity pair with the relation meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, by minimizing the loss over the supporting set with the score of all positive and negative triples, the gradients of parameters can indicate how they should be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Following this gradient update rule, MetaR can make quick updates on relation meta and use the updated one to score the query set with the same scoring function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model is trained to minimize the sum of query loss over all the tasks in one batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Compared with GMatching [78], which relies on a background knowledge graph, our MetaR is independent of them, thus, it is more robust as background knowledge graphs might not be available for few-shot link prediction in real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GANA [43], taking a step further, extends MetaR by refining embedding and relation meta computation with attention mechanism and an LSTM aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The motivation here is that noise neighbor information may hurt the model when the neighbors are spare or even if no proper neighbor is available to represent the few-shot relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GANA proposes a global-local framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' At the global stage, a gated and attentive neighbor aggregator is built to accurately integrate the semantics of a few-shot relation’s neighborhood, which helps filter the noise neighbors even if a KG contains extremely sparse neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The head and tail entities associated with the few-shot relation and their neighborhoods are combined to eliminate noise neighbor information due to the sparse neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A gating mechanism could determine the importance of the neighborhood representation to represent a few-shot relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, a graph attention network (GAT)-based neighbor encoder is developed at the global stage to capture different impacts of neighbors to improve the quality of entity embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The encoder generates the attention weight for each neighbor based on a trainable linear transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GANA employs a gate value with linear transformation to eliminate noise neighbors due to the sparse neighborhood to automatically determine the impact of the neighbor of an entity for the few-shot task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' An entity is then represented by combining its entity embedding with its neighbor representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The final triple neighbor representation of the supporting set is the concatenation of the head and tail representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the supporting set encoded, GANA employs an attentive Bi-LSTM encoder to integrate multiple neighborhood representations of a query relation in the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The query relation representation is a weighted Manuscript submitted to ACM 12 Haodi Ma sum of the final hidden states of the Bi-LSTM by combining all the neighbor embeddings in the supporting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For the local stage, a meta-learning-based TransH(MTransH) method is designed to model complex relations and train our model in a few-shot learning fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The reason to use TransH is its ability to model complex relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A similar loss function is applied with MAML approach to learning well-initialized parameters over all few-shot (query) relations in the meta-training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another similar FKGC model HiRe [76], can be seen as an extension of GANA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It proposes to jointly capture three levels of relational information: entity-level, triple-level, and context-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Contrastive learning is used to encode the union of the neighbors of the head and tail entities together in a triple to encode a wider context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' HiRe proposes a context encoder for the target triplet to learn the embeddings of its true/false contexts based on the self-attention mechanism so that important neighbors within the context would be given higher weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Furthermore, a contrastive loss is employed to pull close the triplet towards its actual context and push it apart from its false context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then at the triple-level relational learning stage, instead of LSTM, HiRe develops a transformer-based meta-relation learner to capture interactions among reference triples and generates meta relational representation of target relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, HiRe employs a TransD-based [29] meta score function to capture the diversity of entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' MAML-based training strategy is also applied similarly to GANA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the three-level relational information, HiRe performs better on NELL-One and Wiki-One compared with state-of-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The ablation study further proved that all three levels of relational information are crucial to the performance of HiRe, which future models can further leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-iKG, another recent work on this track, proposes to utilize local subgraphs to transfer subgraph-specific information and rapidly learn transferable patterns through meta-gradients with meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Graph neural network is recently incorporated into inductive relation reasoning to capture multi-hop information around the target triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, GraIL [63] proposes a subgraph-based relation reasoning framework to process unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' CoMPILE [40] extends the idea by introducing a node-edge communicative message-passing mechanism to model the directed subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-iKG can be interpreted as an extension of CoMPILE method to FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Instead of being limited to transductive settings and unable to process unseen entities, Meta-iKG targets a few-shot inductive KGC task, including new entities in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model splits relations into few-shot and large-shot relations with a threshold K on relation instances number and meta-train with large-shot relations to find well0initialized parameters and adapt the model on triples with few-shot relations following the framework of MAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Inheriting the structure from MetaR, Meta-iKG first extracts direct enclosing subgraphs between target and tail entities at the relation-specific learning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then an inductive node labeling function is applied to identify the different roles of entities in the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The node embedding is initialized by the distances to the target entities to embed the relative position of each node in the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-iKG then follows the idea of CoMPILE to use communicative message-passing neural network to score each subgraph to encode its plausibility of the target triple as the task loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Regular meta-learning steps promise performance on few-shot relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, they may introduce bias to the updated parameters since the task relation query set only updates the final parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To guarantee the performance of Meta-iKG on large-shot relations as well, it introduces the large-shot relation update procedure, which further updates the final parameters using the support set with a lower learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This operation enables Meta-iKG to generalize well on the whole inductive dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To tackle the KG-dependent problem and further exploit negative samples in the training stage, a meta pattern learning framework, MetaP [30], is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Patterns in data are representative regularities to classify data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Triples in KGs also follow relation-specific patterns, which can be used to measure the validity of triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The pattern of a relation refers to the regularity of feature co-occurrence of the head entity, relation, and tail entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' MetaP designs a pattern learner based on convolutional filters to extract patterns of triples directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It can learn latent representations of Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 13 relation-specific patterns from limited references and thus is independent of the background KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, by leveraging negative references, MetaP can measure the validity of query triples more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A pattern matcher with a validity balance mechanism (VBM) is proposed to predict the probabilities of whether patterns of query triples are positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2 Optimization-based Method Optimization-based FKGC models rely on MAML for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In other words, such models tackle the challenge of the few-shot relation prediction problem by optimizing GNN with MAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since methods like GMatching and FSRL only focus on fact prediction and exploit only one-hop neighbors, they miss more structure information provided in KGs, and results lack interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Accordingly, multi-hop relation reasoning was proposed to infer facts using multi-hop reasoning paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', (The Beatles, FoundIn, Liverpool) ∧ (Liverpool, PartOf, British) → (The Beatles, BaseIn, British).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Recent models [14, 34] propose multi-hop reasoning methods, which leverage the symbolic composition information of relations in KGs to achieve explainable reasoning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These works also state multi-hop reasoning as a sequential decision process and exploit reinforcement learning to tackle such tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A meta-based algorithm for multi-hop reasoning (Meta-KGR) sharing similar ideas is then proposed to provide explainable and effective few-shot relations reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, Meta-KGR introduces a reinforcement learning framework to model the multi-hop reasoning process, where a recurrent neural network encodes the search path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then adopts MAML to learn effective meta-parameters from high-frequency relations that could quickly adapt to few-shot relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, deriving the on-policy reinforcement learning (RL) from [34], the multi-hop reasoning is treated as a Markov Decision Process (MDP): give the query relation 𝑟𝑞, the model starts from the source entity 𝑒𝑠, sequentially step through several relations and entities until it arrives at the target entity 𝑒𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The MDP module includes (1) state, which is the entity at the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' All the states share the source entity and task relation as the global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' (2) action: the action space at state 𝑠𝑡 includes all the current entity’s outgoing relation and entity tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' (3) reward: the model will receive a terminal reward equal to 1 if it reaches the correct target entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Otherwise, a reward will be given based on the similarity between the target entity and the current entity using pre-trained KG embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The policy network is then used to determine action at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, Meta-KGR applies the policy network considering the search history over background KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The search history before the current step is encoded with LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The action space is represented by stacking all the action embeddings in the action space at the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The policy network is trained to maximize the expected reward over all triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In meta-learning, Meta-KGR employs a meta-policy network similar to MAML so that Meta-KGR can quickly adapt to a relation-aware policy network for every query relation with well-initialized parameters learned in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' FIRE [87] extends Meta-KGR with a heterogeneous neighbor aggregator and a search space pruning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, FIRE leverages on-policy reinforcement learning to model the path of multi-hop reasoning and encodes entity embedding using multi-hop heterogeneous structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then prunes the reasoning search space using knowledge graph embedding to improve the reasoning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-learning is also applied in the optimization procedure so that the learned parameters can be fast adapted for few-shot task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, since the original RL module in Meta-KGR does not encode the heterogeneous structure information into the entity embedding, FIRE keeps the structure encoding module as in FSRL to enhance entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Note that some entities in KGs have a large number of neighbors, making the action space redundant at specific steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Different from [14, 34] that cut edges based on centrality score, FIRE takes structural correlation between states as an important Manuscript submitted to ACM 14 Haodi Ma feature to guide action search and applies a knowledge-aware search space pruning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The model keeps only top-𝑚 most correlated entities at each step based on the structural correlation between entities at step 𝑡 and possible candidates at step 𝑡 + 1 with pre-trained KGE like TransE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Fast adaptation with meta-learning is then utilized to learn well-initialized parameters that can quickly generalize to few-shot relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' More recently, ADK-KG [89]further improves FIRE by developing a text-enhanced heterogeneous graph neural network to encode node embeddings, where entity and relation embeddings are pre-trained using content information and augmenting MAML with task weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is the first work to leverage a pre-trained language model to capture content information in the FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The reinforcement learning module is similar to the ones in FIRE and Meta-KGR and generates the encoding of each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The problem is that RL only encodes the reasoning process but ignores the content and structural information in KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ADK-KG thus develops a text-enhanced heterogeneous graph neural network to enrich entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Firstly, for each entity and relation in KG, ADK-KG extracts their text information as their content features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then merges all text features of entities and relations and feeds them into the pre-trained BERT language model [16]to obtain the corresponding content feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For enumerated content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', entity and relation ids in Wikidata), ADK-KG applies one-hot encoding to convert it to a binary feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' After that, a neural network is utilized for encoding and aggregating content embeddings of selected neighbors for each relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The selected neighbors include first-order neighbors and relations and also high-order neighbors sharing the same relation and the first-order ones from a random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, because different relation types of neighbors will make different contributions to the final entity representation, the model employs the attention mechanism to utilize these relation-type-based neighborhood embeddings to generate the final embedding of each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such embeddings are then used in RL-based reasoning to replace pre-trained entity embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In the meta-learning step, since relations in KG usually have different meanings, AKD-KG assign different weights to them with self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another recent work extending Meta-KGR and FIRE is THML [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' THML argues that RL-based models’ generalization is usually limited by low reasoning performance on hard relations (relations with high training loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' THML challenges this problem in FKGC by identifying the hard relations at each training batch and then further training the model on those effectively generated new hardness-aware training batches from both relation and relation cluster levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' THML also formulates the reasoning process as an MDP as in Meta-KGR and FIRE at the hardness-aware meta-reinforcement learning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The main difference is that to solve the sparse reward caused by false reasoning paths and efficiency concerns, THML splits the reward into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The terminal reward is the same as in FIRE, except that THML uses ConvE [15] for pre-trained embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The path reward encodes the reasoning chain length to encourage the model to find the target entity with a relation chain that is as short as possible since shorter paths often provide more reliable reasoning than longer paths [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A path may be declined if the length exceeds 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another problem with KG reasoning is that models tend to infer paths with similar semantic meaning in the training stage, which may lead the model into a local-optimal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' THML thus proposes that the diversity reward encourages the model to find different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, instead of random sampling triple queries at the training stage, THML applies a two-level hardness-aware sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Relation level hardness-aware sampling ranks the reasoning accuracy of all relations in a batch online to select hard relations for the next batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' At relational-cluster level hardness-aware sampling, THML obtains pre-trained TransE embeddings for all relations, then performs the k-means algorithm to form relation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then selects the cluster with the hard relation with the lowest accuracy at the current batch as the hard cluster and adds it to the next batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Apart from the methods discussed above, there are also other types of optimization-based solutions on FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, ZSGAN [47] studies zero-shot KGC by establishing the connection between text and knowledge graph with generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The motivation is that the semantic features of new classes can be reflected by their Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 15 textual descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Moreover, textual descriptions contain rich and unambiguous information, which is critical for large-scale recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The core of ZAGAN is the design of a conditional generative model to learn the qualified relation embeddings from raw text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ZSGAN leverages a feature encoder for real data representations to generate reasonable real data distribution from KG embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The feature encoder is trained in advance from the training set and fixed during the adversarial training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The neighbor encoder only considers one-hop neighbors of each entity and used GCN [51] to generate the structure-based representation of each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then a feed-forward layer is used as the entity encoder to extract the information from each head, tail entity pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relation fact representation is finally formulated as the concatenation of head and tail neighbor embeddings and the entity pair embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Given text representations, the generator is to generate reasonable relation embeddings that capture the corresponding relational semantic information in the knowledge graph feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Based on this, the prediction of query relations is converted to a supervised classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, text embedding is the vector sum of word embeddings weighted by TF-IDF values after removing stop-words and punctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The text embedding is passed to the generative adversarial model (GAN) to generate the relation embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' On the contrary, the discriminator seeks to separate the fake data from the real data distribution and identifies the relation type as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The input features are first transformed via a fully-connected (FC) layer with LeakyReLU [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Two network branches follow after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The first branch is a FC layer that acts as a binary classifier to separate real data from fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The other branch is classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In order to stabilize training behavior and eliminate mode collapse, ZSGAN also adopts the gradient penalty, which penalizes the model if the gradient norm moves away from the target norm value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Similarly, RAN [88] worked out a general feature generation-based framework for addressing unseen relations in both few-shot KG completion with unseen relations and few-shot relation extraction from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Unlike previous works that leverage entity pair matching, P-INT [79] utilizes the paths from the head to the tail entities to represent an entity pair and computes the interactions between paths for the FKGC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The motivation is still to involve more structural and expressive information and exclude noise neighbors in few-shot reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To extract the support subgraph for each support entity pair, P-INT employs a two-side BFS algorithm [77] to prune the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The intersected neighbors of the left and right paths are used to generate paths from head to tail entities with different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relations in these paths are then combined as a set of supporting relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Pre-trained TransE embeddings are used to compute the similarity between each pair of relations in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then to reason the query subgraph, P-INT extends the limited number of neighbors with a fixed number of steps and, for each of them, gets the maximum similarity with the supporting relations and returns top-𝐿 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' After 𝑇 hops, the model extends a query subgraph with at most 𝑇 × 𝐿 entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Simultaneous to reasoning, P-INT can trace all paths from the head entity of the query to every extended entity in the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In the matching component, P-INT calculates the similarity between every two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then the RBF aggregation function in [77] is used to extract similarity features of the similarity matrix as the interaction between paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Inspired by FAAN [54], P-INT computes relation-aware attentions for different paths to model their impact on the matching result based on the relevance of a path to the query relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='3 Ontology-based Methods Apart from the FSL methods that are mentioned above, there is an emerging interest in extracting knowledge from large language models as pre-training/transforming fine-tuning models have become a default paradigm for natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Thus, how to effectively transfer between structured relational knowledge and natural language knowledge has become a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Recent works attempt to integrate structural knowledge like ontology to Manuscript submitted to ACM 16 Haodi Ma enhance language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' One of the representatives on this track is OntoZSL [21], which proposes a novel zero-shot learning framework that not only enhances the class semantics with an ontological schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It also employs an ontology-based generative adversarial network (GAN), as in ZSGAN, to synthesize training samples for unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' OntoZSL first designs an ontology encoder for learning relation representations from the ontological schema by considering the structural relation between concepts and their correlations in the textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Pre-trained TransE is used to generate the default embedding of all concepts in the ontological schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then a text-aware semantic embedding model is employed by projecting the structural and textual representations into the same embedding space and learning them simultaneously with the same scoring function as in TransE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With two types of representation learned with the ontology encoder, OntoZSL then follows GAN [47] to learn and train the real relation embeddings in bags containing all the one-hop neighbor triples of the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The embeddings of all entity pairs in the bag consist of the real embedding of the task relation, which contains semantic and structural features of the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' OntoZSL finally generates plausible relation embedding for each unseen relation with its text description using the well-trained generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For prediction, the model calculates the similarity between the relation embedding and the candidate’s head, tail entity pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='4 FKGC with Commonsense Knowledge Previous models primarily focus on structural information in KGs like one-hop neighbors and paths but relatively ignore semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Models like ADK-KG [89] consider content information but still rely on neighbors to generate embeddings for entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As mentioned, due to the recent broad investigation of pre-trained language models such as BERT [16] and GPT [9], some methods are developed by fine-tuning these models to exploit textual information for few/zero-shot KG completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This section covers several representatives in this category to discuss how to leverage commonsense/textual information in KGs effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ConMask [55] is one of the first models that is proposed to tackle the zero-shot KGC problem by encoding unseen entities with their names and text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In general, it feeds the text embeddings of the entities and the relation of a triple into a model composed of an attention-based relation-dependent text masking module and a CNN-based target fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To capture text information that is relevant to task relation, ConMask uses a relation-dependent content masking module to reduce noise in the given descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The component first pre-processes the input description to select small relevant snips based on the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ConMask utilizes the attention mechanism to mask the irrelevant task, which assigns a relation-dependent similarity score to words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A common problem here is that the words with the highest scores are not the target entity but words with similar semantic meaning to the task relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since actual target words are always around these indicator words, ConMask adjusts the similarity score of each word based on its context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then the model extracts relation-based entity embeddings via target fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Three fully convolutional neural networks (FCN) layers without de-convolution operations are developed for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since directly generating entity embeddings with the target fusion module may be costly, ConMask employs a semantic averaging function that aggregates word embeddings to represent entity names and generates representations of other textual features for each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Although ConMask successfully learns embeddings of the entity’s name and parts of its text description to connect unseen entities to the KGs, it does not take full advantage of the rich feature information in the text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, the proposed relationship-dependent content masking method in ConMask may quickly fail to find the target words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To challenge such problems, a Multiple Interaction Attention (MIA) model [44] is proposed to acquire the interactions between the head entity description, head entity name, the relationship name, and the candidate tail entity descriptions for more graphic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, MIA similarly uses the additional textual features of head Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 17 entity descriptions to enhance the head entity representation and apply the attention mechanism between candidate tail entities to enhance their representation of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, MIA first transforms each word in the head entity description, question, and candidate tail entity description into several continuous representations, including GloVe, POS, NER, and BERT embeddings, and concatenates them to form the input representations of each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, to enhance entity representations with the interaction with all the relevant textual information, MIA leverages the same word-level sequence alignment attention mechanism for each interaction since words in the same description are not equally important, and relevant descriptions usually mention each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The third component of the model is an RNN layer which uses Bi-LSTM to encode text context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The exact attention mechanism is applied between multiple candidate tail entity descriptions to enhance their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' MIA also explores different scoring functions to enhance the convergence of the model, which achieves significant improvements against other state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, the problem with this approach is that it relies heavily on entity descriptions and only works when necessary information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' InductiveE [67] proposes a commonsense KG link prediction method that can deal with unseen entities by utilizing textual entity descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It enables inductive learning by directly building representations from entity descriptions instead of leveraging textual entity representations as training initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically, It first represents an entity using the concatenation of its text embeddings by the fastText word embedding model [4] and the last layer for [𝐶𝐿𝑆] token of the pre-trained BERT [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To further enhance semantic entity representations with entity neighbor information, InductiveE adds similarity links for unseen entities to initialize their neighbor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then feeds the entity representations of the densified graph into a model composed of a gated-relational GCN encoder and a simplified ConvE [15] decoder to predict each triple’s score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The gated encoder is employed to guarantee adaptively control over the amount of information fused to the center node from their neighboring connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For the decoder, Conv-TransE [53] has been proven effective and efficient in scoring triples in KGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The decoder of InducctiveE further improved it by adding a shuffling operation before convolution to allow more interaction between embeddings and improve the convolutional model’s expressive ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' InductiveE only explores inductive learning on unseen entities, while inductive learning on unseen/new relations is also valuable for real-world commonsense FKGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A KGC model, KG-BERT [82], is developed to target such a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It transforms a triple head entity, relation, and tail entity into a text sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then makes triple prediction as a downstream text classification task, where BERT is fine-tuned with given training triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For unseen entities and relations with name information, the candidate triples associated with them can be directly predicted by transforming them into text sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Following this approach, BERTRL [85] also proposes to predict triples as a downstream text classification task of BERT, utilizing the text information of entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, it fine-tunes BERT using single triples and possible paths connecting two entities where reasoning is conducted explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Given a query triple (ℎ, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=',𝑡) to exploit the neighborhood knowledge of head and tail entities and select proper neighbors for efficiency concern, BERTRL entirely relies on BERT to encode such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To formalize structural knowledge in KGs to fit into BERT models, BERTRL collects all the length-𝑘 reasoning paths between the head and tail entity in the query triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It then takes each path as a separate input to BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Each path individually induces the query triple with a confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The problem is then transformed into a binary classification problem where the score of the linear layer on top of [𝑐𝑙𝑠] indicates the correctness of the query triple given a reasoning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The maximum aggregation of bag scoring is used at inference time to generalize the score of all reasoning paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The path with the highest score can be used to explain the reasoning process of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM 18 Haodi Ma As all these works manage to consider textual information simultaneously with structural information, they still treat them as two types of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' On the other hand, prompt-tuning proposed in GPT3 [9] as an arising methodology has been used for tasks like relation extraction names, entity recognition, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Recent works have tried to integrate external knowledge into prompt designing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' OntoPrompt [83] is one of the representatives of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It utilizes prompts to bridge commonsense knowledge from pre-trained language models (LMs) and structural knowledge from knowledge graphs for the FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' OntoPrompt first employs ontology transformation to enrich and convert structure knowledge to text format, where it utilizes pre-defined templates to convert knowledge to text as prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically for KGC, the model leverages head entity types and tail entity types from the ontology representation as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It uses corresponding items obtained from the external Wikidata query as the source of ontology and extracts the textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It follows KG-BERT to consider KGC as a triple classification task and concatenate entities and relations as an input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It also uses the learnable virtual token to enhance the prompt representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Next, OntoPrompt proposes span-sensitive knowledge injection to select informative knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Considering that irrelevant and noisy knowledge may lead to changes in the meaning of the original sentence, OntoPrompt leverages a visible matrix based on spans to limit the impact of corresponding knowledge on the knowledge injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In this way, not all tokens in the input sentences will be affected by external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Third, OntoPrompt develops a collective training algorithm to optimize representations jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Note that the injected external knowledge should be associated with the surrounding context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' learnable tokens are added with random initialization and optimized along with injected ontology tokens with a fixed language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Inspired by the previous study [23] that prompt-tuning in the low-data regime is unstable and may obtain poor performance, the model further optimizes all parameters to train the ontology text and input text representations collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With the components above, OntoPrompt can enrich task-relevant knowledge using pre-trained LMs, prevent negative knowledge fusion, and integrate commonsense knowledge into structural which solve challenging problems in knowledge missing, knowledge noise, and knowledge heterogeneity and achieve promising performance in FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ZS-SKA [22] also utilizes prompt to tackle the FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Instead of leveraging ontology knowledge, they directly work on semantic knowledge augmentation for zero-shot relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' ZS-SKA first generates augmented instances with unseen relations from instances with seen relations following a word-level sentence translation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To encode every instance, ZS-SKA tokenizes all the words in a sentence and feeds them to BERT to generate a contextual representation for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' A CNN layer is used after obtaining the tokenized input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then, they design prompts based on ConceptNet [57] to integrate semantic knowledge information learned from seen relations and exploit such knowledge to infer the features of unseen relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They consider multiple types of semantic knowledge, including relation descriptions and name entities, to learn unseen relations effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The input sequence is wrapped with a natural language snip template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Instead of using the actual label sets in the prompt template, they automatically construct weighted virtual label words based on the knowledge graph of each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Prompts are represented by embedding the super-class of the input words and the virtual label embedding for unseen relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' By generating the representations of both seen and unseen relations with augmented instances and prompts through prototypical networks [56], distance is calculated to predict unseen relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The works above have already shown how pre-trained language models, as external sources, can help with FKGC tasks for both entity and relation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Taking one step further, instead of completing a zero/few-shot query tuple, embraced by the power of pre-trained LMs, another trending topic is to directly complete KGs by constructing triples for unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Representative models on this task include COMET [8] and BERTNet [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 19 COMET [8] is the first comprehensive study on automatic commonsense knowledge graph completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is a generative transformer model over commonsense knowledge which learns to generate detailed and diverse commonsense descriptions in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As mentioned in Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2, one main challenge on commonsense knowledge graph reasoning is that commonsense knowledge, represented by open-text, usually does not fit into a fixed schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Generally, COMET tackles this problem by constructing commonsense KG/KB by training a transformer over existing tuples as a seed set of knowledge to learn an adaptive representation of commonsense knowledge with a pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then the LM can be used to produce novel tuples with unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relations are identified by generating phrases that can semantically complete an existing seed phrase and relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In detail, the transformer language model in COMET follows the structure of GPT [48], which consists of multiple transformer blocks of multi-headed attention and fully connected layers to encode input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The input of the model is concatenated sequence of words for knowledge tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To encode the order information between tokens that is ignored by the transformer, COMET adds a position embedding for each position in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The position embedding and word embedding of each word is added for the final representation, which is the input to the first transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' COMET is trained to produce the tail entity given the tuple’s head entity and relation, which follows the setting of the KGC task but is expected to generate novel tuples that do not exist in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The limitation of COMET is that it can only generate triples for new entities with seen relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The ideal zero-shot KGC model should be able to construct tuples with unseen entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' BERTNet [26] is then proposed to harvest KGs with implicit knowledge from pre-trained LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' BERTNet only requires a few-shot seed set, including an initial prompt and seed entities for each relation as input, and can extract knowledge for unseen relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In general, the model automatically generates different prompts and searches within a given LM for novel knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' BERTNet tackles two challenges in KGC/KG construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' First, LMs have shown to be inconsistent even with a slightly different prompt, making it difficult to extract knowledge reliably from LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' An intuitive solution is to learn the optimal prompts automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such methods require extensive training data, which is unavailable in few-shot or zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To this end, BERTNet employs unsupervised paraphrasing on an initial prompt to generate a set of various prompts with their confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then entity pairs that consistently satisfy these prompts are extracted to generate novel triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another challenge then comes into space when searching for proper entity pairs due to the ample candidate space in LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' BERTNet devises an efficient search-and-rescoring strategy that strikes the balance between knowledge accuracy and coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' a prompt/entity pair compatibility function is designed to dynamically reassign weights for both candidate prompts and entities at each knowledge searching step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Specifically for entity searching, BERTNet first uses individual compatible score, which is more accessible to threshold and prune, to weighted average across all prompts to generate a large set of candidate entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These candidates are then re-ranked by the total compatible score to select the output entity pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since BERTNet only requires a set of seed prompts and few-shot entities for relations other than a pre-trained LM, the model guarantees the flexibility to extract novel knowledge even for relations that have complex structures or include multiple entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Besides, the resulting triples can be considered as an interpretation of the respective black-box LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another novelty of BERTNet is that instead of only looking at matrices like hits@10 and BLUE score, it directly integrates the generated tuples into background KGs and applies the new KGs for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The performance on those tasks indicates that BERTNet can generate novel high-quality tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM 20 Haodi Ma 4 APPLICATIONS AND RESOURCES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='1 Applications FKGC models have been applied to problems other than prediction tasks on knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' One main application is to leverage the few-shot link prediction technique for other graph-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, Meta-Graph [7]investigates few-shot link prediction on different networks, including protein-protein interaction (PPI) networks [98], 3D point cloud data [42] and academic social networks [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' MetaTNE [32] also leverages a meta transformer commonly used in FKGC tasks to predict protein-protein interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Similarly, molecular property prediction has always been a demanding problem since manual prediction can be costly and inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-MGNN [24]and Pre-PAR [73] have been proposed to solve such tasks with few-shot link prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-MGNN takes each molecule as a graph and uses a graph-level-GNN to encode each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It further introduces an attention mechanism to make MAML aware of molecular property differences to improve model encoding further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Moreover, Pre-PAR improves Meta-MGNN by capturing relational structure among different molecular properties to effectively and efficiently propagate limited labels among similar molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Similarly, GEN [3] has also been used on drug-drug interaction prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another challenging real-world task related to link prediction is recommendation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, as proposed in the Yelp challenge [2], user reviews have become a significant part of web services like Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Since users can post their opinions about businesses, products, and services through reviews consisting of free-form text and a numeric star rating, the interaction between users, services/products, and reviews intuitively form structural and textual knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Recent techniques in collecting user-related information, like GPS-enabled devices, help form location-based social networks that provide the location information that is valuable for the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' One challenge for such systems is that when a new user joins, there is little existing knowledge in the platform besides basic information and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Thus, few-shot link prediction models like SEATLE [33] and heterogeneous information network-based models [36, 93] aim to tackle cold-start recommendation problems over graphs with meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Other models [17, 18, 35, 61, 68, 80, 81] tackles recommendation problems with methods like attribute matching with previous users, transforming knowledge from other platforms with cross-network meta-learning, modeling user with updated information with dynamic meta-graph reasoning, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' As mentioned in BERTNet [26], since FKGC models help to improve the quality and coverage of original KGs, the output can be leveraged by downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, affordance reasoning and extraction [96] can used few-shot KGC models to generate affordance for unseen entities with pre-trained LMs or similarity-matching with seen entities in the background KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' More recently, in large-scale video extraction/recognition datasets like EPIC [13] and STAR [75] or action planning tasks, procedural reasoning is required due to the rapid changing of reasoning scenarios and dynamic environment knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' FKGC models can be powerful on such tasks since they can easily capture new few-shot information and swiftly adapt and optimize themselves to fit each reasoning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='2 Resources With the bursting growth of zero-shot KG completion methods, various benchmarks have been proposed for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They are usually constructed based on existing commonly used typical KG completion datasets, including FB15k [5], FB15k-237 [46], NELL-995 [77], and some other sub-KGs extracted from popular KGs such as Wikidata [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such benchmarks usually obtain entity information from the original KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' For example, DBpedia50k, FB20k, and Wiki- data5M collect correspondingly text descriptions of entities from DBpedia and Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Here we introduce several representatives for zero/few-shot KGC: Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 21 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Statistics of FKGC Benchmarks (a) Relation prediction Benchmarks Ent # Rel # Triples # Task rel # (train/valid/test) Source NELL-One [78] 68,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='545 358 181,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='109 51/5/11 NELL Wiki-One [78] 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='838,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='244 822 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='859,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='240 133/16/34 Wikidata NELL-ZS [47] 65,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='567 181 181,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='109 139/10/32 NELL Wiki-ZS [47] 605,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='812 537 724,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='967 469/20/48 Wikidata (b) Entity prediction Benchmarks Ent # Rel # Triples(train/valid/test) # Unseen Ent # (train/valid/test) # Source WN18RR-sub [3] 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='478 11 93,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='003 (-) 3000 WN188RR FB15K-237-sub [3] 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='938 237 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='065/6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='246/9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='867 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='500/1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='000/1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='500 FB15K-237 NELL-995-sub [3] 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='694 200 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='345/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='676/5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='852 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='500/600/900 NELL-995 FB15K-237-OWE [52] 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='405 235 242,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='489/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='963/36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='250 2081 (-) FB15K-237 Wikidata5M [72] 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='594,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='485 1222 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='496,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='514v/6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='699/6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='894 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='579,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='609/7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='374/7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='475 Wikidata FB15k-237-OWE [52] is built on FB15k-237 for zero-shot KGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They first sample a set of tail entities and randomly pick associated head entities from FB15K-237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then all triples with their head in the associated entity set are moved to the testing set, which forms the testing set for tail entity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' At the same time, the ones with their tail in the associated entity set are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The testing set for head entity prediction is similarly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These two sets form the final testing set by further removing triples whose relations are not included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This testing set is further split into a validation set and the final testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The dataset contains 2,081 unseen entities, 12,324 seen entities, and 235 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The numbers of triples for training/validation/testing are 242,489/10,963/36,250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Wikidata5M [72] was originally constructed for evaluating text-aware KGE models but is now widely used for zero-shot KGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is developed based on Wikidata and English Wikipedia dump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Each entity uses the first section of its Wikipedia page as its description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The dataset excludes entities without Wikipedia pages or descriptions shorter than 5 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Next, they extract all the triples from Wikidatadump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The dataset keeps triples with qualified head and tail entities, leaving it with 4,594,485 entities, 822 relations, and 20,624,575 triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To support the zero-shot setting, they randomly extract two sub-KGs as the validation and testing sets and use the remaining as the training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They ensure that the entities and triples are mutually disjoint across the training, validation, and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Detailed information on each set can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Subsets of WN18RR, FB15k-237, and NELL-995 are constructed by [3] for out-of-graph completion between seen, unseen entities, and unseen entities themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These subsets are systematically extracted from their original benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' They randomly sample a group of entities with less than 100 associated triples as unseen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These triples are further separated to compose three meta sets (meta-training, meta-validation, and Manuscript submitted to ACM 22 Haodi Ma meta-testing sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The rest of the original benchmarks are considered as the background KGs/In-Graph, and entities inside are respectively seen entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The meta sets are then cleaned to guarantee that each of their triples has at least one unseen entity and that all the triples are out of In-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' More statistics about seen/unseen entities are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' NELL/Wiki-ZS are two zero-shot KG completion benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Each benchmark contains three relation-disjoint sets: the training set holds seen relations, validation/testing set holds unseen relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Associated triples are separated accordingly, while entities included in the testing/validation set are all involved in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' NELL-ZS has 139/10/32 relations in the training/validation/testing set , while Wiki-ZS involves 469/20/48 relations for training/validation/testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GEN [47] uses relation textual descriptions as the textual information, while OntoZSL [21] constructs ontological schemas, which contain not only textual information but also other relation knowledge, including relation hierarchies and relation domains for both benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' NELL/Wiki-One are originally developed in GMatching [78] for evaluating few-shot KG completion with unseen relations only one supporting instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To construct the testing set, they extract relations with less than 500 but more than 50 associated triples from the original benchmarks and use those as task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' With this preprocessing, 67 relations are extracted in NELL-One, and the benchmark further partitions them into 51/5/11 to further extracted associated triples and compose the training/validation/testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Similarly, in Wiki-One, 183 relations are extracted and partitioned into 133/16/34 for constructing triples in the training/validation/testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' On the entity side, 68,545 entities are extracted for NELL-One, and 4,838,244 for Wiki-One.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Additionally, another 291 and 639 relations are extracted as background relations to construct more triples for the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The relations in the training/validation/testing set are guaranteed to be disjoint so that the two benchmarks can be used for both zero-shot KGC and few-shot KGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 5 ANALYSIS Starting with GMatching [78], since the problem setting comes from KGC problem, FKGC models intuitively enhance KGE or GNN models which originally capture the structural information with meta-learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Early metric- based models like MetaR, FSRL [86] and FAAN [54], focusing on how to effectively leverage neighbor information by assigining static or task-aware attentions to different neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Optimization models like Meta-KGR [37], ADK-GK [89], ZS-GAN [47], explore to improve entity and relation embeddings with multi-hop path information and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These paths can also be used to explain reasoning logic which improve the interpretablility of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Meta-learning provides the opportunity for these models to quickly adapt to few-shot tasks at testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' However, these models ignore the semantic information in background KGs like entity/relation names, text description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' To combine the two types of information, prompt-based models like OntoPrompt [83] and ZS-SKA [22] are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' These models employs prompts to translate triples in KGs into sentence-like knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Pre-trained language models like BERT are then used to generate entity and relation embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Semantic information like text description of entities and relations helps to enrich the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' All these models are dependent on large-scale KGs since they provide structural information for entity and relation embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' On the other hand, GPT models [9, 48] show that pre-trained language models originally contain structural knowledge as well, some models take a step further to totally depend on pre-trained LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' COMET [8] and BERTNet [26] are the representatives of this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' COMET follows the framework of GPT models and BERTNet uses prompts to Manuscript submitted to ACM A Survey On Few-shot Knowledge Graph Completion with Structural and Commonsense Knowledge 23 GMatching [78] FSRL [86] MetaR [12] OntoPrompt [83] InductiveE [54] COMET!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' [8] BERTNet [26] NELL-One [78] ✓ ✓ ✓ ✗ ✗ ✗ ✗ FB15K-237-sub [3] ✓ ✓ ✓ ✓ ✗ ✗ ✗ WN18RR-sub [3] ✓ ✓ ✓ ✓ ✗ ✗ ✗ NELL-ZS [47] ✗ ✗ ✗ ✗ ✗ ✗ ✗ ConceptNet [57] ✗ ✗ ✗ ✗ ✓ ✓ ✓ ATOMIC [50] ✗ ✗ ✗ ✗ ✓ ✓ ✗ Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Evalution of Representative FKGC Models generate novel triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Comparing with models with structural knowledge, these models can usually work on both structural and commonsense KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This approach free FKGC models from background KGs and extend the usage to downstream tasks in more area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 6 FUTURE DIRECTIONS As discussed, earlier FKGC models are primarily extensions of meta-learning and transfer learning methods to FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another type of knowledge in meta-learning is the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In the future, representing knowledge on learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', previous reasoning process [60]) as KGs and integrating them with meta-learning or transfer learning algorithms could lead to more general neural-symbolic paradigms that apply to different FSL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Propagation-based methods like GEN [3] and REFORM [70] solve few-shot KG completion by utilizing the few-shot samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=', triples model the correlation of unseen entities with the distribution and correlation of seen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It would be a promising solution to utilize these few-shot links and the unseen entities’ correlations auxiliary information such as textual descriptions, attributes, and schemas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Another track of FKGC models like OntoPrompt [83] has proved that exploiting ontology/rule structured knowledge is a promising approach to infer symbolic knowledge like triples for unseen entities/relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' On the other hand, generation-based models like OntoZSL [21] are not biased to seen or unseen classes in prediction compared with the widely explored mapping-based methods [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Therefore, generation-based ZSL methods conditioned on the embeddings of KGs could be a future direction for FKGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is till recently that semantic information has gained attention in FKGC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' GANA [43]first proposes to integrate the semantics of a few-shot relation’s neighborhood, and ZSGAN [47] generates reasonable relation embeddings with text representations of task relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Works like COMET [8], KG-BERT [82], and BERTNet [26] further present the effectiveness of learning for representation of unseen/few-shot entities and relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' The performance of these models indicates that multi-modal knowledge can also be beneficial for FKGC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Challenging the problem of efficiently integrating data from more modalities into current FKGC models can be a promising direction to tackle not only knowledge graph completion tasks but also zero/few-shot reasoning in other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' According to Table 4, currently there is not enough evaluation to compare the performance between FKGC models with large-scale KGs and the ones leveraging pre-trained LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' It is still at theoretic level that models like BERTNet and COMET are effective for strucural KGs like NELL and Freebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Such evaluations will also be valuable for few-shot knowledge graph completion with multi-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Manuscript submitted to ACM 24 Haodi Ma 7 CONCLUSION As knowledge graphs are a popular source for tasks in various domains, few-shot knowledge graph completion models provide a chance to efficiently integrate new knowledge into existing KGs to improve the quality and coverage of KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' In this survey, we first introduce the major challenges and bases of FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we comprehensively reviewed previous studies on FKGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We categorize these methods into two groups: FKGC with structural knowledge and FKGC with semantic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' This categorization shows the trending of FKGC methods from meta-learning and attention-based models to leveraging semantic and textual information or even directly extracting structural triples from pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Then we discuss how FKGC models can be transferred or applied in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' Finally, we summarize the remaining challenges for different FKGC models and possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' We hope this survey will serve as a valuable guide for others who are interested in few-shot knowledge graph completion and advance future works in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' REFERENCES [1] Ralph Abboud, Ismail Ceylan, Thomas Lukasiewicz, and Tommaso Salvatori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfOvuh/content/2301.01172v1.pdf'} 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sha256:0f0f6665e0f163532f95bcc945348658d4d754f839e78929ef88bf731f665c7a +size 93411 diff --git a/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/2301.11742v1.pdf.txt b/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/2301.11742v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..074755e0e9ad20f7995ef88de3df69436ae8262f --- /dev/null +++ b/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/2301.11742v1.pdf.txt @@ -0,0 +1,2202 @@ +Graph-Free Learning in Graph-Structured Data: A More Efficient +and Accurate Spatiotemporal Learning Perspective +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +1University of Science and Technology of China, Hefei, China +2Shanghai AI Laboratory, Shanghai, China +{wx309,gpf9061,pengkun,wbw1995,zzy0929}@mail.ustc.edu.cn +baisanshi@gmail.com,angyan@ustc.edu.cn* +ABSTRACT +Spatiotemporal learning, which aims at extracting spatiotemporal +correlations from the collected spatiotemporal data, is a research +hotspot in recent years. And considering the inherent graph struc- +ture of spatiotemporal data, recent works focus on capturing spatial +dependencies by utilizing Graph Convolutional Networks (GCNs) +to aggregate vertex features with the guidance of adjacency matri- +ces. In this paper, with extensive and deep-going experiments, we +comprehensively analyze existing spatiotemporal graph learning +models and reveal that extracting adjacency matrices with carefully +design strategies, which are viewed as the key of enhancing perfor- +mance on graph learning, are largely ineffective. Meanwhile, based +on these experiments, we also discover that the aggregation itself +is more important than the way that how vertices are aggregated. +With these preliminary, a novel efficient Graph-Free Spatial (GFS) +learning module based on layer normalization for capturing spa- +tial correlations in spatiotemporal graph learning. The proposed +GFS module can be easily plugged into existing models for replac- +ing all graph convolution components. Rigorous theoretical proof +demonstrates that the time complexity of GFS is significantly better +than that of graph convolution operation. Extensive experiments +verify the superiority of GFS in both the perspectives of efficiency +and learning effect in processing graph-structured data especially +extreme large scale graph data. +1 +INTRODUCTION +In recent years, massive amount of spatiotemporal data have been +collected in various fields, e.g., urban computing, meteorology, and +atmosphere quality. Such collected spatiotemporal data is a set of +correlated time series where each individual sequence is about the +temporal variation of the monitored information at a specific physi- +cal location. And Spatiotemporal learning, which aims at extracting +spatiotemporal correlations from the collected spatiotemporal data, +has attracted more and more attentions [2, 4, 25, 29]. +Early efforts in spatiotemporal learning mostly focus on extract- +ing both spatial and temporal correlations respectively with Convo- +lution Neural Networks (CNNs) [31, 36, 41] and Recurrent Neural +Networks (RNNs) [21, 24, 35]. However, Such CNN based methods, +which divide the whole space into grids to extract spatial correla- +tions, has never considered the irregular spatial distributions of spa- +tiotemporal data. Therefore, this will definitely lead to the inevitable +missing of topology information and non-Euclidean correlations in +spatial learning. To tackle the above-mentioned issues of CNN based +methods, recent spatiotemporal graph learning works [18, 22, 33] +Yang Wang and Lei Bai are the corresponding authors with equal contribution. +tend to model spatiotemporal data into graph structure by mod- +elling spatial points as vertices. They first employ various strategies +to extract adjacency matrices for comprehensively and precisely +modeling the spatial correlations among vertices, and then use +graph convolution to aggregate the features of vertices based on +the extracted adjacency matrices. +Existing works on spatiotemporal graph learning can be divided +into two categories, predefined adjacency matrix based methods +and learnable adjacency matrix based methods. Regarding the first +category, they incorporate predefined distance based [18, 33, 39] +or temporal similarity based [11, 17, 20] adjacency matrices with +graph convolution. However, such methods assume that vertices, +which are in closer distances or with similar temporal series, are +highly correlated, and the adjacency matrices are predefined before +training and keep fixed during training. This determines that these +methods cannot effectively extract the time-varying correlations +among vertices due to their invariable adjacency matrices. Consid- +ering the lacking of the representation ability of these predefined +adjacency matrix based methods, the second category focus on +extracting adjacency matrices in a learnable manner during train- +ing [3, 27, 28] or throughout [14, 16]. The previous methods can +effectively enhance their representation abilities in spatial perspec- +tive with their learnable adjacency matrices, and the learning of +adjacency matrices are only within the training period. The latter +methods can even represent dynamic spatial dependencies into +dynamic adjacency matrices with their embedded self-attention +mechanism. +Throughout all existing works on spatiotemporal graph learning, +we discover that how to effectively extract spatial correlations is +one of the core concerns of them. Therefore, a very simple intu- +ition is that the performances of existing works improve with the +increasing of the complexities of models, and this easily leads all +researchers to an unsubstantiated conclusion: The extraction of spa- +tial adjacency matrix is critical to the success of such spatiotemporal +learning approaches. However, by comprehensively analyzing the +evolution of existing works, we discover that they are usually im- +proved in three aspects, i) new approach for extracting spatial adja- +cency matrix, ii) new method for extracting temporal dependencies, +and iii) new mechanism for more comprehensive fusion between +spatial and temporal correlations. Therefore, it is baseless to simply +attribute the improvements of these models to the construction of +their adjacency matrices. To explicitly verify this deduction, we first +classify all existing methods into four disaggregated classifications, +select STGCN [33], STFGNN [17], AGCRN [3], ASTGNN [16] as +the four representative methods respectively, and carry out a series +of experiments by replacing the learned adjacency matrices of the + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +four representative approaches respectively with a random matrix, +a matrix filled with a specific value, and some variant matrices by +combining the previous two matrices with an identity matrix, and +the result indicates a first fact: the performances of such approaches +can hardly be influenced by replacing their adjacency matrices with +generated ones, and this indicates that using different complex strate- +gies to enhance the extraction of adjacency matrices, which is both +resource-consuming and time-consuming, seem to be unnecessary and +cancelable. Meanwhile, this observation brings another question, +i.e., is graph convolution based data aggregation still useful in en- +hancing the final performance of spatiotemporal graph learning? +To answer this question, we design another series of experiments to +verify the impacts of graph convolution based data aggregation by +replacing the adjacency matrices of those approaches with an iden- +tity matrix, and the result indicates a second fact: graph convolution +based data aggregation appears useful in enhancing spatiotemporal +graph learning for most existing approaches except STGCN. To fig- +ure out whether graph convolution is useful or not in STGCN, we +detailedly analyze the architecture of STGCN and discover that it +has a unique module, layer normalization, which doesn’t exist in +the other three representative methods. Intuitively, employ layer +normalization in vertex dimension leads to aggregations of vertices +due to its integrated mean and covariance calculation operations, +therefore, we conduct an additional experiment by removing the +layer normalization operation from vertex dimension, and the result +indicate a third fact: aggregating vertices in spatial perspective is +essential and important for graph learning, and layer normalization +is also be of the functionality of data aggregation. So far, in summary, +we can obtain a simple conclusion, i.e., the aggregation of neigh- +boring vertices is crucial and effective for spatiotemporal graph +learning, but the way of adjacency matrix based graph convolution +aggregating vertices has little additional effect on spatiotemporal +learning. +Based on such finding, we think that existing sophisticated adja- +cency matrix and graph convolution based approaches are improv- +able, and then propose a novel efficient Graph-Free Spatial (GFS) +learning module based on layer normalization for capturing spa- +tial correlations in spatiotemporal graph learning. Specifically, we +first theoretically prove that the operation of layer normalization +in vertex dimension, which can also be transferred to a graph- +convolution-like form, is equivalent to a series of complex matrix +multiplications which are the core operation of data aggregation +in the operation of graph convolution. Next, we propose a novel +GFS learning module which consists of only a linear projection +and ReLU activation function combined component and a layer +normalization module. Rigorous theoretical proof demonstrates +that the time complexity of GFS is significantly better than that +of graph convolution operation. The proposed GFS module can be +easily plugged into existing models for replacing all graph convolu- +tion components, as far as the dimensions are aligned. Extensive +experiments verify the superiority of GFS in both the perspectives +of efficiency and learning effectiveness. +The contributions of this paper can be summarized as follows, +• To the best of our knowledge, this paper first reveals the fact +that adjacency matrix plays unimportant role in learning +spatial correlations from graph structured data, and also for +the fist time, this paper confirms that aggregating vertices +in spatial perspective is essential and important for graph +learning. +• We cross-verify that layer normalization is effective in ag- +gregating data among vertices in both theoretical and ex- +perimental perspectives. Based on this, we design a novel +efficient GFS learning module based on layer normaliza- +tion for capturing spatial correlations in spatiotemporal +graph learning. The proposed GFS module can be plugged +into existing spatiotemporal graph learning models as an +alternative to graph convolution layer. +• We conduct extensive experiments on several widely-used +real-world graph-structured spatiotemporal datasets as well +as two very large-scale graph datasets. To evaluate the +effectiveness and efficiency of GFS module, we select a +series of baselines and use GFS to replace their embedded +graph convolution layer. Experimental result shows that +GFS module is superior to traditional graph convolution in +terms of both efficiency and learning effect. +The paper is organized as follows. In Section 2, we analyze and +conclude related works, and then re-investigate existing spatiotem- +poral graph learning works with extensive experiments in Section +3. Based on all learned facts, in Section 4, we introduce the design +and implementation of GFS learning module. Section 5 describes +the experiments for evaluating our propose module and Section 6 +concludes this paper. +2 +RELATED WORKS +Spatiotemporal learning has attracted extensive research atten- +tions [12, 19, 22, 26, 37] in recent years. Early works [15, 24, 31, 32, +34–36, 41] mainly focus on employing CNN based networks and +RNN based networks to respectively extract both spatial and tem- +poral correlations. Nevertheless, in spatial perspective, such CNN- +based spatiotemporal learning methods mesh road network into +regular grids and employ convolutional neural networks to capture +spatial dependencies among grids. And those grid-partition based +methods are incapable of capturing the inherent Non-Euclidean +structured characteristics, hence are short in fully capture spatial +correlations. +To this end, recent works, i.e., spatiotemporal graph learning, aim +at modeling such these Non-Euclidean characteristics with graph +structure [7, 13, 18, 38–40], and existing efforts on this branch can +be roughly divided into two categories, predefined adjacency matrix +based methods and learnable adjacency matrix based methods. +Regarding the first category, in spatial perspective, they incor- +porate predefined distance based [18, 33, 39] or temporal simi- +larity based [11, 17, 20] adjacency matrices with graph convolu- +tion. Specifically, STGCN [33] applies graph convolution on fixed +distance-based adjacency matrix, and convolutional network for +modeling temporal dependencies. DCRNN [18] captures spatial +dependencies with bidirectional random walks on graph, and cap- +tures temporal dependencies with a encoder-decoder framework +and scheduled sampling. T-GCN [39] combines GCN with GRU to +exploit the spatiotemporal correlations of urban traffics. On the +other hand, STAG-GCN [20] applies classic DTW algorithm [5] to +construct adjacency matrices based on the similarities of vertices + +Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective +temporal trends and combines temporal aware adjacency matrices +with distance based adjacency matrices to enhance the modeling +of spatial correlations. STFGNN [17] modifies DTW to address +time efficiency concerns and involves temporal connections in the +construction of adjacency matrices. However, such methods as- +sume that vertices, which are in closer distances or with similar +temporal series, are highly correlated, and the adjacency matri- +ces are predefined before training and keep fixed during training. +This determines that these methods cannot effectively extract the +time-varying correlations among vertices due to their invariable +adjacency matrices. +Given the fact that all above-mentioned predefined adjacency +matrix based methods are poor at representing the time-varying +correlations, recent works in spatiotemporal graph learning mostly +construct adjacency matrices in a learnable manner during train- +ing [3, 28] or throughout [14, 16]. In particular, GraphWaveNet [28] +maintains learnable embeddings for each vertex, and constructs ad- +jacency matrices by calculating the embedded similarities between +vertices. AGCRN [3] follows the idea of GraphWaveNet and further +utilizes the embedded similarities between vertices to modify graph +convolution for learning node-specific patterns. Moreover, AST- +GCN [14] utilizes attention mechanism in both spatial and temporal +dimension to capture the dynamic spatiotemporal correlations at +different time intervals. By multiplying the attention values with +distance based adjacency matrices, ASTGCN actually gets dynamic +adjacency matrix changing at different time steps. ASTGNN [16] +is an upgraded version of ASTGCN, which modifies the attention +mechanism in ASTGCN and involves short-term temporal trends +when calculating attention values. +As analyzed, existing works in spatiotemporal graph learning +have devoted themselves into enhancing the effect of graph con- +volution. by generating various adjacency matrices with different +strategies. Indeed, with the increasing of the complexity of adja- +cency matrix construction, the performances of such spatiotemporal +graph learning methods increase correspondingly. This gives us +an illusion that the performance enhancement mainly due to the +improvement in adjacency matrix construction, and this is why +researchers never get tired of designing new strategy to improve +the adjacency matrix of graph convolution. However, by compre- +hensively analyzing the evolution of existing works, we discover +that they are usually improved in three aspects, i) new approach for +extracting spatial adjacency matrix, ii) new method for extracting +temporal dependencies, and iii) new mechanism for more compre- +hensive fusion between spatial and temporal correlations. So, there +naturally raises a question: how much can the modification of ad- +jacency matrix helps on the final performance of spatiotemporal +graph learning? +3 +RE-INVESTIGATION ON EXISTING +SPATIOTEMPORAL GRAPH LEARNING +To answer the above-proposed question, in this section, we first +select some representative spatiotemporal graph learning methods +based on the above analysis on spatiotemporal graph learning, and +then design a series of experiment to comprehensively analyze +them. +Output +Input +Strategy for +Defining Adjacency +Matrices +Predefined Simple Matrices +Stacked Spatial-Temporal Block +Graph +Convolutional +Module +Temporal +Modeling +Module +Adjacency +Matrices +𝑴 +𝑹 +𝑰 + 𝑴 +𝑰 + 𝑹 +Figure 1: Illustration of replacing adjacency matrices of ex- +isting works. +3.1 +Selection of representative spatiotemporal +graph learning methods +In previous section, existing spatiotemporal graph learning works +have been roughly divided into two major categories. Here, to specif- +ically distinguish and evaluate the impacts of adjacency matrix, we +further classify all existing works into four subclasses: +• Methods with distance based predefined adjacency +matrix: The adjacency between two vertices is determined +based on the physical distance between them, and we here +select STGCN [33] as the representative method of this +subclass. +• Methods with temporal similarity based predefined +adjacency matrix: The adjacency between two vertices +is determined based on the similarity of the temporal pat- +terns of these two vertices, and select STFGNN [17] as the +representitive method of this subclass. +• Methods with adjacency matrix learned during train- +ing: The adjacency matrix is determined based on the simi- +larities of embeddings which are learned from vertices dur- +ing training period, such adjacency matrices are expected +to model latent correlations among vertices. We here select +AGCRN [3] as the representative model of this subclass. +• Methods with dynamic adjacency matrix: The adja- +cency matrix, which is constructed dynamically based on +attention or some other mechanisms, are data-driven gen- +erated and changes over time. +3.2 +Re-investigation on the impacts of +adjacency matrix +Regarding the selected four representative works, to re-investigate +the impacts of adjacency matrix on their performances, we con- +duct a series of experiments by replacing their adjacency matrices +with some artificially designed adjacency matrices, and the de- +tailed operation of replacing adjacency matrices of existing works +is demonstrated in Figure 5. The artificially generated adjacency +matrices are, +• R: The matrix filled with random values. Using R as adja- +cency matrix results in randomly aggregating of all vertices. +Here R is generated by +R = softmax(R′) +(1) + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +10 +15 +20 +25 +30 +35 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(a) STGCN PEMSD4 +10 +15 +20 +25 +30 +35 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(b) STFGNN PEMSD4 +10 +15 +20 +25 +30 +35 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(c) AGCRN PEMSD4 +10 +15 +20 +25 +30 +35 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(d) ASTGNN PEMSD4 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(e) STGCN PEMSD8 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(f) STFGNN PEMSD8 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(g) AGCRN PEMSD8 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +MAE +RMSE +MAPE(%) +origin +R +M +I+R +I+M +(h) ASTGNN PEMSD8 +Figure 2: Performances of four representative methods with generated adjacency matrices on PEMSD4 and PEMSD8. +where +R′ = �𝑟𝑖𝑗 +� +𝑁 ×𝑁 +(2) +Notice that R′ +𝑖𝑗 ∼ N (0, 1) which indicates that all elements +of R′ are sampled from standard Gaussian distribution, and +𝑁 corresponds to the number of vertices. +• M: Matrix filled with a specific value 1/𝑁. Using M as adja- +cency matrix results in equally aggregating of all vertices. +• I + R: Add self-loop on matrix R. Here I corresponds an +identity matrix which is filled with 1. +• I + M: Add self-loop on matrix M. Here I corresponds an +identity matrix which is filled with 1. +Notice that all these four alternative matrices are all 𝑁 × 𝑁 dimen- +sional matrices. By comparing the original performances of the +four representative methods and their performances in case that +their adjacency matrices are respectively replaced with R, M, I + R, +and I + M, we can investigate the impacts of different adjacency ma- +trices on them, and the detailed implementation how the adjacency +matrices are replaced are listed as follows, +• STGCN: We utilizes the publicly available implementation +of STGCN 1, and simply replace the adjacency matrix cal- +culated by STGCN with the generated matrices. +• STFGNN: The official implementation of STFGNN is uti- +lized 2. STFGNN proposes a spatial-temporal fusion graph +which is constructed by combining both distance based +and temporal similarity based adjacency matrices. Here +both two matrices are replaced with our proposed matrices +and the construction procedure of spatial-temporal fusion +graph is kept. +• AGCRN: The adjacency matrix of AGCRN is determined +based on the similarity of vertex embeddings. We sidestep +1https://github.com/hazdzz/STGCN +2https://github.com/MengzhangLI/STFGNN +the calculation of similarity and directly utilize the gener- +ated matrices as the adjacency matrix of AGCRN. All other +settings of AGCRN are retained. The official implementa- +tion 3 is used. +• ASTGNN: A modified self-attention module is proposed +in ASTGNN, which generates attention weights among +vertices. The final adjacency matrix of ASTGNN is the dot- +product of attention weights and the distance-based ad- +jacency matrix. Therefore, we replace the final adjacency +matrix of ASTGNN with our generated matrices. Similarly, +the official implementation of ASTGNN is used 4. +The used datasets, metrics, and task settings are as follows, +• Datasets: All the experiments are conducted on PEMSD4 +and PEMSD8 [6]. More details about the datasets and data +preprocessing will be introduced in Section "EXPERIMENTS". +• Task: The task is to predict the spatiotemporal data over +the next 12 time steps based on the data over the past 12 +time steps. +• Metrics: Three widely used evaluation metrics are em- +ployed to measure the prediction accuracy. We here detail +the definitions of all the three metrics. Let V ∈ R𝑇×𝑁 ×1 +denote the ground truth future data of all 𝑁 vertices during +𝑇 time steps and ˆV ∈ R𝑇×𝑁×1 denote the predicted values. +The metrics can be formulated as follows. +3https://github.com/LeiBAI/AGCRN +4https://github.com/guoshnBJTU/ASTGNN + +Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective +MAE(V, ˆV) = +1 +𝑇𝑁 +𝑇 +∑︁ +𝑖=1 +𝑁 +∑︁ +𝑗=1 +|V𝑖𝑗 − ˆV𝑖𝑗 | +(3) +RMSE(V, ˆV) = +� +� +� +� 1 +𝑇𝑁 +𝑇 +∑︁ +𝑖=1 +𝑁 +∑︁ +𝑗=1 +(V𝑖𝑗 − ˆV𝑖𝑗)2 +(4) +MAPE(V, ˆV) = +1 +𝑇𝑁 +𝑇 +∑︁ +𝑖=1 +𝑁 +∑︁ +𝑗=1 +| V𝑖𝑗 − ˆV𝑖𝑗 +V𝑖𝑗 +| +(5) +By replacing the adjacency matrices of the four representative meth- +ods with the generated matrices, R, M, I + R, and I + M, we can +evaluate the effects and validity of the extracted adjacency matrices +of the four representative methods, and the results are demon- +strated in Figure 2. As demonstrated, while the extracted adjacency +matrices of these four methods are replaced by R, M, I + R, and +I + M, respectively, the performances of them barely changed with +two different datasets. Specifically, while the extracted adjacency +matrices are replaced by R, M, I + R, and I + M, on PEMSD4, the +performances of STGCN, STFGNN, AGCRN, ASTGNN change by +{0.41%, 1.06%, 2.14%} at worst and {2.54%, 1.79%, 2.82%} in average +in terms of MAE, RMSE and MAPE respectively, and change respec- +tively by {1.18%, 0.65%, 0.87%} at worst and {1.20%, 1.50%, 1.63%} in +average on PEMSD8. This reveals a very important fact: the effect +of extracting an adjacency matrix with carefully designed strategy +in spatiotemporal graph learning is limited and even non-existent, +and the complex and time-consuming strategies for extracting and +constructing adjacency matrices within those existing spatiotempo- +ral learning efforts seem to be unnecessary and cancelable. On the +other hand, considering the effect of enhancing adjacency matrix is +limited, is graph convolution based data aggregation still useful in +enhancing the final performance of spatiotemporal graph learning? +3.3 +Investigation on the impacts of graph +convolution based data aggregation +To answer the question proposed in the previous subsection, in this +subsection, we design another series of experiments to evaluate the +impacts of graph convolution based data aggregation. Specifically, +regarding the four representative spatiotemporal graph learning +methods, we use the previous defined identity matrix I to make +sure that the feature of each vertex is isolated and will never been +aggregated with the feature of any other vertices. The datasets, task, +metrics and all other basic settings in this series of experiments are +the same to those in the experiments in the previous subsection, +and the results are reported in Table 1. Notice here the column +with Origin corresponds the performances of the four models with +their original adjacency matrices and the column with I means the +performances of them while their adjacency matrices were replaced +with I. As reported in this table, once the adjacency matrices were +replaced with I, on both PEMSD4 and PEMSD8, the performances of +STFGNN, AGCRN and ASTGNN decrease significantly in terms of +all metrics. Specifically, when the adjacency matrices are replaced +with I, on PEMSD4, the performances of STFGNN, AGCRN and +ASTGNN decrease by {17.25%, 15.32%, 17.48%}, {18.95%, 16.23%, +18.80%} and {19.07%, 17.06%, 20.61%} in terms of MAE, RMSE and +MAPE, and {9.11%, 10.64%, 7.09%}, {16.93%, 16.10%, 17.28%} and +Spatial +Graph +Conv +Temporal +Block +Temporal +Block +LayerNorm +across +Vertex and +Feature +dimension +Stacked ST-Conv Block +Output +Input +time +Figure 3: Architecture of STGCN. +{12.52%, 12.44%, 11.42%} on PEMSD8. This can easily lead us to the +conclusion that graph convolution based data aggregation are still +useful in enhancing spatiotemporal graph learning. However, on +the other hand, we discover that the performances of STGCN are +barely change whether its adjacency matrix is replaced or not. This +seems counter the earlier conclusion that we just obtained from the +experiments on STFGNN, AGCRN and ASTGNN. Therefore, here a +big confusion comes naturally, whether graph convolution based +data aggregation is effective or not in enhancing spatiotemporal +learning? +Table 1: Performances of four representative methods with I +on PEMSD4 and PEMSD8. +Datasets +Metrics +STGCN +STFGNN +AGCRN +ASTGNN +Origin +I +Origin +I +Origin +I +Origin +I +PEMSD4 +MAE +21.34 +21.67 +20.29 +23.79 +19.74 +23.48 +18.93 +22.54 +RMSE +34.07 +34.47 +32.23 +37.17 +32.16 +37.38 +31.36 +36.71 +MAPE(%) +14.18 +14.05 +13.33 +15.66 +13.14 +15.61 +12.52 +15.10 +PEMSD8 +MAE +18.15 +19.17 +16.68 +18.20 +16.18 +18.92 +14.94 +16.81 +RMSE +28.67 +29.52 +26.23 +29.02 +25.65 +29.78 +25.08 +28.20 +MAPE(%) +11.61 +12.18 +10.72 +11.48 +10.30 +12.08 +9.81 +10.93 +3.4 +Further investigation on STGCN +In the previous subsection, we discover an interesting issue that +graph convolution based data aggregation seems essential in STFGNN, +AGCRN and ASTGNN, while it is almost useless in STGCN, and +this greatly encourages us to further investigate STGCN. As illus- +trated in Figure 3, we discover that STGCN includes an additional +layer normalization [1] module which doesn’t exist in the other +three representative methods. Specifically, STGCN applies layer +normalization on both vertices dimension and feature dimension. +Given input of all the features of all vertices at a specific time point, +V ∈ R𝑁×𝐷, the layer normalization can be briefly formulated as, +LN(V) = V − E[V] +√︁ +Var[V] +� +W + B +(6) +W, B ∈ R𝑁 ×𝐷 are learnable affine parameters and � is element- +wise multiplication. E[V] and Var[V] are the mean and covariance +of V respectively. Due to the operations of computing mean and +covariance, applying layer normalization on vertex dimension im- +plicitly leads to aggregation of all vertices. Therefore, this module + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +is also of the functionality of data aggregation. So far, we conject +that whether the data aggregation among vertices is what really +works? To clarify such conjecture, we design an additional exper- +iment. Here we directly modify STGCN by only employing layer +normalization on feature dimension and removing the implicit ag- +gregation of vertices, and the result are reported in Table 2. As +Table 2: Impacts of data aggregation operation in STGCN. +Datasets +Metrics +STGCN +STGCN* +Origin +I +Origin +I +PEMSD4 +MAE +21.34 +21.67 +31.02 +36.98 +RMSE +34.07 +34.47 +46.79 +54.80 +MAPE(%) +14.18 +14.05 +21.79 +26.53 +PEMSD8 +MAE +18.15 +19.17 +23.58 +30.30 +RMSE +28.67 +29.52 +35.89 +44.78 +MAPE(%) +11.61 +12.18 +14.55 +19.29 +shown in this table, in case that the data aggregation operation is +removed from STGCN, the performances of STGCN* are signifi- +cantly worse than those of STGCN. In particular, in case that the +STGCN uses its original extracted adjacency matrix, comparing +with the performances of STGCN, the performances of STGCN* +decreases by {50.89%, 42.68%, 62.41%} on PEMSD4 in terms of MAE, +RMSE and MAPE, and decrease by {29.92%, 25.18%, 25.32%} respec- +tively on PEMSD8. And while the adjacency matrix is replaced with +I, the performances of STGCN* further decrease by {19.21%, 17.12%, +21.75%} on PEMSD4 and {28.50%, 24.77%, 32.58%} on PEMSD8 re- +spectively. In summary, these experiments explicitly and undoubtedly +verify the effectiveness and importance of aggregating vertices in spa- +tial learning, and simultaneously indicates that layer normalization +is also be of the functionality of data aggregation. +3.5 +Lessons learned in re-investigating existing +spatiotemporal graph learning methods +In this subsection, we will briefly summarize the lessons learned in +re-investigating existing spatiotemporal graph learning with three +series of carefully designed experiments. +• LL1. The core idea of existing spatiotemporal graph learn- +ing methods, i.e., using different complex strategies to en- +hance the extraction of adjacency matrices, which is both +resource-consuming and time-consuming, is noneffective +to the performance of spatiotemporal graph learning. +• LL2. Data aggregation among vertices, which can effec- +tively capture spatial correlations among vertices, is the +key to the success of spatiotemporal graph learning. Even +with a matrix with a fixed value or random values, tradi- +tional graph convolution is still an effective way to achieve +data aggregation, however it time complexity is also an +issue that should be concerned about. +• LL3. Layer normalization, which is also be of the function- +ality of data aggregation, can also significantly enhance +the performance of spatiotemporal graph learning by ef- +fectively extracting the spatial correlations among vertices. +And compared with traditional graph convolution, the time +complexity of layer normalization has obvious advantages. +4 +GRAPH-FREE SPATIAL MODULE IN +SPATIOTEMPORAL GRAPH LEARNING +In this section, based on the previous learned lessons, we analyze +the possible technical solution of spatiotemporal graph learning, +and propose a graph-free spatial learning module as an alternative +to graph convolution. +4.1 +Rethinking of spatial learning in +spatiotemporal graph learning +Based on the learned lessons LL1 and LL2, we discover that graph +convolution based spatial learning is highly inefficient. Specifically, +the strategies for defining better adjacency matrices contribute little +to the final effect of spatiotemporal graph learning, and though, by +replacing the learned adjacency matrix with the fore-mentioned +matrix M, the efficiency of traditional graph convolution can signif- +icantly improved on the premise of without significantly losing the +performance of spatiotemporal graph learning, the computation +complexity of graph convolution itself is also a serious issue that +should be concerned about. Therefore, regarding spatial learning in +spatiotemporal graph learning, we should pay more attentions to +aggregating vertices in a more efficient manner rather than seeking +strategies for defining adjacency matrices. +On the other hand, the learned lesson LL3 indicates that layer +normalization across vertex dimension is also valid to data aggre- +gation, however, in theory, whether or why layer normalization +has the additional functionality of data aggregation still needs to +be further explored and analyzed. +4.2 +Theoretical analysis on the effect of layer +normalization in spatial learning +As analyzed in Section 3.4, layer normalization has shown its par- +tial effectiveness on capturing spatial dependencies among vertices. +Here, we further explore the data aggregation operation in layer +normalization and analyze the correlations between layer normal- +ization and graph convolution. The operation of layer normalization +is defined in Equation 6. To simplify the following analysis, we set +the feature dimension as 1 and omit the bias term B in layer nor- +malization. Also, the temporal dimension is omitted since it is not +relevant for the calculation. Based on the simplification, we have +input V ∈ R𝑁 ×1, and layer normalization first calculates the mean +of V, i.e., +E[V] = 𝔑 × V +(7) +where 𝔑 ∈ R𝑁×𝑁 is a matrix filled with 1 +𝑁 . Thus, the term V − +E[V] in Equation 6 can be calculated by, +V − E[V] = (I − 𝔑) × V +(8) +I ∈ R𝑁 ×𝑁 corresponds to the identity matrix. Considering the +affine parameter W ∈ R𝑁 ×1 in Equation 6, the element-wise multi- +plication between V − E[V] and W can be transferred to a matrix +multiplication, i.e., +(V − E[V]) +� +W = Diag(W) × (I − 𝔑) × V +(9) + +Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective +where Diag(W) is a diagonal matrix whose diagonal element in +each row corresponds to the element in the corresponding row of W, +and � is element-wise product. Finally, by defining 𝜎V = Var[V] +which represents the standard deviation of V, the calculation of +layer normalization can be written as, +LN(V) = [ Diag(W) +𝜎V +(I − 𝔑)]V = A × V +(10) +Where A = [ Diag(W) +𝜎V +(I − 𝔑)]. Regarding the calculation of layer +normalization, the diagonal matrix Diag(W) ensures the diversity +among the values in different rows in A, and A contains data- +driven term 𝜎V and learnable parameters W, which determine that +layer normalization is effective and flexible in aggregating vertices. +In case that the dimensionality of feature is 𝐷, we should calculate +each dimension individually with the above-mentioned method. +Figure 4 illustrates the layer normalization calculation for the 𝑖-th +dimension where V𝑖 the 𝑖-th dimensional feature. +Matrix +Multiplication +𝓥 +𝓥𝒊 +෡𝓥𝒊 +෡𝓥 +𝓐𝒊 = [𝐃𝐢𝐚𝐠 𝑾 +𝝈𝓥 +(𝑰 − 𝕹)] +Figure 4: Graph-convolution-like form of layer normaliza- +tion in data aggregation among vertices. +Comparing layer normalization with the calculation of graph +convolution defined as, +GraphConv(V, A) = A × V × W′ +(11) +where A and W′ correspond to the adjacency matrix and the learn- +able parameters respectively, we discover that layer normalization +aggregates vertices in a way similar to graph convolution. Actually, +the aggregation of vertices in layer normalization can be seen as +a graph convolution layer equipped with adjacency matrix as A +in Equation 10 and the difference between the two is that layer +normalization has no linear projection on the feature dimension, +i.e., W′ in Equation 11. +So far, we theoretically analyze the effectiveness of layer nor- +malization on data aggregating and prove that the operation of +layer normalization is similar to graph convolution without linear +projection on feature dimension. Considering the superiority of +layer normalization in terms of computational complexity, the idea +that how to achieve a more efficient spatial learning module by +taking advantage of layer normalization is worth exploring. +4.3 +Layer normalization based graph-free +spatial learning +Based on the analysis in the previous subsection, we propose a +novel and efficient Graph-Free Spatial (GFS) learning module by +equipping layer normalization with some add-on components. The +detailed architecture of GFS learning module is illustrated in Figure +5. As shown, we equip layer normalization with a linear projec- +tion and ReLU activation function to extract spatial correlations +among vertices. Considering that residual connection is effective +to increase the representation power of graph convolution [10], we +also employ a residual connection in GFS learning to increase the +representation ability of layer normalization. +Layer +Normalization +Linear & +ReLU +Output +Input +Linear +Figure 5: Architecture of layer normalization based graph- +free spatial learning module. +Specifically, given input V ∈ R𝑄×𝑁×𝑑𝑖𝑛, GFS first applies a linear +projection and a ReLU activation function on feature dimension, +i.e., +V′ = ReLU(VWs1 + bs1) +(12) +where Ws1 ∈ R𝑑𝑖𝑛×𝑑𝑜𝑢𝑡 and bs1 ∈ R𝑑𝑜𝑢𝑡 are learnable parameters, +and V′ ∈ R𝑄×𝑁 ×𝑑𝑜𝑢𝑡 is the result of projection. Next, a layer nor- +malization is applied on V′ on both vertex and feature dimensions, +i.e., +ˆ +V = V′ − E[V′] +√︁ +Var[V′] +� +Ws2 + Bs2 +(13) +where Ws2 ∈ R𝑁 ×𝑑𝑜𝑢𝑡 and Bs2 ∈ R𝑁 ×𝑑𝑜𝑢𝑡 are learnable affine +parameters, ˆ +V is the output of layer normalization. To generate +the final output V𝑜𝑢𝑡 ∈ R𝑄×𝑁×𝑑𝑜𝑢𝑡 of GFS, a residual connection +integrated with linear projection is applied on the input and added +to ˆ +V, i.e., +V𝑟𝑒𝑠 = ReLU(VWres + bres) +(14) +V𝑜𝑢𝑡 = V𝑟𝑒𝑠 + ˆ +V +(15) +where Wres ∈ R𝑑𝑖𝑛×𝑑𝑜𝑢𝑡 and bres ∈ R𝑑𝑜𝑢𝑡 are learned parameters. +Notice that GFS module can be easily plugged into existing mod- +els for replacing all graph convolution components, as far as the +dimensions are aligned. +4.4 +Time complexity comparison between GFS +and graph convolution +To validate the efficiency of GFS theoretically, in this subsection, +we compare the time complexity of GFS with that of standard graph +convolution defined in Equation 11. +• Time complexity of GFS: As illustrated in Figure 5, GFS +contains two linear projection operations on feature di- +mension, and each linear projection operation has the time +complexity of O(𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡). Regarding the opera- +tion of layer normalization, its embedded mean calcula- +tion, standard deviation calculation, and element-wise mul- +tiplication are all in linear time complexity in both the +dimensionalities of vertex and feature, i.e., O(𝑁 × 𝑑𝑜𝑢𝑡). +Therefore, the overall time complexity of GFS, which is + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +O(𝑁 × 𝑑𝑜𝑢𝑡 + 𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡), is exactly linear to the num- +ber of vertices. +• Time complexity of standard graph convolution: As +defined in Equation 11, graph convolution contains two +matrix multiplications. The first multiplication between +adjacency matrix and features of vertices has the time com- +plexity of O(𝑁 2 × 𝑑𝑖𝑛), and The latter multiplication with +learnable parameters has the same time complexity with +linear projection in GFS, i.e., O(𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡). There- +fore, the overall time complexity of graph convolution is +O(𝑁 2 × 𝑑𝑖𝑛 + 𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡), which is quadratic to the +number of vertices. +As theoretically analyzed, the proposed GFS is more efficient than +the operation of graph convolution, and we reckon that such differ- +ence in efficiency will be reflected most vividly in processing very +large scale graph. +5 +EXPERIMENTS +5.1 +Experimental scheme and datasets +To evaluate the performance of our proposed GFS module, we select +a series of representative graph convolution based spatiotempo- +ral graph learning works and replace their integrated graph con- +volution module with GFS module. The experiments consist of +three parts: i) Performances on spatiotemporal graph learn- +ing: Based on four widely-used real-world spatiotemporal datasets, +PEMSD3, PEMSD4, PEMSD7 and PEMSD8 [6]5, we conduct a series +of experiments to compare the performances of GFS and graph +convolution on different backbones in terms of traffic prediction, ii) +Performances on graph learning with extreme large graph: +Based on two graph datasets with extreme large numbers of ver- +tices, i.e., PubMed [30] and Coauthor Physics [23], regarding the +task of node classification, we further investigate the time efficiency +and capability of GFS in modeling spatial correlations, and iii) In- +vestigation on graph-free architecture: To further verify the +effectiveness of each individual component of the graph-free archi- +tecture, we carry out a series of ablative studies on PeMSD4. The +statistical information of all used datasets is summarized in Table 3. +Table 3: Dataset descriptions. +Dataset +#Vertices +#Features +Time Range +PeMSD3 +358 +1 +09/01/2018 - 11/30/2018 +PeMSD4 +307 +1 +01/01/2018 - 02/28/2018 +PeMSD7 +883 +1 +05/01/2017 - 08/31/2017 +PeMSD8 +170 +1 +07/01/2016 - 08/31/2016 +PubMed +19717 +500 +N/A +Coauthor Physics +495924 +8415 +N/A +5.2 +Data preprocess +For spatiotemporal datasets, linear interpolation is utilized to fill +the missing values in the datasets. Then, we apply min-max nor- +malization to normalize all data into the range of [−1, 1] to stabilize +5These four datasets, which are about the highway traffic flow in California, are +collected by Caltrans Performance Measurement System. +the training process. Regarding all experiments on spatiotemporal +graph learning, all spatiotemporal datasets are divided into training, +validation and testing sets with the ratio of 6:2:2 in chronological +order, i.e., the earliest 60% are used for training, the subsequent 20% +are used for validation, and the last samples are for testing. Notice +that the raw traffic flow data within spatiotemporal datasets is ag- +gregated with the interval of 5 minutes, therefore the aggregated +datasets contain 288 data points for each day. For graph datasets, +we directly use the data provided by torch_geometric 6 and split +PubMed and Coauthor Physics with the ratio of 9:1 and 7:3 respec- +tively for training and testing. +5.3 +Backbones and experimental settings +Backbones: to compare the performances of GFS and graph con- +volution, we select a series of graph convolution based backbones +including: +• STGCN [33]: deploys graph convolution and temporal con- +volution for capturing spatial and temporal dependencies, +respectively. +• DCRNN [18]: combines diffusion graph convolution with +recurrent units for multi-step prediction. +• GraphWaveNet [28]: proposes node embeddings for con- +structing adjacency matrices and combines GCN with di- +lated casual convolution for traffic forecasting. +• ASTGCN [14]: is a self-attentive traffic forecasting model, +and captures the dynamics in a flexible manner. +• AGCRN [3]: proposes node-adaptive graph convolution, +generates node-specific parameters according to learnable +node embeddings, and combines it with GRU [8]. +• STFGNN [17]: constructs temporal graphs based on the +similarities between time series of vertices by utilizing DTW +algorithm. The temporal graphs are fused with distance- +based graphs for better modeling spatial dependencies. +• STG-NCDE [9]: extends the concept of neural controlled +differential equations and designs two novel NCDEs for +spatial and temporal processing, respectively. +• ASTGNN [16]: is an upgraded version of ASTGCN by mod- +ifying the attention mechanism in ASTGCN and adding +positional embedding into the model. +Experimental settings: Regarding the experiments of spatiotem- +poral forecasting, to evaluate the effect of GFS module, we replace +the graph convolution component of all backbones with our pro- +posed GFS module and keep all other settings of those backbones +unchanged. The metric of MAE are chosen as the loss, and two +more metrics, RMSE and MAPE, are additionally evolved to com- +prehensively evaluate all models. In case that GFS is employed on +existing models, we strictly follow the training settings of the orig- +inal models for fair comparison, including optimizers, batch size, +maximum epochs, etc. Regarding the learning on extreme large +graphs, a stack of three layer GFSs and a stack of three layer GCNs +are trained on randomly selected training samples with NLLLoss 7. +Notice that all experiments are executed with one E5-2620 v4 @ +2.10GHz CPU and one Nvidia Tesla V100 16GB GPU. +6https://pytorch-geometric.readthedocs.io/en/latest/modules/datasets.html +7https://pytorch.org/docs/stable/generated/torch.nn.NLLLoss.html + +Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective +Table 4: Performance comparison between backbones and their GFS variants. # denotes that GFS is integrated. +Model +PEMSD3 +PEMSD4 +PEMSD7 +PEMSD8 +MAE +RMSE +MAPE(%) +MAE +RMSE +MAPE(%) +MAE +RMSE +MAPE(%) +MAE +RMSE +MAPE(%) +STGCN +17.55 +30.42 +17.34 +21.34 +34.07 +14.18 +25.33 +39.34 +11.21 +18.15 +28.67 +11.61 +STGCN# +16.98 +28.60 +16.01 +20.74 +32.69 +13.22 +24.61 +38.46 +10.49 +17.31 +27.94 +11.26 +DCRNN +17.99 +30.31 +18.34 +21.22 +33.44 +14.17 +25.22 +38.61 +11.82 +16.82 +26.36 +10.92 +DCRNN# +17.21 +28.97 +17.49 +20.37 +32.14 +13.28 +23.99 +37.02 +11.18 +15.93 +25.24 +10.10 +GraphWaveNet +19.12 +32.77 +18.89 +24.89 +39.66 +17.29 +26.39 +41.50 +11.97 +18.28 +30.05 +12.15 +GraphWaveNet# +18.95 +31.85 +18.20 +23.06 +37.25 +16.80 +25.44 +39.81 +11.13 +17.68 +28.33 +11.72 +ASTGCN +17.34 +29.56 +17.21 +22.93 +35.22 +16.56 +24.01 +37.87 +10.73 +18.25 +28.06 +11.64 +ASTGCN# +16.77 +28.91 +16.80 +20.51 +32.84 +14.00 +22.79 +36.53 +10.22 +17.84 +26.38 +11.02 +AGCRN +15.98 +28.25 +15.23 +19.74 +32.16 +13.14 +22.37 +36.55 +9.12 +16.18 +25.65 +10.30 +AGCRN# +15.46 +27.83 +14.90 +19.69 +32.02 +12.87 +22.01 +36.12 +9.03 +16.00 +25.43 +10.17 +STFGNN +16.77 +28.34 +16.30 +20.29 +32.23 +13.33 +23.46 +36.60 +9.21 +16.68 +26.23 +10.72 +STFGNN# +16.33 +27.96 +16.11 +19.97 +31.82 +13.01 +23.10 +36.44 +9.20 +16.29 +25.94 +10.36 +STG-NCDE +15.57 +27.09 +15.06 +19.21 +31.09 +12.76 +20.53 +33.84 +8.80 +15.45 +24.81 +9.92 +STG-NCDE# +15.21 +26.77 +14.74 +19.00 +31.20 +12.46 +20.80 +33.55 +8.82 +15.38 +24.53 +9.78 +ASTGNN +14.80 +24.81 +14.89 +18.93 +31.36 +12.52 +20.03 +33.43 +8.41 +14.94 +25.08 +9.81 +ASTGNN# +14.21 +24.77 +13.74 +18.40 +30.20 +11.46 +19.80 +33.55 +8.92 +14.88 +24.69 +9.38 +5.4 +Experiments on spatiotemporal graph +learning +Main experiments: To evaluate the effectiveness of GFS, we in- +corporate it with those graph convolution backbones by replacing +their embedded graph convolution component. Specifically, we use +those backbones and their GFS variants to predict the urban traffics +during the next hour with the traffics during the previous hour, +and the average result over the next 12 prediction steps is shown +in Table 4. Notice that # denotes that GFS is integrated with cor- +responding backbones. The results of each individual backbone +and its corresponding GFS variant are grouped by double line for +clearer comparison. Regarding each group, the better one is marked +in bold, and the best performance over all models is highlighted +with underlines. As demonstrated in Table 4, in most cases, GFS +based variants outperform the corresponding backbones, and this +indicates the superiority of our proposed GFS module in capturing +spatial correlations. And the performances of the GFS variant based +on ASTGNN are the best in most experiments, and utilizing GFS +in STGCN, DCRNN, GraphWaveNet, ASTGCN, AGCRN, STFGNN, +STG-NCDE, ASTGNN can respectively gain the improvements on +all metrics by {4.35%, 4.92%, 4.24%, 5.63%, 1.39%, 1.61%, 0.92%, 2.33%} +in average, and this indicates that our proposed GFS module is ap- +plicable to all existing graph convolution based spatiotemporal +learning works. On the other hand, regarding all backbones, re- +placing their integrated graph convolution with GFS can gain the +performance improvements by {2.92%, 2.63%, 3.95%} respectively on +MAE, RMSE and MAPE. In summary, the results on spatiotemporal +graph learning undoubtedly verify the effectiveness of GFS in spa- +tial learning. For more fine-grained analysis, we further compare +the step-wise performances of AGCRN, STFGNN, ASTGNN and +their corresponding GFS variants for each individual prediction +step on PeMSD4 and PeMSD8, and results are shown in Figure 6. +The horizontal axis corresponds to different time steps and the +vertical axis corresponds to the performances in different metrics +and with different datasets. First, as observed, the performances +of the GFS based variants are better than the performances of the +corresponding backbones at almost all time steps, this verifies the +superiority of GFS in capturing spatial correlations. Second, we +discover that the performances of the GFS based variants decrease +slower than the performances of the corresponding backbones with +the increasing of time steps, this indicates that GFS module can +significantly improve traditional spatiotemporal learning on multi- +step predictions. +Time consumption: As analyzed, the time complexity of GFS is +significantly better than that of graph convolution. In this part, +we compare the experiment time consumption of GFS with that +of graph convolution. Similarly, for each backbone, GFS is used to +replace their graph convolution component, all models are trained +and tested on PEMSD4 and PEMSD8, and the results are listed in +Table 7. As shown, for all models, utilizing GFS can significantly +reduce the time consumptions on both training and testing by 20% +averagely. Considering that different backbones have diverse archi- +tectures, and such divergence may interfere with the experiments +to a certain extent. For the sake of fairness, we construct a series of +generated datasets with different numbers of vertices ranging from +100 to 2000 to further investigate the time consumption issue. Each +dataset contains 1000 batches and each batch contains 64 samples. +Based on these generated datasets, we test the time consumptions +of single-layer GFS and single-layer GCN (using the definition in +Equation 11), respectively. The results are shown in Figure 7a. As +illustrated, the difference between the time consumptions of these + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +16 +17 +18 +19 +20 +21 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +MAE on PEMSD4 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(a) +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +RMSE on PEMSD4 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(b) +10 +10.5 +11 +11.5 +12 +12.5 +13 +13.5 +14 +14.5 +15 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +MAPE on PEMSD4 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(c) +12 +13 +14 +15 +16 +17 +18 +19 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +MAE on PEMSD8 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(d) +19 +21 +23 +25 +27 +29 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +RMSE on PEMSD8 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(e) +8 +8.5 +9 +9.5 +10 +10.5 +11 +11.5 +12 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +MAPE on PEMSD8 +AGCRN +AGCRN# +STFGNN +STFGNN# +ASTGNN +ASTGNN# +(f) +Figure 6: Step-wise performances of AGCRN, STFGNN, ASTGNN and their corresponding GFS variants on PeMSD4 and PeMSD8. +Table 5: Time consumption comparison between backbones +and their GFS based variants on PEMSD4 and PEMSD8 +Model +PEMSD4 +PEMSD8 +Test +Train +Test +Train +STGCN +2.89 +22.38 +1.62 +13.48 +STGCN# +2.76 +18.05 +1.57 +10.48 +DCRNN +3.64 +25.39 +2.53 +23.94 +DCRNN# +1.74 +14.71 +1.32 +21.17 +GraphWaveNet +1.70 +28.31 +1.66 +23.57 +GraphWaveNet# +1.65 +24.96 +1.56 +19.04 +ASTGCN +3.98 +21.69 +3.15 +18.39 +ASTGCN# +1.87 +7.56 +1.97 +12.33 +AGCRN +2.79 +20.51 +1.99 +19.07 +AGCRN# +1.17 +13.79 +1.14 +14.90 +STFGNN +9.56 +103.79 +7.82 +88.62 +STFGNN# +8.90 +90.12 +6.04 +77.91 +STG-NCDE +16.54 +167.58 +9.91 +103.92 +STG-NCDE# +13.90 +142.69 +9.12 +95.67 +ASTGNN +100.23 +239.21 +32.14 +70.47 +ASTGNN# +89.94 +217.36 +30.78 +66.89 +two networks is relatively small while the number of vertices is +small. However, since the time complexity of GCN is quadratic to +the number of vertices, the time consumption of GCN increases +much faster than that of GFS with the increasing of vertex number. +Furthermore, We also test the time consumptions of AGCRN and +0 +5 +10 +15 +20 +25 +30 +35 +100 +300 +500 +700 +900 +1100 +1300 +1500 +1700 +1900 +time consumption +number of vertices +GCN +GFS +(a) Single-layer GFS and single-layer +GCN +0 +100 +200 +300 +400 +500 +600 +700 +800 +100 +300 +500 +700 +900 +1100 +1300 +1500 +1700 +1900 +time consumption +number of vertices +AGCRN +AGCRN# +(b) AGCRN and its GFS variants +Figure 7: Time consumptions of Single-layer GFS and single- +layer GCN with different numbers of vertices. +its GFS based variant on the constructed datasets, and the results +are shown in Figure 7b. As demonstrated, the time complexity of +AGCRN# scales linearly with the increasing of vertex number, while +the time complexity of original AGCRN increases quadratically with +the increasing of vertex number. The experimental results on the +generated dataset validate the temporal efficiency of GFS. +Detailed Performances of state-of-the-art solutions on sin- +gle vertex: Regarding state-of-the-art solutions, ASTGNN and its +corresponding GFS based variant ASTGNN#, we select two random +vertices in one random day from the testing set, and evaluate the +performances of these two state-of-the-art solutions for each indi- +vidual time point during the selected two vertices, and the results +are shown in Figure 8. Note that the data of vertices from about +17:30 to 20:30 is missed. As shown, both two models can achieve +satisfying accuracies all the time. However, regarding some extreme +scenarios including peak values and wild fluctuations, as has been +highlighted with amplification rectangles, the performance curves + +Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective +-20 +30 +80 +130 +180 +230 +280 +330 +00:00 +03:20 +06:40 +10:00 +13:20 +16:40 +20:00 +23:20 +traffic flow +time +Truth +ASTGNN# +ASTGNN +(a) Vertex 1 +0 +50 +100 +150 +200 +250 +300 +00:00 +03:20 +06:40 +10:00 +13:20 +16:40 +20:00 +23:20 +traffic flow +time +Truth +ASTGNN# +ASTGNN +(b) Vertex 2 +Figure 8: Detailed performances of state-of-the-art solutions +at different vertices on PEMSD4. +of the GFS based variant can approximate the curves of truths more +accurately, and this further illustrates the effectiveness of GFS. +5.5 +Node classification on extreme large graph +Main experiments: In this subsection, we investigate the perfor- +mance of GFS in processing extreme large graph, and a very simple +three-layer stacked architecture is proposed in the experiments for +both GFS and GCN. For training these two stacked networks, we +randomly select training samples, train for 200 epochs, and test +for 1000 rounds. The average classification accuracies are reported +in Table 6. As observed, compared with GCN, utilizing GFS can +improve the classification accuracy by 2.15% and 0.7% respectively +on PubMed and Coauthor Physics. The results demonstrate the +generalizability of GFS on general graph learning task and the scal- +ability of GFS on large graph with massive vertices. Further, we also +investigate the training accuracy of each epoch of GFS and GCN on +two datasets, and the results are show in Figure 9. As can be easily +observed, both two modules have similar convergence speeds and +achieve satisfying fitting on Coauthor Physics. However, regarding +PubMed, GFS is able to fit training samples better and thus has +better representation capability than GCN. Such results witness the +capability of GFS on modeling spatial correlations. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1 +20 +39 +58 +77 +96 +115 +134 +153 +172 +191 +Acc(%) +Epoch +GFS +GCN +(a) PubMed +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1 +20 +39 +58 +77 +96 +115 +134 +153 +172 +191 +Acc(%) +Epoch +GFS +GCN +(b) Coauthor Physics +Figure 9: Training accuracy of each epoch of GFS and GCN. +Time comsuption: Regarding the previous node classification ex- +periment, for each individual approach and dataset, we also record +its time comsumption, and the results are also reported in Table 6. +As shown, compared with GCN, GFS can reduce the time consump- +tion by 1.64× and 9.6× respectively on on PubMed and Coauthor +Physics. Compared with the previous spatiotemporal graph learn- +ing experiments on PEMSD3, PEMSD4, PEMSD7 and PEMSD8, our +approach has more advantages over GCN in terms of time consump- +tion with large graphs. The larger the graph is, the more advantage +our GFS has in terms of time consumption, and this reflects that +the great prospects of GFS in processing extreme large graph. +Table 6: Accuracy and time consumption of GFS and GCN. +Dataset +Model +Acc(%) +Total time +PubMed +GFS +0.8667343 +26.429018 +GCN +0.8452333 +43.343687 +Coauthor Physics +GFS +0.9670854 +72.546249 +GCN +0.9593931 +698.05498 +5.6 +Investigation on graph-free architecture +In this section, to further verify the effectiveness of each individual +component of the graph-free architecture, we carry out a series of +ablative studies on PeMSD4, and the variants of GFS include: +• Mean: to evaluate the effect of layer normalization, we +design this variant by replacing the layer normalization +module of GFS with mean function, which is the equal to +combine residual connection with a graph convolution with +adjacency matrix as matrix 𝔑 defined in Equation10. +• MeanP: to further evaluate the performance gap between +mean function and normalization operation, we design this +variant by replacing the layer normalization of GFS with +mean function but retains the affine parameters of layer +normalization. +• NoLNP: to evaluate the effect of normalization operation +itself, we design this variant by directly removing the layer +normalization in GFS but keep the affine parameters of +layer normalization in Equation 6, which equals to apply +affine parameters directly on a graph convolution layer + +Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ +with adjacency matrix I. As affine parameters can be applied +individually, we retain them in this variant. +• NoRes: to evaluate the effect of residual connection, the +residual connection in GFS is removed in this variant. +• LNNoP: to evaluate the effect of affine parameters, we +design this variant by removing the affine parameters of +layer normalization. +12 +17 +22 +27 +32 +37 +MAE +RMSE +MAPE +Mean +MeanP +NoLNP +NoRes +LNNoP +Origin +(a) GFS variants with STGCN +12 +17 +22 +27 +32 +37 +MAE +RMSE +MAPE +Mean +MeanP +NoLNP +NoRes +LNNoP +Origin +(b) GFS variants with STFGNN +12 +17 +22 +27 +32 +37 +MAE +RMSE +MAPE +Mean +MeanP +NoLNP +NoRes +LNNoP +Origin +(c) GFS variants with AGCRN +10 +15 +20 +25 +30 +35 +40 +MAE +RMSE +MAPE +Mean +MeanP +NoLNP +NoRes +LNNoP +Origin +(d) GFS variants with ASTGNN +Figure 10: Performances of different GFS variants with +STGCN, STFGNN, AGCRN and ASTGNN on PEMSD4. +To comprehensively evaluate the impacts of different components +of GFS, we conduct a series of ablative experiments on PeMSD4 by +incorporating all variants with the four representative approaches, +i.e., STGCN, STFGNN, AGCRN, and ASTGNN, and the results are +illustrated in Figure 10. As can be easily observed, GFS itself out- +performs all variants with all backbones, and this first indicates +that all components in GFS is effective to the final performances +of GFS. And the variant of MeanP outperforms all other variants +with all backbones except STFGNN, and this cross-verifies two +points: i) the aggregation itself is more important than the way of +aggregation, and ii) layer normalization is more effective than mean +function on aggregating vertices. And the performances of NoLNP +also indicates the first point which is consistent to the conclusion +that we have obtained in Section 3. Comparing the performances +of two solution pairs, MeanP vs. Mean and GFS vs. LNNoP, we +discover that, no matter mean or layer normalization is used for +aggregating vertices, affine parameters are vital and essential for +GFS. Furthermore, based the performances of NoLNP and LNNoP, it +is obvious that, the normalization operation itself, which introduces +the aggregation of vertices, is more important than the mechanism +of affine parameters, even though affine parameters can change +the way that layer normalization aggregates vertices. This triple +verifies that the aggregation itself is more important than the way +of aggregation. And finally, the performance comparison between +NoRes and GFS also verifies the importance and necessity of the +residual connection component. +6 +CONCLUSION AND DISCUSSION +Conclusion: In this paper, with extensive and deep-going experi- +ments, we comprehensively analyze existing spatiotemporal graph +learning models and reveal that extracting adjacency matrices with +carefully design strategies, which are viewed as the key of en- +hancing performance on graph learning, are largely ineffective. +Meanwhile, based on these experiments, we also discover that the +aggregation itself is more important than the way that how ver- +tices are aggregated. With these preliminary, a novel efficient GFS +learning module based on layer normalization for capturing spa- +tial correlations in spatiotemporal graph learning. The proposed +GFS module can be easily plugged into existing models for replac- +ing all graph convolution components. Rigorous theoretical proof +demonstrates that the time complexity of GFS is significantly better +than that of graph convolution operation. Extensive experiments +verify the superiority of GFS in both the perspectives of efficiency +and learning effect in processing graph-structured data especially +extreme large scale graph data. +Discussion: In future works, there are some more interesting issues +can be further discussed, +• The effectiveness of GFS further indicates that spending too +much efforts on extracting adjacency matrix is the wrong +region to spatiotemporal learning, and the reason why adja- +cency matrix is almost useless needs to be further explored. +And Instead of relying on designing new adjacency matrix +and incorporating it with graph convolution, how to effec- +tively capture spatial correlations from spatiotemporal data +need to further thought. +• Even though GFS has achieved promising performances on +spatiotemporal graph learning, its performance is largely +owed to the affine parameters of layer normalization. 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In IJCAI. 3732–3738. + diff --git a/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/load_file.txt b/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f35267755c425b97e3f4e5e417acd132776224c1 --- /dev/null +++ b/tNFKT4oBgHgl3EQfKi0w/content/tmp_files/load_file.txt @@ -0,0 +1,1358 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf,len=1357 +page_content='Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ 1University of Science and Technology of China, Hefei, China 2Shanghai AI Laboratory, Shanghai, China {wx309,gpf9061,pengkun,wbw1995,zzy0929}@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='cn baisanshi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='com,angyan@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='cn* ABSTRACT Spatiotemporal learning, which aims at extracting spatiotemporal correlations from the collected spatiotemporal data, is a research hotspot in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And considering the inherent graph struc- ture of spatiotemporal data, recent works focus on capturing spatial dependencies by utilizing Graph Convolutional Networks (GCNs) to aggregate vertex features with the guidance of adjacency matri- ces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In this paper, with extensive and deep-going experiments, we comprehensively analyze existing spatiotemporal graph learning models and reveal that extracting adjacency matrices with carefully design strategies, which are viewed as the key of enhancing perfor- mance on graph learning, are largely ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Meanwhile, based on these experiments, we also discover that the aggregation itself is more important than the way that how vertices are aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' With these preliminary, a novel efficient Graph-Free Spatial (GFS) learning module based on layer normalization for capturing spa- tial correlations in spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The proposed GFS module can be easily plugged into existing models for replac- ing all graph convolution components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Rigorous theoretical proof demonstrates that the time complexity of GFS is significantly better than that of graph convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Extensive experiments verify the superiority of GFS in both the perspectives of efficiency and learning effect in processing graph-structured data especially extreme large scale graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 1 INTRODUCTION In recent years, massive amount of spatiotemporal data have been collected in various fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', urban computing, meteorology, and atmosphere quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Such collected spatiotemporal data is a set of correlated time series where each individual sequence is about the temporal variation of the monitored information at a specific physi- cal location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And Spatiotemporal learning, which aims at extracting spatiotemporal correlations from the collected spatiotemporal data, has attracted more and more attentions [2, 4, 25, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Early efforts in spatiotemporal learning mostly focus on extract- ing both spatial and temporal correlations respectively with Convo- lution Neural Networks (CNNs) [31, 36, 41] and Recurrent Neural Networks (RNNs) [21, 24, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, Such CNN based methods, which divide the whole space into grids to extract spatial correla- tions, has never considered the irregular spatial distributions of spa- tiotemporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, this will definitely lead to the inevitable missing of topology information and non-Euclidean correlations in spatial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To tackle the above-mentioned issues of CNN based methods, recent spatiotemporal graph learning works [18, 22, 33] Yang Wang and Lei Bai are the corresponding authors with equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' tend to model spatiotemporal data into graph structure by mod- elling spatial points as vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' They first employ various strategies to extract adjacency matrices for comprehensively and precisely modeling the spatial correlations among vertices, and then use graph convolution to aggregate the features of vertices based on the extracted adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Existing works on spatiotemporal graph learning can be divided into two categories, predefined adjacency matrix based methods and learnable adjacency matrix based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding the first category, they incorporate predefined distance based [18, 33, 39] or temporal similarity based [11, 17, 20] adjacency matrices with graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, such methods assume that vertices, which are in closer distances or with similar temporal series, are highly correlated, and the adjacency matrices are predefined before training and keep fixed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This determines that these methods cannot effectively extract the time-varying correlations among vertices due to their invariable adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Consid- ering the lacking of the representation ability of these predefined adjacency matrix based methods, the second category focus on extracting adjacency matrices in a learnable manner during train- ing [3, 27, 28] or throughout [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The previous methods can effectively enhance their representation abilities in spatial perspec- tive with their learnable adjacency matrices, and the learning of adjacency matrices are only within the training period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The latter methods can even represent dynamic spatial dependencies into dynamic adjacency matrices with their embedded self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Throughout all existing works on spatiotemporal graph learning, we discover that how to effectively extract spatial correlations is one of the core concerns of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, a very simple intu- ition is that the performances of existing works improve with the increasing of the complexities of models, and this easily leads all researchers to an unsubstantiated conclusion: The extraction of spa- tial adjacency matrix is critical to the success of such spatiotemporal learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, by comprehensively analyzing the evolution of existing works, we discover that they are usually im- proved in three aspects, i) new approach for extracting spatial adja- cency matrix, ii) new method for extracting temporal dependencies, and iii) new mechanism for more comprehensive fusion between spatial and temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, it is baseless to simply attribute the improvements of these models to the construction of their adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To explicitly verify this deduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' we first classify all existing methods into four disaggregated classifications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' select STGCN [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' AGCRN [3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ASTGNN [16] as the four representative methods respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and carry out a series of experiments by replacing the learned adjacency matrices of the Xu Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Pengfei Gu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Pengkun Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Binwu Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Zhengyang Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Lei Bai2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Yang Wang1∗ four representative approaches respectively with a random matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' a matrix filled with a specific value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and some variant matrices by combining the previous two matrices with an identity matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and the result indicates a first fact: the performances of such approaches can hardly be influenced by replacing their adjacency matrices with generated ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and this indicates that using different complex strate- gies to enhance the extraction of adjacency matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' which is both resource-consuming and time-consuming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' seem to be unnecessary and cancelable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Meanwhile, this observation brings another question, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', is graph convolution based data aggregation still useful in en- hancing the final performance of spatiotemporal graph learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To answer this question, we design another series of experiments to verify the impacts of graph convolution based data aggregation by replacing the adjacency matrices of those approaches with an iden- tity matrix, and the result indicates a second fact: graph convolution based data aggregation appears useful in enhancing spatiotemporal graph learning for most existing approaches except STGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To fig- ure out whether graph convolution is useful or not in STGCN, we detailedly analyze the architecture of STGCN and discover that it has a unique module, layer normalization, which doesn’t exist in the other three representative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Intuitively, employ layer normalization in vertex dimension leads to aggregations of vertices due to its integrated mean and covariance calculation operations, therefore, we conduct an additional experiment by removing the layer normalization operation from vertex dimension, and the result indicate a third fact: aggregating vertices in spatial perspective is essential and important for graph learning, and layer normalization is also be of the functionality of data aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' So far, in summary, we can obtain a simple conclusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', the aggregation of neigh- boring vertices is crucial and effective for spatiotemporal graph learning, but the way of adjacency matrix based graph convolution aggregating vertices has little additional effect on spatiotemporal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Based on such finding, we think that existing sophisticated adja- cency matrix and graph convolution based approaches are improv- able, and then propose a novel efficient Graph-Free Spatial (GFS) learning module based on layer normalization for capturing spa- tial correlations in spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, we first theoretically prove that the operation of layer normalization in vertex dimension, which can also be transferred to a graph- convolution-like form, is equivalent to a series of complex matrix multiplications which are the core operation of data aggregation in the operation of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Next, we propose a novel GFS learning module which consists of only a linear projection and ReLU activation function combined component and a layer normalization module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Rigorous theoretical proof demonstrates that the time complexity of GFS is significantly better than that of graph convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The proposed GFS module can be easily plugged into existing models for replacing all graph convolu- tion components, as far as the dimensions are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Extensive experiments verify the superiority of GFS in both the perspectives of efficiency and learning effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The contributions of this paper can be summarized as follows, To the best of our knowledge, this paper first reveals the fact that adjacency matrix plays unimportant role in learning spatial correlations from graph structured data, and also for the fist time, this paper confirms that aggregating vertices in spatial perspective is essential and important for graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We cross-verify that layer normalization is effective in ag- gregating data among vertices in both theoretical and ex- perimental perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Based on this, we design a novel efficient GFS learning module based on layer normaliza- tion for capturing spatial correlations in spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The proposed GFS module can be plugged into existing spatiotemporal graph learning models as an alternative to graph convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We conduct extensive experiments on several widely-used real-world graph-structured spatiotemporal datasets as well as two very large-scale graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To evaluate the effectiveness and efficiency of GFS module, we select a series of baselines and use GFS to replace their embedded graph convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Experimental result shows that GFS module is superior to traditional graph convolution in terms of both efficiency and learning effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In Section 2, we analyze and conclude related works, and then re-investigate existing spatiotem- poral graph learning works with extensive experiments in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Based on all learned facts, in Section 4, we introduce the design and implementation of GFS learning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Section 5 describes the experiments for evaluating our propose module and Section 6 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 2 RELATED WORKS Spatiotemporal learning has attracted extensive research atten- tions [12, 19, 22, 26, 37] in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Early works [15, 24, 31, 32, 34–36, 41] mainly focus on employing CNN based networks and RNN based networks to respectively extract both spatial and tem- poral correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Nevertheless, in spatial perspective, such CNN- based spatiotemporal learning methods mesh road network into regular grids and employ convolutional neural networks to capture spatial dependencies among grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And those grid-partition based methods are incapable of capturing the inherent Non-Euclidean structured characteristics, hence are short in fully capture spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To this end, recent works, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', spatiotemporal graph learning, aim at modeling such these Non-Euclidean characteristics with graph structure [7, 13, 18, 38–40], and existing efforts on this branch can be roughly divided into two categories, predefined adjacency matrix based methods and learnable adjacency matrix based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding the first category, in spatial perspective, they incor- porate predefined distance based [18, 33, 39] or temporal simi- larity based [11, 17, 20] adjacency matrices with graph convolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, STGCN [33] applies graph convolution on fixed distance-based adjacency matrix, and convolutional network for modeling temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' DCRNN [18] captures spatial dependencies with bidirectional random walks on graph, and cap- tures temporal dependencies with a encoder-decoder framework and scheduled sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' T-GCN [39] combines GCN with GRU to exploit the spatiotemporal correlations of urban traffics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' On the other hand, STAG-GCN [20] applies classic DTW algorithm [5] to construct adjacency matrices based on the similarities of vertices Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective temporal trends and combines temporal aware adjacency matrices with distance based adjacency matrices to enhance the modeling of spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN [17] modifies DTW to address time efficiency concerns and involves temporal connections in the construction of adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, such methods as- sume that vertices, which are in closer distances or with similar temporal series, are highly correlated, and the adjacency matri- ces are predefined before training and keep fixed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This determines that these methods cannot effectively extract the time-varying correlations among vertices due to their invariable adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Given the fact that all above-mentioned predefined adjacency matrix based methods are poor at representing the time-varying correlations, recent works in spatiotemporal graph learning mostly construct adjacency matrices in a learnable manner during train- ing [3, 28] or throughout [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In particular, GraphWaveNet [28] maintains learnable embeddings for each vertex, and constructs ad- jacency matrices by calculating the embedded similarities between vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' AGCRN [3] follows the idea of GraphWaveNet and further utilizes the embedded similarities between vertices to modify graph convolution for learning node-specific patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Moreover, AST- GCN [14] utilizes attention mechanism in both spatial and temporal dimension to capture the dynamic spatiotemporal correlations at different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' By multiplying the attention values with distance based adjacency matrices, ASTGCN actually gets dynamic adjacency matrix changing at different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ASTGNN [16] is an upgraded version of ASTGCN, which modifies the attention mechanism in ASTGCN and involves short-term temporal trends when calculating attention values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As analyzed, existing works in spatiotemporal graph learning have devoted themselves into enhancing the effect of graph con- volution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' by generating various adjacency matrices with different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Indeed, with the increasing of the complexity of adja- cency matrix construction, the performances of such spatiotemporal graph learning methods increase correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This gives us an illusion that the performance enhancement mainly due to the improvement in adjacency matrix construction, and this is why researchers never get tired of designing new strategy to improve the adjacency matrix of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, by compre- hensively analyzing the evolution of existing works, we discover that they are usually improved in three aspects, i) new approach for extracting spatial adjacency matrix, ii) new method for extracting temporal dependencies, and iii) new mechanism for more compre- hensive fusion between spatial and temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' So, there naturally raises a question: how much can the modification of ad- jacency matrix helps on the final performance of spatiotemporal graph learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3 RE-INVESTIGATION ON EXISTING SPATIOTEMPORAL GRAPH LEARNING To answer the above-proposed question, in this section, we first select some representative spatiotemporal graph learning methods based on the above analysis on spatiotemporal graph learning, and then design a series of experiment to comprehensively analyze them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Output Input Strategy for Defining Adjacency Matrices Predefined Simple Matrices Stacked Spatial-Temporal Block Graph Convolutional Module Temporal Modeling Module Adjacency Matrices 𝑴 𝑹 𝑰 + 𝑴 𝑰 + 𝑹 Figure 1: Illustration of replacing adjacency matrices of ex- isting works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1 Selection of representative spatiotemporal graph learning methods In previous section, existing spatiotemporal graph learning works have been roughly divided into two major categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here, to specif- ically distinguish and evaluate the impacts of adjacency matrix, we further classify all existing works into four subclasses: Methods with distance based predefined adjacency matrix: The adjacency between two vertices is determined based on the physical distance between them, and we here select STGCN [33] as the representative method of this subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Methods with temporal similarity based predefined adjacency matrix: The adjacency between two vertices is determined based on the similarity of the temporal pat- terns of these two vertices, and select STFGNN [17] as the representitive method of this subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Methods with adjacency matrix learned during train- ing: The adjacency matrix is determined based on the simi- larities of embeddings which are learned from vertices dur- ing training period, such adjacency matrices are expected to model latent correlations among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We here select AGCRN [3] as the representative model of this subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Methods with dynamic adjacency matrix: The adja- cency matrix, which is constructed dynamically based on attention or some other mechanisms, are data-driven gen- erated and changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 Re-investigation on the impacts of adjacency matrix Regarding the selected four representative works, to re-investigate the impacts of adjacency matrix on their performances, we con- duct a series of experiments by replacing their adjacency matrices with some artificially designed adjacency matrices, and the de- tailed operation of replacing adjacency matrices of existing works is demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The artificially generated adjacency matrices are, R: The matrix filled with random values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Using R as adja- cency matrix results in randomly aggregating of all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here R is generated by R = softmax(R′) (1) Xu Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Pengfei Gu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Pengkun Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Binwu Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Zhengyang Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Lei Bai2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Yang Wang1∗ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='I+M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(f) STFGNN PEMSD8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='16 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='I+R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='I+M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(g) AGCRN PEMSD8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='I+R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='I+M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(h) ASTGNN PEMSD8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Figure 2: Performances of four representative methods with generated adjacency matrices on PEMSD4 and PEMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' where R′ = �𝑟𝑖𝑗 � 𝑁 ×𝑁 (2) Notice that R′ 𝑖𝑗 ∼ N (0, 1) which indicates that all elements of R′ are sampled from standard Gaussian distribution, and 𝑁 corresponds to the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' M: Matrix filled with a specific value 1/𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Using M as adja- cency matrix results in equally aggregating of all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' I + R: Add self-loop on matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here I corresponds an identity matrix which is filled with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' I + M: Add self-loop on matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here I corresponds an identity matrix which is filled with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice that all these four alternative matrices are all 𝑁 × 𝑁 dimen- sional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' By comparing the original performances of the four representative methods and their performances in case that their adjacency matrices are respectively replaced with R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' I + R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and I + M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' we can investigate the impacts of different adjacency ma- trices on them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and the detailed implementation how the adjacency matrices are replaced are listed as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STGCN: We utilizes the publicly available implementation of STGCN 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and simply replace the adjacency matrix cal- culated by STGCN with the generated matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN: The official implementation of STFGNN is uti- lized 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN proposes a spatial-temporal fusion graph which is constructed by combining both distance based and temporal similarity based adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here both two matrices are replaced with our proposed matrices and the construction procedure of spatial-temporal fusion graph is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' AGCRN: The adjacency matrix of AGCRN is determined based on the similarity of vertex embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We sidestep 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='com/hazdzz/STGCN 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='com/MengzhangLI/STFGNN the calculation of similarity and directly utilize the gener- ated matrices as the adjacency matrix of AGCRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' All other settings of AGCRN are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The official implementa- tion 3 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ASTGNN: A modified self-attention module is proposed in ASTGNN, which generates attention weights among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The final adjacency matrix of ASTGNN is the dot- product of attention weights and the distance-based ad- jacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, we replace the final adjacency matrix of ASTGNN with our generated matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Similarly, the official implementation of ASTGNN is used 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The used datasets, metrics, and task settings are as follows, Datasets: All the experiments are conducted on PEMSD4 and PEMSD8 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' More details about the datasets and data preprocessing will be introduced in Section "EXPERIMENTS".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Task: The task is to predict the spatiotemporal data over the next 12 time steps based on the data over the past 12 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Metrics: Three widely used evaluation metrics are em- ployed to measure the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We here detail the definitions of all the three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Let V ∈ R𝑇×𝑁 ×1 denote the ground truth future data of all 𝑁 vertices during 𝑇 time steps and ˆV ∈ R𝑇×𝑁×1 denote the predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The metrics can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='com/LeiBAI/AGCRN 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='com/guoshnBJTU/ASTGNN Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective MAE(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ˆV) = 1 𝑇𝑁 𝑇 ∑︁ 𝑖=1 𝑁 ∑︁ 𝑗=1 |V𝑖𝑗 − ˆV𝑖𝑗 | (3) RMSE(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ˆV) = � � � � 1 𝑇𝑁 𝑇 ∑︁ 𝑖=1 𝑁 ∑︁ 𝑗=1 (V𝑖𝑗 − ˆV𝑖𝑗)2 (4) MAPE(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ˆV) = 1 𝑇𝑁 𝑇 ∑︁ 𝑖=1 𝑁 ∑︁ 𝑗=1 | V𝑖𝑗 − ˆV𝑖𝑗 V𝑖𝑗 | (5) By replacing the adjacency matrices of the four representative meth- ods with the generated matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' I + R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and I + M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' we can evaluate the effects and validity of the extracted adjacency matrices of the four representative methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and the results are demon- strated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As demonstrated, while the extracted adjacency matrices of these four methods are replaced by R, M, I + R, and I + M, respectively, the performances of them barely changed with two different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, while the extracted adjacency matrices are replaced by R, M, I + R, and I + M, on PEMSD4, the performances of STGCN, STFGNN, AGCRN, ASTGNN change by {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='41%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='06%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14%} at worst and {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='54%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='82%} in average in terms of MAE, RMSE and MAPE respectively, and change respec- tively by {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='65%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='87%} at worst and {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='50%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='63%} in average on PEMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This reveals a very important fact: the effect of extracting an adjacency matrix with carefully designed strategy in spatiotemporal graph learning is limited and even non-existent, and the complex and time-consuming strategies for extracting and constructing adjacency matrices within those existing spatiotempo- ral learning efforts seem to be unnecessary and cancelable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' On the other hand, considering the effect of enhancing adjacency matrix is limited, is graph convolution based data aggregation still useful in enhancing the final performance of spatiotemporal graph learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='3 Investigation on the impacts of graph convolution based data aggregation To answer the question proposed in the previous subsection, in this subsection, we design another series of experiments to evaluate the impacts of graph convolution based data aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, regarding the four representative spatiotemporal graph learning methods, we use the previous defined identity matrix I to make sure that the feature of each vertex is isolated and will never been aggregated with the feature of any other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The datasets, task, metrics and all other basic settings in this series of experiments are the same to those in the experiments in the previous subsection, and the results are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice here the column with Origin corresponds the performances of the four models with their original adjacency matrices and the column with I means the performances of them while their adjacency matrices were replaced with I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As reported in this table, once the adjacency matrices were replaced with I, on both PEMSD4 and PEMSD8, the performances of STFGNN, AGCRN and ASTGNN decrease significantly in terms of all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, when the adjacency matrices are replaced with I, on PEMSD4, the performances of STFGNN, AGCRN and ASTGNN decrease by {17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='25%, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32%, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='48%}, {18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='95%, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23%, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80%} and {19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='07%, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='06%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='61%} in terms of MAE, RMSE and MAPE, and {9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='11%, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='64%, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='09%}, {16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='93%, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10%, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='28%} and Spatial Graph Conv Temporal Block Temporal Block LayerNorm across Vertex and Feature dimension Stacked ST-Conv Block Output Input time Figure 3: Architecture of STGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' {12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='52%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='44%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='42%} on PEMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This can easily lead us to the conclusion that graph convolution based data aggregation are still useful in enhancing spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, on the other hand, we discover that the performances of STGCN are barely change whether its adjacency matrix is replaced or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This seems counter the earlier conclusion that we just obtained from the experiments on STFGNN, AGCRN and ASTGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, here a big confusion comes naturally, whether graph convolution based data aggregation is effective or not in enhancing spatiotemporal learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Table 1: Performances of four representative methods with I on PEMSD4 and PEMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Datasets Metrics STGCN STFGNN AGCRN ASTGNN Origin I Origin I Origin I Origin I PEMSD4 MAE 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='67 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='29 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='74 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='48 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='93 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='54 RMSE 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='07 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='47 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='16 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='38 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='36 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='71 MAPE(%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='05 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='33 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='66 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='61 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='52 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10 PEMSD8 MAE 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='15 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='68 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='94 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='81 RMSE 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='67 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='52 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='02 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='65 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='78 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 MAPE(%) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='61 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='72 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='48 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='81 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4 Further investigation on STGCN In the previous subsection, we discover an interesting issue that graph convolution based data aggregation seems essential in STFGNN, AGCRN and ASTGNN, while it is almost useless in STGCN, and this greatly encourages us to further investigate STGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As illus- trated in Figure 3, we discover that STGCN includes an additional layer normalization [1] module which doesn’t exist in the other three representative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, STGCN applies layer normalization on both vertices dimension and feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Given input of all the features of all vertices at a specific time point, V ∈ R𝑁×𝐷, the layer normalization can be briefly formulated as, LN(V) = V − E[V] √︁ Var[V] � W + B (6) W, B ∈ R𝑁 ×𝐷 are learnable affine parameters and � is element- wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' E[V] and Var[V] are the mean and covariance of V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Due to the operations of computing mean and covariance, applying layer normalization on vertex dimension im- plicitly leads to aggregation of all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, this module Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ is also of the functionality of data aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' So far, we conject that whether the data aggregation among vertices is what really works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To clarify such conjecture, we design an additional exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here we directly modify STGCN by only employing layer normalization on feature dimension and removing the implicit ag- gregation of vertices, and the result are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As Table 2: Impacts of data aggregation operation in STGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Datasets Metrics STGCN STGCN* Origin I Origin I PEMSD4 MAE 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='67 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='02 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='98 RMSE 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='07 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='47 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 MAPE(%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='05 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='53 PEMSD8 MAE 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='15 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='58 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='30 RMSE 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='67 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='52 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='89 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='78 MAPE(%) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='61 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='55 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='29 shown in this table, in case that the data aggregation operation is removed from STGCN, the performances of STGCN* are signifi- cantly worse than those of STGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In particular, in case that the STGCN uses its original extracted adjacency matrix, comparing with the performances of STGCN, the performances of STGCN* decreases by {50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='89%, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='68%, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='41%} on PEMSD4 in terms of MAE, RMSE and MAPE, and decrease by {29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='18%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32%} respec- tively on PEMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And while the adjacency matrix is replaced with I, the performances of STGCN* further decrease by {19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='21%, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12%, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='75%} on PEMSD4 and {28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='50%, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='77%, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='58%} on PEMSD8 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In summary, these experiments explicitly and undoubtedly verify the effectiveness and importance of aggregating vertices in spa- tial learning, and simultaneously indicates that layer normalization is also be of the functionality of data aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 Lessons learned in re-investigating existing spatiotemporal graph learning methods In this subsection, we will briefly summarize the lessons learned in re-investigating existing spatiotemporal graph learning with three series of carefully designed experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' LL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The core idea of existing spatiotemporal graph learn- ing methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', using different complex strategies to en- hance the extraction of adjacency matrices, which is both resource-consuming and time-consuming, is noneffective to the performance of spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' LL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Data aggregation among vertices, which can effec- tively capture spatial correlations among vertices, is the key to the success of spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Even with a matrix with a fixed value or random values, tradi- tional graph convolution is still an effective way to achieve data aggregation, however it time complexity is also an issue that should be concerned about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' LL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Layer normalization, which is also be of the function- ality of data aggregation, can also significantly enhance the performance of spatiotemporal graph learning by ef- fectively extracting the spatial correlations among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And compared with traditional graph convolution, the time complexity of layer normalization has obvious advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 4 GRAPH-FREE SPATIAL MODULE IN SPATIOTEMPORAL GRAPH LEARNING In this section, based on the previous learned lessons, we analyze the possible technical solution of spatiotemporal graph learning, and propose a graph-free spatial learning module as an alternative to graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1 Rethinking of spatial learning in spatiotemporal graph learning Based on the learned lessons LL1 and LL2, we discover that graph convolution based spatial learning is highly inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' the strategies for defining better adjacency matrices contribute little to the final effect of spatiotemporal graph learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and though,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' by replacing the learned adjacency matrix with the fore-mentioned matrix M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' the efficiency of traditional graph convolution can signif- icantly improved on the premise of without significantly losing the performance of spatiotemporal graph learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' the computation complexity of graph convolution itself is also a serious issue that should be concerned about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, regarding spatial learning in spatiotemporal graph learning, we should pay more attentions to aggregating vertices in a more efficient manner rather than seeking strategies for defining adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' On the other hand, the learned lesson LL3 indicates that layer normalization across vertex dimension is also valid to data aggre- gation, however, in theory, whether or why layer normalization has the additional functionality of data aggregation still needs to be further explored and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 Theoretical analysis on the effect of layer normalization in spatial learning As analyzed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4, layer normalization has shown its par- tial effectiveness on capturing spatial dependencies among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Here, we further explore the data aggregation operation in layer normalization and analyze the correlations between layer normal- ization and graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The operation of layer normalization is defined in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To simplify the following analysis, we set the feature dimension as 1 and omit the bias term B in layer nor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Also, the temporal dimension is omitted since it is not relevant for the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Based on the simplification, we have input V ∈ R𝑁 ×1, and layer normalization first calculates the mean of V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', E[V] = 𝔑 × V (7) where 𝔑 ∈ R𝑁×𝑁 is a matrix filled with 1 𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Thus, the term V − E[V] in Equation 6 can be calculated by, V − E[V] = (I − 𝔑) × V (8) I ∈ R𝑁 ×𝑁 corresponds to the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Considering the affine parameter W ∈ R𝑁 ×1 in Equation 6, the element-wise multi- plication between V − E[V] and W can be transferred to a matrix multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', (V − E[V]) � W = Diag(W) × (I − 𝔑) × V (9) Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective where Diag(W) is a diagonal matrix whose diagonal element in each row corresponds to the element in the corresponding row of W, and � is element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Finally, by defining 𝜎V = Var[V] which represents the standard deviation of V, the calculation of layer normalization can be written as, LN(V) = [ Diag(W) 𝜎V (I − 𝔑)]V = A × V (10) Where A = [ Diag(W) 𝜎V (I − 𝔑)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding the calculation of layer normalization, the diagonal matrix Diag(W) ensures the diversity among the values in different rows in A, and A contains data- driven term 𝜎V and learnable parameters W, which determine that layer normalization is effective and flexible in aggregating vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In case that the dimensionality of feature is 𝐷, we should calculate each dimension individually with the above-mentioned method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Figure 4 illustrates the layer normalization calculation for the 𝑖-th dimension where V𝑖 the 𝑖-th dimensional feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Matrix Multiplication 𝓥 𝓥𝒊 \u0de1𝓥𝒊 \u0de1𝓥 𝓐𝒊 = [𝐃𝐢𝐚𝐠 𝑾 𝝈𝓥 (𝑰 − 𝕹)] Figure 4: Graph-convolution-like form of layer normaliza- tion in data aggregation among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Comparing layer normalization with the calculation of graph convolution defined as, GraphConv(V, A) = A × V × W′ (11) where A and W′ correspond to the adjacency matrix and the learn- able parameters respectively, we discover that layer normalization aggregates vertices in a way similar to graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Actually, the aggregation of vertices in layer normalization can be seen as a graph convolution layer equipped with adjacency matrix as A in Equation 10 and the difference between the two is that layer normalization has no linear projection on the feature dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', W′ in Equation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' So far, we theoretically analyze the effectiveness of layer nor- malization on data aggregating and prove that the operation of layer normalization is similar to graph convolution without linear projection on feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Considering the superiority of layer normalization in terms of computational complexity, the idea that how to achieve a more efficient spatial learning module by taking advantage of layer normalization is worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='3 Layer normalization based graph-free spatial learning Based on the analysis in the previous subsection, we propose a novel and efficient Graph-Free Spatial (GFS) learning module by equipping layer normalization with some add-on components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The detailed architecture of GFS learning module is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As shown, we equip layer normalization with a linear projec- tion and ReLU activation function to extract spatial correlations among vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Considering that residual connection is effective to increase the representation power of graph convolution [10], we also employ a residual connection in GFS learning to increase the representation ability of layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Layer Normalization Linear & ReLU Output Input Linear Figure 5: Architecture of layer normalization based graph- free spatial learning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, given input V ∈ R𝑄×𝑁×𝑑𝑖𝑛, GFS first applies a linear projection and a ReLU activation function on feature dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', V′ = ReLU(VWs1 + bs1) (12) where Ws1 ∈ R𝑑𝑖𝑛×𝑑𝑜𝑢𝑡 and bs1 ∈ R𝑑𝑜𝑢𝑡 are learnable parameters, and V′ ∈ R𝑄×𝑁 ×𝑑𝑜𝑢𝑡 is the result of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Next, a layer nor- malization is applied on V′ on both vertex and feature dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', ˆ V = V′ − E[V′] √︁ Var[V′] � Ws2 + Bs2 (13) where Ws2 ∈ R𝑁 ×𝑑𝑜𝑢𝑡 and Bs2 ∈ R𝑁 ×𝑑𝑜𝑢𝑡 are learnable affine parameters, ˆ V is the output of layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To generate the final output V𝑜𝑢𝑡 ∈ R𝑄×𝑁×𝑑𝑜𝑢𝑡 of GFS, a residual connection integrated with linear projection is applied on the input and added to ˆ V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', V𝑟𝑒𝑠 = ReLU(VWres + bres) (14) V𝑜𝑢𝑡 = V𝑟𝑒𝑠 + ˆ V (15) where Wres ∈ R𝑑𝑖𝑛×𝑑𝑜𝑢𝑡 and bres ∈ R𝑑𝑜𝑢𝑡 are learned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice that GFS module can be easily plugged into existing mod- els for replacing all graph convolution components, as far as the dimensions are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4 Time complexity comparison between GFS and graph convolution To validate the efficiency of GFS theoretically, in this subsection, we compare the time complexity of GFS with that of standard graph convolution defined in Equation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Time complexity of GFS: As illustrated in Figure 5, GFS contains two linear projection operations on feature di- mension, and each linear projection operation has the time complexity of O(𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding the opera- tion of layer normalization, its embedded mean calcula- tion, standard deviation calculation, and element-wise mul- tiplication are all in linear time complexity in both the dimensionalities of vertex and feature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', O(𝑁 × 𝑑𝑜𝑢𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Therefore, the overall time complexity of GFS, which is Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ O(𝑁 × 𝑑𝑜𝑢𝑡 + 𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡), is exactly linear to the num- ber of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Time complexity of standard graph convolution: As defined in Equation 11, graph convolution contains two matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The first multiplication between adjacency matrix and features of vertices has the time com- plexity of O(𝑁 2 × 𝑑𝑖𝑛), and The latter multiplication with learnable parameters has the same time complexity with linear projection in GFS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', O(𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' There- fore, the overall time complexity of graph convolution is O(𝑁 2 × 𝑑𝑖𝑛 + 𝑁 × 𝑑𝑖𝑛 × 𝑑𝑜𝑢𝑡), which is quadratic to the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As theoretically analyzed, the proposed GFS is more efficient than the operation of graph convolution, and we reckon that such differ- ence in efficiency will be reflected most vividly in processing very large scale graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 5 EXPERIMENTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1 Experimental scheme and datasets To evaluate the performance of our proposed GFS module, we select a series of representative graph convolution based spatiotempo- ral graph learning works and replace their integrated graph con- volution module with GFS module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The experiments consist of three parts: i) Performances on spatiotemporal graph learn- ing: Based on four widely-used real-world spatiotemporal datasets, PEMSD3, PEMSD4, PEMSD7 and PEMSD8 [6]5, we conduct a series of experiments to compare the performances of GFS and graph convolution on different backbones in terms of traffic prediction, ii) Performances on graph learning with extreme large graph: Based on two graph datasets with extreme large numbers of ver- tices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', PubMed [30] and Coauthor Physics [23], regarding the task of node classification, we further investigate the time efficiency and capability of GFS in modeling spatial correlations, and iii) In- vestigation on graph-free architecture: To further verify the effectiveness of each individual component of the graph-free archi- tecture, we carry out a series of ablative studies on PeMSD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The statistical information of all used datasets is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Table 3: Dataset descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Dataset #Vertices #Features Time Range PeMSD3 358 1 09/01/2018 - 11/30/2018 PeMSD4 307 1 01/01/2018 - 02/28/2018 PeMSD7 883 1 05/01/2017 - 08/31/2017 PeMSD8 170 1 07/01/2016 - 08/31/2016 PubMed 19717 500 N/A Coauthor Physics 495924 8415 N/A 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 Data preprocess For spatiotemporal datasets, linear interpolation is utilized to fill the missing values in the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Then, we apply min-max nor- malization to normalize all data into the range of [−1, 1] to stabilize 5These four datasets, which are about the highway traffic flow in California, are collected by Caltrans Performance Measurement System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding all experiments on spatiotemporal graph learning, all spatiotemporal datasets are divided into training, validation and testing sets with the ratio of 6:2:2 in chronological order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', the earliest 60% are used for training, the subsequent 20% are used for validation, and the last samples are for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice that the raw traffic flow data within spatiotemporal datasets is ag- gregated with the interval of 5 minutes, therefore the aggregated datasets contain 288 data points for each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' For graph datasets, we directly use the data provided by torch_geometric 6 and split PubMed and Coauthor Physics with the ratio of 9:1 and 7:3 respec- tively for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='3 Backbones and experimental settings Backbones: to compare the performances of GFS and graph con- volution, we select a series of graph convolution based backbones including: STGCN [33]: deploys graph convolution and temporal con- volution for capturing spatial and temporal dependencies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' DCRNN [18]: combines diffusion graph convolution with recurrent units for multi-step prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' GraphWaveNet [28]: proposes node embeddings for con- structing adjacency matrices and combines GCN with di- lated casual convolution for traffic forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ASTGCN [14]: is a self-attentive traffic forecasting model, and captures the dynamics in a flexible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' AGCRN [3]: proposes node-adaptive graph convolution, generates node-specific parameters according to learnable node embeddings, and combines it with GRU [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN [17]: constructs temporal graphs based on the similarities between time series of vertices by utilizing DTW algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The temporal graphs are fused with distance- based graphs for better modeling spatial dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STG-NCDE [9]: extends the concept of neural controlled differential equations and designs two novel NCDEs for spatial and temporal processing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ASTGNN [16]: is an upgraded version of ASTGCN by mod- ifying the attention mechanism in ASTGCN and adding positional embedding into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Experimental settings: Regarding the experiments of spatiotem- poral forecasting, to evaluate the effect of GFS module, we replace the graph convolution component of all backbones with our pro- posed GFS module and keep all other settings of those backbones unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The metric of MAE are chosen as the loss, and two more metrics, RMSE and MAPE, are additionally evolved to com- prehensively evaluate all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In case that GFS is employed on existing models, we strictly follow the training settings of the orig- inal models for fair comparison, including optimizers, batch size, maximum epochs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding the learning on extreme large graphs, a stack of three layer GFSs and a stack of three layer GCNs are trained on randomly selected training samples with NLLLoss 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice that all experiments are executed with one E5-2620 v4 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10GHz CPU and one Nvidia Tesla V100 16GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 6https://pytorch-geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='io/en/latest/modules/datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='html 7https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NLLLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='html Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective Table 4: Performance comparison between backbones and their GFS variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' # denotes that GFS is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Model PEMSD3 PEMSD4 PEMSD7 PEMSD8 MAE RMSE MAPE(%) MAE RMSE MAPE(%) MAE RMSE MAPE(%) MAE RMSE MAPE(%) STGCN 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='55 30.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='29 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='94 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='36 STG-NCDE 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='57 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='09 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='06 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='21 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='09 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='76 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='53 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='84 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='45 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='81 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92 STG-NCDE# 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='21 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='77 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='74 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='00 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='46 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='82 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='38 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='78 ASTGNN 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='89 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='93 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='36 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='52 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='03 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='43 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='41 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='94 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='81 ASTGNN# 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='21 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='77 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='74 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='40 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='46 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='88 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='69 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4 Experiments on spatiotemporal graph learning Main experiments: To evaluate the effectiveness of GFS, we in- corporate it with those graph convolution backbones by replacing their embedded graph convolution component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Specifically, we use those backbones and their GFS variants to predict the urban traffics during the next hour with the traffics during the previous hour, and the average result over the next 12 prediction steps is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Notice that # denotes that GFS is integrated with cor- responding backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The results of each individual backbone and its corresponding GFS variant are grouped by double line for clearer comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Regarding each group, the better one is marked in bold, and the best performance over all models is highlighted with underlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As demonstrated in Table 4, in most cases, GFS based variants outperform the corresponding backbones, and this indicates the superiority of our proposed GFS module in capturing spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And the performances of the GFS variant based on ASTGNN are the best in most experiments, and utilizing GFS in STGCN, DCRNN, GraphWaveNet, ASTGCN, AGCRN, STFGNN, STG-NCDE, ASTGNN can respectively gain the improvements on all metrics by {4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='35%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='24%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='63%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='39%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='61%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='33%} in average, and this indicates that our proposed GFS module is ap- plicable to all existing graph convolution based spatiotemporal learning works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' On the other hand, regarding all backbones, re- placing their integrated graph convolution with GFS can gain the performance improvements by {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='63%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='95%} respectively on MAE, RMSE and MAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In summary, the results on spatiotemporal graph learning undoubtedly verify the effectiveness of GFS in spa- tial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' For more fine-grained analysis, we further compare the step-wise performances of AGCRN, STFGNN, ASTGNN and their corresponding GFS variants for each individual prediction step on PeMSD4 and PeMSD8, and results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The horizontal axis corresponds to different time steps and the vertical axis corresponds to the performances in different metrics and with different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' First, as observed, the performances of the GFS based variants are better than the performances of the corresponding backbones at almost all time steps, this verifies the superiority of GFS in capturing spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Second, we discover that the performances of the GFS based variants decrease slower than the performances of the corresponding backbones with the increasing of time steps, this indicates that GFS module can significantly improve traditional spatiotemporal learning on multi- step predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Time consumption: As analyzed, the time complexity of GFS is significantly better than that of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In this part, we compare the experiment time consumption of GFS with that of graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Similarly, for each backbone, GFS is used to replace their graph convolution component, all models are trained and tested on PEMSD4 and PEMSD8, and the results are listed in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As shown, for all models, utilizing GFS can significantly reduce the time consumptions on both training and testing by 20% averagely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Considering that different backbones have diverse archi- tectures, and such divergence may interfere with the experiments to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' For the sake of fairness, we construct a series of generated datasets with different numbers of vertices ranging from 100 to 2000 to further investigate the time consumption issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Each dataset contains 1000 batches and each batch contains 64 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Based on these generated datasets, we test the time consumptions of single-layer GFS and single-layer GCN (using the definition in Equation 11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The results are shown in Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As illustrated, the difference between the time consumptions of these Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ 16 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10 11 12 MAE on PEMSD4 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (a) 25 26 27 28 29 30 31 32 33 34 35 1 2 3 4 5 6 7 8 9 10 11 12 RMSE on PEMSD4 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (b) 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 15 1 2 3 4 5 6 7 8 9 10 11 12 MAPE on PEMSD4 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (c) 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 MAE on PEMSD8 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (d) 19 21 23 25 27 29 1 2 3 4 5 6 7 8 9 10 11 12 RMSE on PEMSD8 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (e) 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 12 1 2 3 4 5 6 7 8 9 10 11 12 MAPE on PEMSD8 AGCRN AGCRN# STFGNN STFGNN# ASTGNN ASTGNN# (f) Figure 6: Step-wise performances of AGCRN, STFGNN, ASTGNN and their corresponding GFS variants on PeMSD4 and PeMSD8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Table 5: Time consumption comparison between backbones and their GFS based variants on PEMSD4 and PEMSD8 Model PEMSD4 PEMSD8 Test Train Test Train STGCN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='89 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='62 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='48 STGCN# 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='76 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='57 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='48 DCRNN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='64 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='53 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='94 DCRNN# 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='74 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 GraphWaveNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='70 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='66 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='57 GraphWaveNet# 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='65 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='56 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='04 ASTGCN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='98 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='15 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='39 ASTGCN# 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='97 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='33 AGCRN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='99 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='07 AGCRN# 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='90 STFGNN 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='56 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='79 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='82 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='62 STFGNN# 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='90 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='04 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='91 STG-NCDE 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='54 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='58 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='91 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='92 STG-NCDE# 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='90 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='69 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='67 ASTGNN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='21 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='47 ASTGNN# 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='94 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='36 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='78 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='89 two networks is relatively small while the number of vertices is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, since the time complexity of GCN is quadratic to the number of vertices, the time consumption of GCN increases much faster than that of GFS with the increasing of vertex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' We also test the time consumptions of AGCRN and ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='time consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='number of vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='GFS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(a) Single-layer GFS and single-layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='1900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='time consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='number of vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='AGCRN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='AGCRN# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(b) AGCRN and its GFS variants ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Figure 7: Time consumptions of Single-layer GFS and single- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='layer GCN with different numbers of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' its GFS based variant on the constructed datasets, and the results are shown in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As demonstrated, the time complexity of AGCRN# scales linearly with the increasing of vertex number, while the time complexity of original AGCRN increases quadratically with the increasing of vertex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The experimental results on the generated dataset validate the temporal efficiency of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Detailed Performances of state-of-the-art solutions on sin- gle vertex: Regarding state-of-the-art solutions, ASTGNN and its corresponding GFS based variant ASTGNN#, we select two random vertices in one random day from the testing set, and evaluate the performances of these two state-of-the-art solutions for each indi- vidual time point during the selected two vertices, and the results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Note that the data of vertices from about 17:30 to 20:30 is missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As shown, both two models can achieve satisfying accuracies all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' regarding some extreme scenarios including peak values and wild fluctuations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' as has been highlighted with amplification rectangles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' the performance curves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='130 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='280 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='330 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='06:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='13:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='16:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='traffic flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='ASTGNN# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='ASTGNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(a) Vertex 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='06:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='13:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='16:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='23:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='traffic flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='ASTGNN# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='ASTGNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(b) Vertex 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Figure 8: Detailed performances of state-of-the-art solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='at different vertices on PEMSD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' of the GFS based variant can approximate the curves of truths more accurately, and this further illustrates the effectiveness of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='5 Node classification on extreme large graph Main experiments: In this subsection, we investigate the perfor- mance of GFS in processing extreme large graph, and a very simple three-layer stacked architecture is proposed in the experiments for both GFS and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' For training these two stacked networks, we randomly select training samples, train for 200 epochs, and test for 1000 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The average classification accuracies are reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As observed, compared with GCN, utilizing GFS can improve the classification accuracy by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='15% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='7% respectively on PubMed and Coauthor Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The results demonstrate the generalizability of GFS on general graph learning task and the scal- ability of GFS on large graph with massive vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Further, we also investigate the training accuracy of each epoch of GFS and GCN on two datasets, and the results are show in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As can be easily observed, both two modules have similar convergence speeds and achieve satisfying fitting on Coauthor Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' However, regarding PubMed, GFS is able to fit training samples better and thus has better representation capability than GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Such results witness the capability of GFS on modeling spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 1 20 39 58 77 96 115 134 153 172 191 Acc(%) Epoch GFS GCN (a) PubMed 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='2 1 20 39 58 77 96 115 134 153 172 191 Acc(%) Epoch GFS GCN (b) Coauthor Physics Figure 9: Training accuracy of each epoch of GFS and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Time comsuption: Regarding the previous node classification ex- periment, for each individual approach and dataset, we also record its time comsumption, and the results are also reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As shown, compared with GCN, GFS can reduce the time consump- tion by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='64× and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='6× respectively on on PubMed and Coauthor Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Compared with the previous spatiotemporal graph learn- ing experiments on PEMSD3, PEMSD4, PEMSD7 and PEMSD8, our approach has more advantages over GCN in terms of time consump- tion with large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The larger the graph is, the more advantage our GFS has in terms of time consumption, and this reflects that the great prospects of GFS in processing extreme large graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Table 6: Accuracy and time consumption of GFS and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Dataset Model Acc(%) Total time PubMed GFS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8667343 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='429018 GCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='8452333 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='343687 Coauthor Physics GFS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='9670854 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='546249 GCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='9593931 698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='05498 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='6 Investigation on graph-free architecture In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' to further verify the effectiveness of each individual component of the graph-free architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' we carry out a series of ablative studies on PeMSD4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' and the variants of GFS include: Mean: to evaluate the effect of layer normalization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' we design this variant by replacing the layer normalization module of GFS with mean function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' which is the equal to combine residual connection with a graph convolution with adjacency matrix as matrix 𝔑 defined in Equation10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' MeanP: to further evaluate the performance gap between mean function and normalization operation, we design this variant by replacing the layer normalization of GFS with mean function but retains the affine parameters of layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' NoLNP: to evaluate the effect of normalization operation itself, we design this variant by directly removing the layer normalization in GFS but keep the affine parameters of layer normalization in Equation 6, which equals to apply affine parameters directly on a graph convolution layer Xu Wang1, Pengfei Gu1, Pengkun Wang1, Binwu Wang1, Zhengyang Zhou1, Lei Bai2∗, Yang Wang1∗ with adjacency matrix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As affine parameters can be applied individually, we retain them in this variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' NoRes: to evaluate the effect of residual connection, the residual connection in GFS is removed in this variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' LNNoP: to evaluate the effect of affine parameters, we design this variant by removing the affine parameters of layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MeanP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoLNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoRes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='LNNoP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(a) GFS variants with STGCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MeanP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoLNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoRes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='LNNoP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(b) GFS variants with STFGNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MeanP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoLNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoRes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='LNNoP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(c) GFS variants with AGCRN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='RMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MAPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='MeanP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoLNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='NoRes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='LNNoP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='(d) GFS variants with ASTGNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='Figure 10: Performances of different GFS variants with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='STGCN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' STFGNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' AGCRN and ASTGNN on PEMSD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' To comprehensively evaluate the impacts of different components of GFS, we conduct a series of ablative experiments on PeMSD4 by incorporating all variants with the four representative approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=', STGCN, STFGNN, AGCRN, and ASTGNN, and the results are illustrated in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' As can be easily observed, GFS itself out- performs all variants with all backbones, and this first indicates that all components in GFS is effective to the final performances of GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And the variant of MeanP outperforms all other variants with all backbones except STFGNN, and this cross-verifies two points: i) the aggregation itself is more important than the way of aggregation, and ii) layer normalization is more effective than mean function on aggregating vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And the performances of NoLNP also indicates the first point which is consistent to the conclusion that we have obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Comparing the performances of two solution pairs, MeanP vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Mean and GFS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' LNNoP, we discover that, no matter mean or layer normalization is used for aggregating vertices, affine parameters are vital and essential for GFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Furthermore, based the performances of NoLNP and LNNoP, it is obvious that, the normalization operation itself, which introduces the aggregation of vertices, is more important than the mechanism of affine parameters, even though affine parameters can change the way that layer normalization aggregates vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' This triple verifies that the aggregation itself is more important than the way of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And finally, the performance comparison between NoRes and GFS also verifies the importance and necessity of the residual connection component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 6 CONCLUSION AND DISCUSSION Conclusion: In this paper, with extensive and deep-going experi- ments, we comprehensively analyze existing spatiotemporal graph learning models and reveal that extracting adjacency matrices with carefully design strategies, which are viewed as the key of en- hancing performance on graph learning, are largely ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Meanwhile, based on these experiments, we also discover that the aggregation itself is more important than the way that how ver- tices are aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' With these preliminary, a novel efficient GFS learning module based on layer normalization for capturing spa- tial correlations in spatiotemporal graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' The proposed GFS module can be easily plugged into existing models for replac- ing all graph convolution components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Rigorous theoretical proof demonstrates that the time complexity of GFS is significantly better than that of graph convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Extensive experiments verify the superiority of GFS in both the perspectives of efficiency and learning effect in processing graph-structured data especially extreme large scale graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Discussion: In future works, there are some more interesting issues can be further discussed, The effectiveness of GFS further indicates that spending too much efforts on extracting adjacency matrix is the wrong region to spatiotemporal learning, and the reason why adja- cency matrix is almost useless needs to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' And Instead of relying on designing new adjacency matrix and incorporating it with graph convolution, how to effec- tively capture spatial correlations from spatiotemporal data need to further thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Even though GFS has achieved promising performances on spatiotemporal graph learning, its performance is largely owed to the affine parameters of layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Con- sidering that the shape of such parameters are predefined, the scalability of GFS is largely limited since GFS is not applicable to the scenario where the number of vertices is dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Graph-Free Learning in Graph-Structured Data: A More Efficient and Accurate Spatiotemporal Learning Perspective REFERENCES [1] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Gman: A graph multi-attention network for traffic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' In Proceedings of the AAAI conference on artificial intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 1234–1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' [41] Ali Zonoozi, Jung-jae Kim, Xiao-Li Li, and Gao Cong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' Periodic-CRN: A convolutional recurrent model for crowd density prediction with recurring periodic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content='. In IJCAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} +page_content=' 3732–3738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNFKT4oBgHgl3EQfKi0w/content/2301.11742v1.pdf'} diff --git a/v9FQT4oBgHgl3EQfvTYe/content/2301.13397v1.pdf b/v9FQT4oBgHgl3EQfvTYe/content/2301.13397v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c2b8d2b6e63d7615aeae0eeea5b58078ccaec940 --- /dev/null +++ b/v9FQT4oBgHgl3EQfvTYe/content/2301.13397v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ba5539488ee3a4f0a10552ca6279b17b39673edaa40961cd230285581d5332ec +size 765107 diff --git a/v9FQT4oBgHgl3EQfvTYe/vector_store/index.faiss b/v9FQT4oBgHgl3EQfvTYe/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..94a5145ac35289a41747f6be1c7be775dbeee5fa --- /dev/null +++ b/v9FQT4oBgHgl3EQfvTYe/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6b666c23e965332229b4d186ed54713156c59373d42d88939ecea116f2c64e6c +size 3997741 diff --git a/v9FQT4oBgHgl3EQfvTYe/vector_store/index.pkl b/v9FQT4oBgHgl3EQfvTYe/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3890f9976090b97d215aa87d81988f9981df55dd --- /dev/null +++ b/v9FQT4oBgHgl3EQfvTYe/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e878a3e7551641db9c11fe9feba2635f13d973596c3709d3f1b0eeba8c7055cf +size 146272 diff --git a/v9FRT4oBgHgl3EQfgDc7/vector_store/index.faiss b/v9FRT4oBgHgl3EQfgDc7/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f67a11a3da1ffc3542282790fcffe32aafedd421 --- /dev/null +++ b/v9FRT4oBgHgl3EQfgDc7/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1641bb282d5dea9fb6c7b0bd3b455d1a2c91170cd4189a7e699c1f1d3282f29e +size 3211309 diff --git a/vtE3T4oBgHgl3EQfkwrp/content/2301.04601v1.pdf b/vtE3T4oBgHgl3EQfkwrp/content/2301.04601v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..701fe5be990fa2c3b95eaa00efe224ad9b9fb924 --- /dev/null +++ b/vtE3T4oBgHgl3EQfkwrp/content/2301.04601v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:53713ac1b8eac9f690de82c73c91df9295044c5bede5d53844f2ba49e916a4c9 +size 344365 diff --git a/vtE3T4oBgHgl3EQfkwrp/vector_store/index.faiss b/vtE3T4oBgHgl3EQfkwrp/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8a5b2ce46bdece285c9967b6f392905a480ce4f0 --- /dev/null +++ b/vtE3T4oBgHgl3EQfkwrp/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5d7f8764215b6fe42de572c5c37fb58612343d8e042d7d075b4808a1ff11458d +size 5308461 diff --git a/vtE3T4oBgHgl3EQfkwrp/vector_store/index.pkl b/vtE3T4oBgHgl3EQfkwrp/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a363b0904ad5b86bbd537cf38bd5805eb1ac1173 --- /dev/null +++ b/vtE3T4oBgHgl3EQfkwrp/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4207ca3165b81799b4b7fe5200e4510621d2fc5ae6a861bdadc4cbf36bf126be +size 190225 diff --git a/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/2301.13215v1.pdf.txt b/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/2301.13215v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5bdb66fabea30d60a957b1ecdf000dde5d8ef94 --- /dev/null +++ b/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/2301.13215v1.pdf.txt @@ -0,0 +1,1593 @@ +Quasi-extremal primordial black holes are a viable dark matter candidate +Jose A. de Freitas Pacheco,1 Elias Kiritsis,2, 3 Matteo Lucca,4 and Joseph Silk5, 6, 7 +1Universit´e de la Cˆote d’Azur - Observatoire de la Cˆote d’Azur, Bd de l’Observatoire, 06304 Nice Cedex, France +2Universit´e Paris Cit´e, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France +3Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics, +Department of Physics, P.O. Box 2208, University of Crete, 70013, Heraklion, Greece +4Service de Physique Th´eorique, Universit´e Libre de Bruxelles, C.P. 225, B-1050 Brussels, Belgium +5Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne Universities), F-75014, Paris, France +6Department of Physics and Astronomy, The Johns Hopkins University Homewood Campus, Baltimore, MD 21218, USA +7BIPAC, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK +Black hole evaporation is generally considered inevitable for low-mass black holes, yet there is +no confirmation of this remarkable hypothesis. Here, we propose a phenomenological model that +appeals to the possible survival of light quasi-extremal primordial black holes as a significant dark +matter component and show that the related cosmological and astrophysical constraints disappear +for reasonable degrees of quasi-extremality. +The results obtained are general, conservative and +should be taken as a proof of principle for future, model-specific analyses. +I. +INTRODUCTION +Since first postulated in the late ‘60s [1–4], primor- +dial black holes (PBHs) are regarded as a viable dark +matter (DM) candidate, albeit over a highly constrained +mass range (see e.g., [5–8]). The most appealing exam- +ple is given by a range roughly between 1017 and 1022 g, +bounded from below by constraints coming from the im- +pact of the energy injection following Hawking evapora- +tion of the PBHs [9] on observables such as cosmic mi- +crowave background (CMB) anisotropies [10–13], cosmic +rays [14], and 21 cm lines [15]. +These bounds on PBH evaporation are, however, de- +rived under the assumption that the PBHs are non- +rotating and neutral, which maximizes their evapora- +tion rate. +In fact, it is well known that if the BHs +were charged and/or spinning, their Hawking temper- +ature would decrease, and consequently so would their +mass loss rate and luminosity. +In the limit where the +temperature approaches zero one has what are referred +to as quasi-extremal PBHs (qPBHs). +Since increasing +degrees of quasi-extremality can be reached, this implies +that the aforementioned mass range could be extended +to lower masses by decreasing Hawking evaporation. +Nevertheless, the formation and survivability of qPBHs +is clearly a contentious issue. +The acquisition of ex- +treme spin-to-mass ratios is astrophysically forbidden for +BH growth by accretion in thin [16] or thick disks [17]. +Even if the PBHs had a spin at formation, it would be +lost more efficiently than its mass [18], making quasi- +extremality impossible to obtain over extended periods +of time. A similar discussion also applies to the case of +charged BHs [19]. +One can, however, invoke other processes to justify the +existence of qPBHs at lower masses. For instance, accre- +tion and Schwinger pair production can reduce the BH +charge Q, but Hawking evaporation can counter these +effects and augment the charge [20]. Such relics have of- +ten been considered as DM candidates [21–24]. Indeed, +small quantized BHs have a fundamental stable state de- +fined by a mass equal to the Planck value and a spin +J = ℏ [25]. We emphasize furthermore that PBHs can +form deep in the radiation era where stabilized qPBHs +are possible DM candidates if their fractional abundance +at formation is as low as ∼ 10−25, and application of ex- +treme value statistics leads to the expectation that some +of the DM may plausibly include a qPBH component [26]. +The motivation for the rarity of extremely light qPBHs +comes from the equilibrium charge distribution [27] once +the emission of charged particles of random sign is in- +cluded, P(Q) ∼ exp[−4πα(Q/e)2], where α is the elec- +tromagnetic coupling and the PBHs rms charge satisfies +Q/e = 1/ +√ +8πα ≈ 6. +Also PBHs living in an higher- +dimensional space are DM candidates [28, 29]. Provided +that the scale of the extra dimension is of the order of +the gravitational radius, we show below that such BHs +may be quasi-extremal. +It is therefore not unreasonable to expect qPBHs to +exist in a realistic cosmological scenario. Yet, the impact +that such quasi-extremality would have on the commonly +imposed cosmological and astrophysical constraints on +PBH evaporation has not been systematically considered +in the literature so far. Here we address this task and +propose a very general phenomenlogical analysis to be +taken as a proof of principle for future, more specific +studies. As a result, in our simplified and yet conservative +scenario, we find that for values of the quasi-extremality +parameter ε (defined in terms of e.g., the PBH charge +Q and mass M as ε = 1 − Q2/M 2) lower than ≲ 10−3 +all cosmological and astrophysical constraints allow for +PBHs to make up for the totality of the DM, and that +they become stable over cosmological times for masses as +low as ∼ 1011 g. This implies that for sufficiently low +values of ε, qPBHs are a viable DM candidate. +This work is organized as follows. +In Sec. +II we +discuss both how such levels of quasi-extremality can +be reached and maintained over cosmic times, and how +they affect the standard picture of Hawking evaporation. +In Sec. III we overview the many cosmological and as- +trophysical observables affected by the presence of this +ULB-TH/23-01 +arXiv:2301.13215v1 [astro-ph.CO] 30 Jan 2023 + +2 +quasi-extremality and suggest simple ways to recast ex- +isting bounds in the quasi-extremal limit. In Sec. IV we +present the resulting constraints on qPBHs as a function +of both the quasi-extremality parameter and the frac- +tional abundance of the PBHs. In Sec. V we conclude +with a summary and closing remarks. +II. +QUASI-EXTREMAL PRIMORDIAL BLACK +HOLES +In this section we provide a brief overview of scenar- +ios that can generate quasi-extremal BHs. We do so by +highlighting the general features that they share and how +the analysis presented here can be consider as a proof of +principle for other, more specific examples. We further- +more also discuss how this general scenario would affect +the standard picture of BH evaporation. +A. +Representative examples +A first well-known example of qPBHs is given by highly +charged BHs, in particular Reissner-Nordstrom (RN) +BHs. In this case, to parametrize the degree of quasi- +extremality we can define the parameter ε such that1 +1 − Q2 +M 2 = ε2 ≪ 1 , +(1) +where M is the mass and Q is the electric charge of the +BH. RN BHs have two horizons: the outer, corresponding +to the event horizon, and the inner, the Cauchy horizon, +defined by the zeros of the lapse function, that is +r± = M +� +1 ± +� +1 − Q2 +M 2 +� += M(1 ± ε) . +(2) +The size of the outer horizon should never be smaller than +the BH-associated Compton wavelength. This condition +gives the Planck mass as the smallest PBH mass. +The notion of the dual horizon also applies to Kerr +BHs, which represent a second possible qPBH solution +should they spin to a high degree. Here the similar hori- +zon structure is now characterized by +r± = M +� +1 ± +� +1 − a2 +M 2 +� += M(1 ± ε) , +(3) +where a is the dimensionless spin parameter (related to +the angular momentum J via a = J/M), and, analo- +gously to Eq. (1), we can then also define +1 − a2 +M 2 = ε2 , +(4) +1 Here and henceforth we assume G = c = ℏ = kB = 1. +which shows a similar mass dependence of the relation +between ε and the model-specific parameters Q and a. +It is therefore clear that much of the model dependence +of the aforementioned scenarios can be captured by the +single parameter ε, which represents the degree of ex- +tremality of the BH. In the following discussion we will +then only make use of ε so as to be as general as possible +in our conclusions. +In a realistic cosmological scenario, however, it is well +known that any charge or spin would be lost very quickly +by any BH population of primordial origin. +Quasi- +extremal primordial RN or Kerr BHs are therefore not +expected to exist today. Based on these models, it is nev- +ertheless possible to envision a scenario where the BH is +charged, but instead of standard electromagnetism (EM) +the BH is charged under a generic EM-like dark charge +whose carriers are always much heavier than the temper- +ature of the BH [24]. In this way, one obtains the same +mathematical setup as for a RN BH, but with the dif- +ference that the charge Q does not get evaporated away +from the BH and remains therefore constant. +From Eq. (1), however, one can infer that a constant +Q does not necessarily imply a constant ε. In fact, since +initially the charge will always be smaller than the mass +of the BH, ε will always be larger than zero and the BH +will radiate its mass at the rate discussed in the following +section. Nevertheless, the smaller the mass (i.e., the more +the BH evaporates) the more ε will approach the zero +value and the slower the mass loss rate becomes. This +means that a constant charge with Q < M leads to a BH +that naturally approaches extremality over cosmic times. +On the other hand, Eq. (1) shows that if both Q and +M are radiated away at the same rate, ε stays constant. +This means that in a setup where the initial charge-to- +mass ratio is very close to unity and both quantities get +radiated at the same rate, the BH can maintain extremal- +ity indefinitely. However, contrary to the constant-charge +example mentioned above, this scenario would require a +larger degree of fine-tuning, as one would need to fix the +characteristics of the charged dark particles to be exactly +such that the BH loses charge and mass at the same rate. +Another possibility to obtain (and maintain) quasi- +extremality that does not depend on charge or spin but +that presents very similar features in terms of ε is given +by higher dimensional BHs [28, 29]. One could, in fact, +consider the 5D gravitational theory compactified on a +circle of radius R with R ≲ 1 µm, although the notion +of quasi-extremality is generalizable to d dimensions. In +that case, BHs with a horizon size smaller than R behave +as 5D BHs while those with size larger then R as 4D BHs. +One can then show (see App. A) that the evaporation +of all Kaluza-Klein (KK) modes of a d-dimensional BH +leads to an effective degree of extremality +ε−4 +eff = 3(d − 3) +(d − 1) +S4 +Sd +�2πR +rs +�2(d−4) +. +(5) +A “large” BH with rs ≫ R evaporates following the 4D +decay equation until its horizon becomes rs ≃ R. From + +3 +that point on, it decays as a higher-dimensional BH at +a rate that is slower than in 4D. This implies that a +higher-dimensional BH would tend to quasi-extremality +the more it evaporates, qualitatively just like a constant- +charge BH. +In summary, in the discussion above, we have high- +lighted different ways to justify the existence of qPBHs +which can all be phenomenologically described by the +evolution of ε in the respective scenarios. For simplicity, +hereafter we will solely focus on the aforementioned (RN +BH) case with a constant ε value. With respect to the +other possibilities, this choice is conservative in the sense +that, for the same initial value of ε, in the other scenarios +the degree of extremality would only increase and hence +they would be covered by the results obtained for the +constant ε case (see [23, 24, 28, 29] for related but rel- +atively limited discussions). Furthermore, we point out +that, although based on a slightly less realistic scenario, +ours has to be taken as a useful proof of principle to be +applied to more specific examples in the future. +B. +Evaporation +Given these possible sources of quasi-extremality, it is +interesting to consider how one might observe the decay +of these quasi-extremal BHs and set constraints on their +modified evaporation emission. +The key quantity that is modified by the quasi- +extremality of the BHs is their evaporation temperature +T, which now reads (assuming e.g., a RN BH, but with +no loss of generality) +T = +1 +4πr+ +� +1 − Q2 +r2 ++ +� += +1 +8πM +22 ε +(1 + ε)2 , +(6) +where ε encapsulates the deviations from the standard +Hawking temperature. Once the temperature is defined, +it becomes possible to determine the luminosity L of the +BH via to the Stefan-Boltzmann black body formula2, +L = AσT 4 ∝ r2 ++T 4 ∝ +26 ε4 +(1 + ε)6M 2 , +(7) +where A = 4πr2 ++ is the area of the BH and σ is the +Stefan-Boltzmann constant. This simple dependence of +the luminosity is however strictly speaking only valid as +long as the BH evaporates at a constant rate. Neverthe- +less, since different particles can be emitted at different +BH temperatures, it is more convenient to interpret the +2 Near extremality the Stefan-Boltzmann formula is modified by +important grey-body factors that tend to suppress emission. We +do not consider such factors and therefore our formulae should +be considered as upper bounds on Hawking emission near ex- +tremality in the RN case. In the higher-dimensional case such +factors are not important. +luminosity as the energy emission rate with explicit de- +pendence on the mass loss rate dM/dt, such that +L = −dM +dt +26 ε4 +(1 + ε)6 , +(8) +where the ε dependence needs to be introduced for con- +sistency with Eq. (7). +The mass loss rate of an evaporating BH is commonly +defined in terms of the total energy carried away by +the emitted particles (due to energy conservation argu- +ments), i.e., following the relation +dM +dt = − +� � +j +dNj +dtdE EdE , +(9) +where dNj/dtdE is the number of emitted particles j of +spin s in the energy interval between (E, E + dE) and is +defined as +dNj +dtdE = 1 +2π +Γj +e(E−µj)/T − (−1)2sj . +(10) +Here, Γj is the dimensionless absorption probability of +the given emitted species, which, in full generality, de- +pends on both M and ε. The presence of charge and spin +can in fact enhance or reduce the probability of charged +particles or particles with spin (mis-)aligned with that +of the BH to be emitted from the system. +For stan- +dard RN BHs charged under EM, [27] found that the +impact of the charge on Γj is of the order of a few per- +cent. In Eq. (10), µj refers to the chemical potential of +a given emitted particle and generally depends on ε. For +instance, in the case of standard charged BHs, it would +take the form µj ∝ qj +� +(1 − ε)/(1 + ε), where qj is the +charge of the particle. In our simplified scenario, how- +ever, we assume that the particles carrying the charge of +the quasi-extremal BH are not emitted from the BH at +all (or at least very slowly) and we can therefore neglect +these ε-dependent contributions to Γj and µj. +Under this simplifying assumption, the only way ε af- +fects the mass loss rate is via the exponential dependence +of dNj/dtdQ on the BH temperature T. This determines +what particles are kinematically available at a given tem- +perature T and it therefore makes sense for it to be de- +pendent on the temperature of the system only, regard- +less of the characteristics of the BH reaching that tem- +perature. One can then simply extend the validity of the +results found in [30, 31] according to which +dM +dt = −5.34 × 1025 f(T) +M 2 g/s , +(11) +where f(T) defines the number of emitted species and can +be expressed as in Eq. (9) of [31] (see also [11, 12] for +additional details, updated coefficients and contributions + +4 +beyond the QCD phase transition).3 +With this definition of the mass loss rate it is then +possible to compute the lifetime of the BH by integrating +Eq. (8). Following again [31], one obtains +tev = 6.24 × 10−27 M 3 +f(T) +(1 + ε)6 +26 ε4 +s . +(12) +III. +IMPACT ON THE OBSERVABLES +Once the mass loss rate due to the PBH evaporation +and the related luminosity have been defined, it is pos- +sible to analyse how the emission of particles from the +PBH affects various cosmological and astrophysical ob- +servables such as the CMB anisotropies as well as the +cosmic and γ-ray spectra. Since all of these probes are +sensitive to different epochs of the universe, they also con- +strain different mass ranges, allowing us to cover a wide +region of parameter space. In the following sections, we +describe all of the constraints and explain how they are +affected by the presence of evaporating qPBHs. +A. +BBN +The first observable we focus on is Big Bang Nu- +cleosynthesis (BBN), which covers the period of light- +element formation, such as deuterium and helium, in the +early universe [32]. +The predictions of standard BBN +are in extremely good agreement with measurements of +the corresponding abundances in the first galaxies, where +galactic dynamics and star formation have not had the +time to affect the primordial abundances yet [33]. +In- +ferred quantities such as the baryon energy density, the +baryon-to-photon ratio and the primordial helium abun- +dance are also consistent with CMB measurements [34]. +Given the success of the standard BBN model, this +probe has been often employed to constrain beyond-the- +standard-model (BSM) physics, such as annihilating or +decaying DM [35, 36] or PBH evaporation [13, 37], typ- +ically delivering the most stringent constraints on these +types of models prior to recombination. In fact, BBN +can constrain BSM models in a variety of ways, from the +impact that they might have on the expansion of the uni- +verse (changing e.g., the number of relativistic degrees of +freedom) to the photo-disintegration of the light elements +after BBN is completed. +Precisely this richness of constraints, however, prevents +us from deriving simple and general limits that can be +3 Concretely, focusing for instance on the left panel of Fig. +10 +of [12] for a graphical representation, the only aspect of the plot +that would be modified by the presence of a non-zero ε would be +the relation between the two horizontal axis, reporting the PBH +mass M and the corresponding T values, with the latter being +shifted more and more to the left the higher the value of ε. +recast for any value of ε. This is due to the fact that, +for instance, ε affects both the overall and the relative +amount of injected species (via the modification to f(T)) +as well as the lifetime of the PBHs. Therefore, different +aspects of the standard BBN picture might be modified +in non-trivial ways, affecting the magnitude and shape +of the constraints. BBN bounds on qPBH evaporation +would then have to be derived with dedicated analyses. +For this reason, the accurate inclusion of the BBN con- +straints in the following discussion goes beyond the proof- +of-principle type of study conducted here and will not be +considered any further. We note, however, that, albeit +the scaling of the constraints is not directly proportional +to ε as for the probes discussed below, we do expect a +significant suppression of the constraints the lower the +value is of ε and that the results of this work will not be +affected by the non-inclusion of the BBN constraints. +B. +CMB +1. +CMB anisotropies +CMB anisotropies are very well known to be affected +by exotic energy injections during the dark ages (see +e.g., [10] for a thorough discussion). +In fact, in that +period of the thermal history of the universe, the cos- +mic medium was almost perfectly neutral, allowing the +CMB photons to travel straight from the last scatter- +ing surface (at z ≃ 1100) to us. Any injection of parti- +cles with enough energy to ionize the abundant hydrogen +atoms would have increased the amount of free electrons, +thereby enhancing the probability of further scattering +of the CMB photons. This modification of the so-called +visibility function would in turn affect the shape of the +CMB anisotropy power spectra (both temperature and +polarization). Since the observed spectra are in perfect +agreement with the ΛCDM model in the absence of any +energy injection [34], the CMB anisotropies can be used +to constrain processes such as the evaporation of PBHs. +In order to estimate the extent to which PBH evapora- +tion affects the CMB anisotropies one needs to determine +the energy injection rate, which in this case is given by +dE +dtdV +���� +inj += ρcdmfPBH +L +M , +(13) +where fPBH is the (primordial) fractional abundance of +PBHs with respect to the DM. This injected energy does +not, however, necessarily coincide with the effectively de- +posited energy. In fact, for instance, part of the injected +energy might be in form of non-electromagnetically in- +teracting particles and not all of it is spent to ionize the +medium (some of this energy would heat up or excite +the plasma). These contributions are commonly taken +into account by deposition efficiency feff and deposition +fraction per channel χc, respectively [10–12, 38–40]. Ex- + +5 +plicitly, this implies +dE +dtdV +���� +dep,c += +dE +dtdV +���� +inj +feff χc . +(14) +A graphical representation of the heating rate due to +PBH evaporation is displayed in Fig. 5 of [12]. The con- +sequent impact of PBH evaporation on the free electron +fraction can be seen in e.g., Fig. 6 of [10]4, while that in +relation to the CMB power spectra can be found in Fig. 6 +of [11]. Some important remarks can be drawn from the +figures. First of all, the majority of the energy injection +takes place around the lifetime of the PBH, similarly to +the DM decay scenario. This means that the injection +time can be roughly approximated to coincide with the +lifetime of the PBHs. Secondly, only PBHs with masses +larger than 1013 g evaporate after recombination and can +therefore be constrained with CMB anisotropies. +Based on the discussion in Sec. II B, the energy depo- +sition rate has a dependence on the PBH parameters of +the form +dE +dtdV +���� +dep +∝ fPBH +f(T) 26 ε4 +M 3 (1 + ε)6 . +(15) +In the ε = 1 case, f(T) is almost constant for masses +above 1013 g (it varies at most by a factor 3, see e.g., +Fig. 10 of [12]) meaning that the energy deposition rate +has a simple dependence of the form fPBH/M 3. +This +implies a proportionality of the constraints on the PBH +abundance as M 3, which is perfectly recovered in the +bounds shown in e.g., [11–13]. +On the other hand, the simple proportionality of +Eq. (15) also allows us to take into account the contri- +bution of ε by simply rescaling the bounds on fPBH by a +factor f(Tε=1)(1 + ε)6/(f(T)26ε4), where f(Tε=1) is the +value of f(T) in the ε = 1 case (i.e., for a Schwarzschild +BH). The overall order of magnitude of the rescaling is +given by the chosen value of ε, with the f(T) ratio in- +troducing a further enhancement of at most an order of +a few (and never more than ten). +This enables us to +recast existing constraints, such as the ones derived in +e.g., [11–13], for any value of ε. +Here we rely on the bounds derived in [13], which are +based on Planck 2015 data [41]. +Corresponding con- +straints employing Planck 2018 data [34] have been de- +rived in [12] and seem to be approximately one order of +magnitude more constraining than those of [13]. Never- +theless, [12] employed a simplified thermal history and +made use of a mock likelihood instead of real data. For +this reason, and for sake of being conservative, we choose +to focus on the results of [13]. Furthermore, compared to +other results based on Planck 2015 data such as [11, 42], +4 From the left panel of the figure it becomes clear that heating +and ionization rates are rather correlated, so that the heating +rate shown in [12] is also indicative of the ionization rate. +the findings of [13] overlap well for PBH masses cor- +responding to lifetimes longer than recombination but +improve upon them at lower masses, where the PBHs +evaporate before recombination. This is due to a better +analysis of the delay between the energy injection and its +deposition which extends the constraints down to evap- +oration redshifts of the order of z ≃ 5 × 103. +2. +CMB spectral distortions +In brief, CMB spectral distortions (SDs) are any type +of deviation of the CMB energy spectrum from a pure +black body [12, 43–45]. They are typically created by +the injection of energy or photons in the thermal bath, +although they can also be produced by effects such as the +dissipation of acoustic waves and adiabatic cooling. Com- +plementary to the CMB anisotropies, CMB SDs are very +sensitive to the thermal history of the universe prior to +recombination, up to redshifts of the order of z ≃ 2 × 106. +In the context of PBH evaporation, as in the case of +the CMB anisotropies, their shape is determined by the +amount of injected energy defined in Eq. (14). A key +difference is, however, that for CMB SDs it is the heating +rate that needs to be considered and not the ionization +rate (which would anyway be zero before recombination). +This implies that the same rescaling of existing bounds +(see e.g., [12, 46–48]) discussed in the previous section can +be employed for SDs as well. Here we follow the results +of [47] based on FIRAS data [49], which perfectly overlap +with the more recent and exact calculations of [48] at very +high evaporation redshifts (or, equivalently, for very low +PBH masses), but extend them until recombination. +C. +21 cm +Another cosmological observable that can be employed +to constrain the evaporation of PBHs are the 21 cm ab- +sorption lines [50–52]. These lines are generated when- +ever a neutral hydrogen atom undergoes a spin-flip tran- +sition and are therefore a very important tracker of the +neutral hydrogen distribution across space and time. +Cosmologically, the probability of this transition to hap- +pen is proportional to the relative abundance of the two +spin levels, which in turn depends on what is known as +the spin temperature TS. Since TS is determined by the +CMB and gas temperature, any process that affects the +latter inevitably modifies also the 21 cm signal. +This logic has been applied to constrain several +beyond-ΛCDM models (see e.g., [51] and references +therein for a recent overview), and here we focus on the +case of PBH evaporation [15]. As extensively explained in +the reference, the relation between energy injection and +modified 21 cm signal is dictated by the same equations +discussed in the previous section in the context of CMB +anisotropies. This similarity is qualitatively confirmed, +for instance, in Fig. 4 of the reference, where the same + +6 +fPBH ∝ M 3 proportionality is shown for the 21 cm con- +straints. Therefore, in analogy to the previous section +also in the 21 cm case we can simply recast the existing +bounds of [15] to account for the role of ε. +D. +Diffuse γ-ray background +Next we move to constraints of astrophysical origin, +i.e., focusing on the evaporation of PBHs in the local +environment. +Firstly, we consider the case of the dif- +fuse extra-galactic γ-ray background. The idea in this +context is then to consider the observed γ-ray fluxes, ob- +served by e.g., Fermi LAT [53] and HAWK [54], and to +impose the condition that the flux of photons emitted +from the cosmological PBH population does not exceed +this limit. This exercise has been performed in e.g., [37] +(see Fig. 5 therein) including a number of observations +and found that this probe is particularly constraining for +PBH masses around 1015 g. In a subsequent work [55], +the authors also computed the constraints on the flux of +galactic origin, which however turn out to be subdomi- +nant with respect to the extra-galactic counterpart and +will therefore be neglected here. +Since these constraints depend on the flux of photons +emitted from the PBHs, i.e., +φγ = 1 +4π +� +Lγ nPBH dt ∝ fPBH +tev +, +(16) +where Lγ is the emitted luminosity in form of photons +and nPBH is the PBH number density, also in this case the +constraints have a power-law dependence on ε4/(1 + ε)6, +which allows for a straightforward rescaling of the afore- +mentioned bounds derived in [37]. +E. +Cosmic rays +A similar discussion can be also carried out for galac- +tic cosmic rays, such as electrons and positrons. +The +observational difficulty in this case is, however, that low- +energy charged particles are significantly affected by the +heliosphere of the sun and this limits the amount of infor- +mation that can be extracted from the data. This prob- +lem has been overcome with the exit from the heliosphere +of the Voyager 1 spacecraft [56] and now it is therefore +possible to combine Voyager 1 [57] and AMS-02 [58] data +to constrain the cosmic ray flux over an energy range +between a few MeV and hundreds of GeV. +These data sets have been employed by [14] to bound +the PBH abundance in the mass range between 5×1014− +3 × 1016 g, where they are also the most constraining to +date. As for the γ-rays, also in this case the limits rely +on the definition of the flux of particles from the PBHs, +so that the same ε rescaling applies here as well. +F. +Diffuse high-energy neutrino background +Finally, the observation of the neutrino flux at facilities +such as IceCube [59] and Super-Kamiokande [60] would +enable us to constrain the PBH abundance in the local +environment. However, so far the current sensitivity of +these experiments has not been able to set competitive +bounds with respect to the aforementioned ones [61]. We +will therefore not consider these observations in the fol- +lowing discussion, but point them out as a promising +avenue for the future. +IV. +RESULTS +The collection of the cosmological and astrophysical +constraints on the PBH abundance discussed in the pre- +vious section is summarized in the left panel of Fig. 1. +There, we also display the BBN constraints derived in +[13, 37] for reference (solid green line), although they are +not rescaled as the others and are only to be relied upon +for the ϵ = 1 case. We remark, however, that the con- +clusions drawn below do not depend on this limitation of +the analysis. +In the figure, the solid lines represent the cases with +ε = 1, while dashed, dashed-dotted and dotted lines re- +fer respectively to the ε = 0.1, 0.01 and 0.001 cases. As +expected, the constraints are significantly suppressed the +lower the value of ε. +In fact, as argued in the previ- +ous section the upper bound on the PBH abundance re- +laxes roughly proportionally to ε4/(1+ε)6, which in turns +means that the largest mass allowed by evaporation con- +strains for fPBH = 1 reduces to approximately 2 × 1016 g +for ε = 0.1, to 4×1015 g for ε = 0.01 and all PBH masses +are allowed for ε ≲ 10−3. +We also show as vertical lines the PBH masses whose +lifetime would correspond to the age of the universe (with +the same line style as above). This sets the threshold +above which the PBHs are still present in the universe +today (or, alternatively, below which they are already +evaporated). +While in the ε = 1 this corresponds to +approximately 4.1×1014 g, this value scales as in Eq. (12), +i.e., approximately as (ε4/(1 + ε)6)1/3. This means that +for ε ∼ 10−3 even PBHs as light as ∼ 1011 g would +survive until today. +The combination of these two conclusions, i.e., that +qPBHs with ε ≤ 10−3 can match the correct DM abun- +dance and that they would still be present today, opens +the door to the interesting possibility that such light +qPBHs could be the DM. Of course, this will need to +be developed in the context of more refined qPBH mod- +els, but can still act as a useful (conservative) benchmark +for such scenarios. +Interestingly, for relatively small values of ε, the +bounds presented in the left panel of Fig. 1 can be ap- +proximated to be on the parameter combination fPBH ϵ4 +(neglecting the dependence on f(T) and on the second +order ε term, see Eq. (15)). This allows for a very simple, + +7 +1012 +1014 +1016 +M [g] +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +fPBH +BBN +CMB SDs +CMB ani. +γ-rays +cosmic rays +21 cm +tuni = tev +ε = 1 +ε = 0.1 +ε = 0.01 +ε = 0.001 +1011 +1013 +1015 +1017 +M [g] +10−3 +10−2 +10−1 +100 +ε +fPBH = 1 +fPBH = 10−4 +fPBH = 10−8 +FIG. 1. Left panel: Cosmological and astrophysical constraints on the fractional PBH abundance as a function of the PBH +mass for different values of the quasi-extremality parameter ε. The vertical gray lines represent the PBH masses whose lifetimes +correspond to the age of the universe (with the same line styles). The BBN constraints are only shown for reference in the +ε = 1 case and are not rescaled for the other values of ε for the reasons explained in the main text. Right panel: Same as in +the left panel but on the ε − M plane for different values of fPBH. +order-of-magnitude reinterpretation of the constraints for +any value of ε. Furthermore, it also allows us to present +the limits in the ε − M plane for fixed values of fPBH, +which can be useful for realistic models where the qPBHs +are predicted to be a given sub-component of the DM. +We perform this exercise (with the exact dependence of +Eq. (15)) in the right panel of Fig. 1 for the representa- +tive cases of fPBH = 1, 10−4, 10−8. The figure confirms +the aforementioned discussion. +V. +SUMMARY AND DISCUSSION +The analysis carried out here focuses on PBHs in the +mass range between ∼ 1010 − 1017 g. The allowed abun- +dance of such PBHs is mostly constrained by the impact +of their evaporation on cosmological and astrophysical +observables such as the CMB, the 21 cm lines and cosmic +rays. Nevertheless, these stringent limits are derived as- +suming non-spinning, neutral (i.e., Schwarzschild) BHs, +a scenario that maximizes the evaporation efficiency. If +one assumes instead that the BHs are e.g., charged, spin- +ning or even living in a higher-dimensional space, their +evaporation temperature decreases, and consequently so +does their luminosity. This approach can be pushed to +the limit where the evaporation stops completely, leading +to what are known as extremal BHs. +In this work we consider so-called quasi-extremal +PBHs and show that indeed the assumption of quasi- +extremality can greatly suppress the aforementioned con- +straints on the PBH evaporation. Concretely, we anal- +yse the case of a general and conservative scenario where +the degree of quasi-extremality is captured by a model- +independent parameter ε, which we assume to be con- +stant for simplicity. In the context of charged BHs, for +instance, this quasi-extremality parameter would be de- +fined as ε = 1−Q2/M 2, where Q and M represent charge +and mass of the PBH, respectively. As a result, we find +i) that all constraints vanish for ε ≲ 10−3 and ii) that +for these values of ε all PBHs with masses larger than +∼ 1011 g would still be present today. The combination +of these conclusions implies that such light qPBHs are a +viable DM candidate. +However, the question of observability remains. +In +fact, given the dependencies on ε of the constraints dis- +cussed in Sec. III we do not expect that upcoming ex- +periments such as CMB-S4 [62, 63] and SKA [64] (see +e.g., [12, 65] for related forecasts) would be able to sig- +nificantly change the current picture since qPBHs with +ε ≲ 10−3 would still largely evade them. +Further- +more, even if a survey did observe a signature compati- +ble with the energy injection following the evaporation of +PBHs, it would be impossible to disentangle the case of a +Schwarzschild PBH population from that of a more abun- +dant population of lighter qPBHs. Therefore, cosmolog- +ical and astrophysical probes testing Hawking evapora- +tion are a priori not sensitive enough to uniquely prove +the existence of qPBHs and complementary observations +would become fundamental. +For instance, while microscopic PBHs might make up +for most of the DM if they are quasi-extremal, there +may also be a high mass tail, which would provide a +unique gravitational wave (GW) signature observable by +future observatories such as LISA [66]. In fact, the de- +gree of quasi-extremality has an impact on the GW sig- +nature in the case of a merger and values of 1 − a (and, +similarly, of ε) as small as 10−9 may be detectable in +the waveform measurable by LISA for extreme mass ra- +tio merger events [67]. Such BH-BH interactions might +also leave unique signatures in the early universe, as + +8 +would be the case for opposite charge BH encounters, +although we leave a more accurate investigation of this +possibility for future work. Another avenue to disentan- +gle Schwarzschild and qPBHs in a potential cosmological +observation is to determine the PBH mass independently, +which can be achieved by e.g., gravitational direct detec- +tion [68] and other direct detection techniques [23]. +In summary, in this work we have shown that light, +quasi-extremal PBHs can be the DM and argued that +with the help of complementary GW observations, an +accurate determination of their characteristics might be +within reach. +The results found here are general and +conservative, and should be taken as the basis for future, +model-specific studies. +ACKNOWLEDGEMENTS +We thank Marco Hufnagel for very useful discussions. +ML is supported by an F.R.S.-FNRS fellowship. +Appendix A: More details on the properties of +four-dimensional and higher-dimensional black holes +In this appendix we collect some useful formulae +that pertain to the properties of 4D as well as higher- +dimensional Schwarzschild BHs. +1. +4D Schwarzschild black holes +The Schwarzschild radius is +rs = 2GM +c2 += 2.95 M +M⊙ +km , +(A1) +where the Planck mass is given by +MP = +� +ℏc +G ≃ 1.21019 +c2 +GeV ≃ 2.2 × 10−8 kg . +(A2) +The Hawking temperature is given by +TH = +ℏc3 +8πk GM ≃ 2.5 × 1021 +�1 gr +M +� +eV . +(A3) +The decay rate due to Hawking radiation to massless +constituents is given by the a Stefan-Boltzmann-like for- +mula +dM +dt = −4πσr2 +s T 4 +H = − +ℏc6 +30 · 83πG2 +1 +M 2 , +(A4) +where +σ = π2 +60 +k4 +c2ℏ3 +(A5) +is the standard Stefan-Boltzmann coefficient. This for- +mula is also valid for massive particles emitted, provided +their mass mc2 ≪ TH. For the Standard Model (SM) +of particle physics plus Gravity, this means photons and +gravitons, For M ≫ 2.5 × 1018 g we can neglect there- +fore the emission of other SM particles. For 5 × 1012 g +≪ M ≪ 2.5 × 1018 g, one should also include the three +SM neutrinos. For 2.5 × 1010 g ≪ M ≪ 5 × 1012 g one +should include electron emission, and so on. +Solving Eq. (A4) we obtain for the evolution of the +mass +M(t) = +� +M 3 +0 − +ℏc6 +10 · 83πG2 t +� 1 +3 +(A6) +and therefore the evaporation time tev is given by +ctev = 10 · 83 G2M 3 +ℏc5 += 640 +ℏG r3 +s = 640M 2 +P +ℏ2 r3 +s . +(A7) +The evaporation formulae above are assuming massless +photons. +Including gravitons doubles the rate. +Grey- +body factors are ignored as they are not important for +standard Schwarzschild BHs as absorption cross sections +are geometrical in the IR and suppressed in the UV +regime. +2. +General dimension d ≥ 4 +We now move to d spacetime dimensions with d ≥ 4 +and we set ℏ = c = 1. In this case, the definitions of +Schwarzschild radius and Hawking temperature can be +generalized as +rd−3 +s += 2GdM +and +kTH = d − 3 +4πrs +, +(A8) +while the decay rate due to ”massless” Hawking radiation +is given by +− dM +dt = Ωd−2 rd−2 +s +σd(kTH)d = Sd +r2s += +Sd +(2GdM) +2 +d−3 +(A9) +with +Ωd−2 ≡ 2π +d−1 +2 +Γ +� d−1 +2 +� , +Sd ≡ Ωd−2σd +�d − 3 +4π +�d +(A10) +and σd is the analogue of the Stefan-Boltzman coefficient +in d dimensions. +Solving Eq. (A9) we obtain +(2Gd) +2 +d−1 M(t) = +�� +2GdM +d−1 +2 +0 +� +2 +d−3 − d − 1 +d − 3Sdt +� d−3 +d−1 +(A11) +and +t(d) +ev = d − 3 +d − 1 +� +2GdM +d−1 +2 +0 +� +2 +d−3 +Sd +. +(A12) + +9 +Consider now the d − 4 ≡ n extra dimensions to be com- +pactified on T n with all radii equal to R for simplicity. +In that case the 4D, G, and d-dimensional Newton con- +stants, Gd, are related as +Gd = G (2πR)d−4 . +(A13) +We may rewrite the d-dimensional evaporation time in +this case as +t(d) +evap(M) = d − 3 +d − 1 +� +2G(2πR)d−4M +d−1 +2 +� +2 +d−3 +Sd +(A14) +from which it follows that +t(d) +ev (M) +t(4) +ev (M) += 3(d − 3) +(d − 1) +S4 +Sd +� πR +GM +�2 (d−4) +(d−3) += 3(d − 3) +(d − 1) +S4 +Sd +�2πR +rs +�2(d−4) +. +(A15) +Simple physical arguments indicate that when the BH is +much smaller in size than R, i.e., with +rs ≪ R , +(A16) +then it behaves as a d-dimensional BH. From Eq. (A8) +we obtain that kTH ≫ +1 +R and therefore the BH can ra- +diate all Kaluza-Klein (KK) modes of the graviton and +other fields. Its lifetime from Eq. (A15) is much longer +than a 4D BH of the same mass. Moreover, if rs ≪ R +then during the evaporation process, the horizon radius +becomes smaller and smaller and the whole evaporation +process happens in the d-dimensional regime. +On the other hand for the BH to be semi-classical, we +must have +Gdr2−d +s +≪ 1 +⇒ +G +(2πR)2 +�2πR +rs +�d−2 +≪ 1 , +(A17) +which implies the inequalities +1 ≪ t(d) +ev (M) +t(4) +ev (M) +≪ +�(2πR)2 +G +�2 d−4 +d−2 +→ +�2πcMP R +ℏ +�4 d−4 +d−2 +. +(A18) +In the extreme case, R ≃ 1 µm, we obtain +c +ℏMP R ≃ 1028 . +If on the other hand we have a BH with a horizon +radius rs ≫ R, in this case the BH behaves as 4D BH. +The temperature is much smaller than the KK mass scale +and none of the KK states can be emitted. +While it +is evaporating, it will start doing so by using the 4D +formula, but as its horizon radius becomes smaller than +R, then it starts evaporating as a d-dimensional BH. The +transition mass M∗ is given by rs = R, +2G M∗ = R +⇒ +M∗ = MP R +2 +MP +(A19) +with +MP R ≲ 1028 . +(A20) +In the extreme case of R = 1 µm, we obtain +M∗ ≃ 1023 gr . +(A21) +Simplifying, we assume that the evaporation process +happens as 4D, until M reduces to M∗ and as higher d +when M < M∗, and we obtain, +M(t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +M 3 − 3S4t +(2G)2 +� 1 +3 +, +0 < t ≤ t∗, +� +M +d−1 +d−3 +∗ +− d − 1 +d − 3 +Sd(t − t∗) +(2Gd) +2 +d−3 +� d−3 +d−1 +, +t∗ < t ≤ T(M) +(A22) +where t∗ is the time for the BH to reach the transition +mass M∗ +MP t∗ = 2 +S4 +� M 3 +M 3 +P +− M 3 +∗ +M 3 +P +� +(A23) +and T(M) is the total evaporation time +T(M) = t∗ + d − 3 +d − 1 +(2Gd) +2 +d−3 M +d−1 +d−3 +∗ +Sd +. +(A24) +When M ≫ M∗, then +t∗ +T − t∗ +≃ (d − 1) +3(d − 3) +Sd +S4 +�2GM +R +�3 += (d − 1) +3(d − 3) +Sd +S4 +�rs +R +�3(d−3) +≫ 1 +(A25) +and essentially, the decay time is given by t∗ which is +approximately equal to the 4D evaporation time +T(M) ≃ t∗ ≃ tevap ≡ (2G)2M 3 +3S4 +. +(A26) +If on the other hand the mass M < M∗ then the evapo- +ration is higher-dimensional and in that case +T(M) = t(d) +ev ≡ d − 3 +d − 1 +(2Gd) +2 +d−3 M +d−1 +d−3 +Sd +(A27) +We conclude that “large” BHs M ≫ M∗ have a 4D decay +time, while “small” BHs, M ≪ M∗ have a higher dimen- +sional decay time given in Eq. 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D 102, 072003 (2020), +arXiv:1903.00492 [hep-ph]. + diff --git a/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/load_file.txt b/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6df61a6a3f0a9c2a3dfc6a7cd0b079e7a7351757 --- /dev/null +++ b/vtFPT4oBgHgl3EQf-TW6/content/tmp_files/load_file.txt @@ -0,0 +1,752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf,len=751 +page_content='Quasi-extremal primordial black holes are a viable dark matter candidate Jose A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' de Freitas Pacheco,1 Elias Kiritsis,2, 3 Matteo Lucca,4 and Joseph Silk5, 6, 7 1Universit´e de la Cˆote d’Azur - Observatoire de la Cˆote d’Azur, Bd de l’Observatoire, 06304 Nice Cedex, France 2Universit´e Paris Cit´e, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France 3Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics, Department of Physics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Box 2208, University of Crete, 70013, Heraklion, Greece 4Service de Physique Th´eorique, Universit´e Libre de Bruxelles, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 225, B-1050 Brussels, Belgium 5Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne Universities), F-75014, Paris, France 6Department of Physics and Astronomy, The Johns Hopkins University Homewood Campus, Baltimore, MD 21218, USA 7BIPAC, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK Black hole evaporation is generally considered inevitable for low-mass black holes, yet there is no confirmation of this remarkable hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Here, we propose a phenomenological model that appeals to the possible survival of light quasi-extremal primordial black holes as a significant dark matter component and show that the related cosmological and astrophysical constraints disappear for reasonable degrees of quasi-extremality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The results obtained are general, conservative and should be taken as a proof of principle for future, model-specific analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' INTRODUCTION Since first postulated in the late ‘60s [1–4], primor- dial black holes (PBHs) are regarded as a viable dark matter (DM) candidate, albeit over a highly constrained mass range (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [5–8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The most appealing exam- ple is given by a range roughly between 1017 and 1022 g, bounded from below by constraints coming from the im- pact of the energy injection following Hawking evapora- tion of the PBHs [9] on observables such as cosmic mi- crowave background (CMB) anisotropies [10–13], cosmic rays [14], and 21 cm lines [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' These bounds on PBH evaporation are, however, de- rived under the assumption that the PBHs are non- rotating and neutral, which maximizes their evapora- tion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, it is well known that if the BHs were charged and/or spinning, their Hawking temper- ature would decrease, and consequently so would their mass loss rate and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the limit where the temperature approaches zero one has what are referred to as quasi-extremal PBHs (qPBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Since increasing degrees of quasi-extremality can be reached, this implies that the aforementioned mass range could be extended to lower masses by decreasing Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Nevertheless, the formation and survivability of qPBHs is clearly a contentious issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The acquisition of ex- treme spin-to-mass ratios is astrophysically forbidden for BH growth by accretion in thin [16] or thick disks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Even if the PBHs had a spin at formation, it would be lost more efficiently than its mass [18], making quasi- extremality impossible to obtain over extended periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' A similar discussion also applies to the case of charged BHs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' One can, however, invoke other processes to justify the existence of qPBHs at lower masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For instance, accre- tion and Schwinger pair production can reduce the BH charge Q, but Hawking evaporation can counter these effects and augment the charge [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Such relics have of- ten been considered as DM candidates [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Indeed, small quantized BHs have a fundamental stable state de- fined by a mass equal to the Planck value and a spin J = ℏ [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We emphasize furthermore that PBHs can form deep in the radiation era where stabilized qPBHs are possible DM candidates if their fractional abundance at formation is as low as ∼ 10−25, and application of ex- treme value statistics leads to the expectation that some of the DM may plausibly include a qPBH component [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The motivation for the rarity of extremely light qPBHs comes from the equilibrium charge distribution [27] once the emission of charged particles of random sign is in- cluded, P(Q) ∼ exp[−4πα(Q/e)2], where α is the elec- tromagnetic coupling and the PBHs rms charge satisfies Q/e = 1/ √ 8πα ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Also PBHs living in an higher- dimensional space are DM candidates [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Provided that the scale of the extra dimension is of the order of the gravitational radius, we show below that such BHs may be quasi-extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' It is therefore not unreasonable to expect qPBHs to exist in a realistic cosmological scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Yet, the impact that such quasi-extremality would have on the commonly imposed cosmological and astrophysical constraints on PBH evaporation has not been systematically considered in the literature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Here we address this task and propose a very general phenomenlogical analysis to be taken as a proof of principle for future, more specific studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' As a result, in our simplified and yet conservative scenario, we find that for values of the quasi-extremality parameter ε (defined in terms of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', the PBH charge Q and mass M as ε = 1 − Q2/M 2) lower than ≲ 10−3 all cosmological and astrophysical constraints allow for PBHs to make up for the totality of the DM, and that they become stable over cosmological times for masses as low as ∼ 1011 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This implies that for sufficiently low values of ε, qPBHs are a viable DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' II we discuss both how such levels of quasi-extremality can be reached and maintained over cosmic times, and how they affect the standard picture of Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' III we overview the many cosmological and as- trophysical observables affected by the presence of this ULB-TH/23-01 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='13215v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='CO] 30 Jan 2023 2 quasi-extremality and suggest simple ways to recast ex- isting bounds in the quasi-extremal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' IV we present the resulting constraints on qPBHs as a function of both the quasi-extremality parameter and the frac- tional abundance of the PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' V we conclude with a summary and closing remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' QUASI-EXTREMAL PRIMORDIAL BLACK HOLES In this section we provide a brief overview of scenar- ios that can generate quasi-extremal BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We do so by highlighting the general features that they share and how the analysis presented here can be consider as a proof of principle for other, more specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We further- more also discuss how this general scenario would affect the standard picture of BH evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Representative examples A first well-known example of qPBHs is given by highly charged BHs, in particular Reissner-Nordstrom (RN) BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In this case, to parametrize the degree of quasi- extremality we can define the parameter ε such that1 1 − Q2 M 2 = ε2 ≪ 1 , (1) where M is the mass and Q is the electric charge of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' RN BHs have two horizons: the outer, corresponding to the event horizon, and the inner, the Cauchy horizon, defined by the zeros of the lapse function, that is r± = M � 1 ± � 1 − Q2 M 2 � = M(1 ± ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (2) The size of the outer horizon should never be smaller than the BH-associated Compton wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This condition gives the Planck mass as the smallest PBH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The notion of the dual horizon also applies to Kerr BHs, which represent a second possible qPBH solution should they spin to a high degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Here the similar hori- zon structure is now characterized by r± = M � 1 ± � 1 − a2 M 2 � = M(1 ± ε) , (3) where a is the dimensionless spin parameter (related to the angular momentum J via a = J/M), and, analo- gously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (1), we can then also define 1 − a2 M 2 = ε2 , (4) 1 Here and henceforth we assume G = c = ℏ = kB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' which shows a similar mass dependence of the relation between ε and the model-specific parameters Q and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' It is therefore clear that much of the model dependence of the aforementioned scenarios can be captured by the single parameter ε, which represents the degree of ex- tremality of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the following discussion we will then only make use of ε so as to be as general as possible in our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In a realistic cosmological scenario, however, it is well known that any charge or spin would be lost very quickly by any BH population of primordial origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Quasi- extremal primordial RN or Kerr BHs are therefore not expected to exist today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Based on these models, it is nev- ertheless possible to envision a scenario where the BH is charged, but instead of standard electromagnetism (EM) the BH is charged under a generic EM-like dark charge whose carriers are always much heavier than the temper- ature of the BH [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In this way, one obtains the same mathematical setup as for a RN BH, but with the dif- ference that the charge Q does not get evaporated away from the BH and remains therefore constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (1), however, one can infer that a constant Q does not necessarily imply a constant ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, since initially the charge will always be smaller than the mass of the BH, ε will always be larger than zero and the BH will radiate its mass at the rate discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Nevertheless, the smaller the mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', the more the BH evaporates) the more ε will approach the zero value and the slower the mass loss rate becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This means that a constant charge with Q < M leads to a BH that naturally approaches extremality over cosmic times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' On the other hand, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (1) shows that if both Q and M are radiated away at the same rate, ε stays constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This means that in a setup where the initial charge-to- mass ratio is very close to unity and both quantities get radiated at the same rate, the BH can maintain extremal- ity indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' However, contrary to the constant-charge example mentioned above, this scenario would require a larger degree of fine-tuning, as one would need to fix the characteristics of the charged dark particles to be exactly such that the BH loses charge and mass at the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Another possibility to obtain (and maintain) quasi- extremality that does not depend on charge or spin but that presents very similar features in terms of ε is given by higher dimensional BHs [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' One could, in fact, consider the 5D gravitational theory compactified on a circle of radius R with R ≲ 1 µm, although the notion of quasi-extremality is generalizable to d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In that case, BHs with a horizon size smaller than R behave as 5D BHs while those with size larger then R as 4D BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' One can then show (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' A) that the evaporation of all Kaluza-Klein (KK) modes of a d-dimensional BH leads to an effective degree of extremality ε−4 eff = 3(d − 3) (d − 1) S4 Sd �2πR rs �2(d−4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (5) A “large” BH with rs ≫ R evaporates following the 4D decay equation until its horizon becomes rs ≃ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' From 3 that point on, it decays as a higher-dimensional BH at a rate that is slower than in 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This implies that a higher-dimensional BH would tend to quasi-extremality the more it evaporates, qualitatively just like a constant- charge BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In summary, in the discussion above, we have high- lighted different ways to justify the existence of qPBHs which can all be phenomenologically described by the evolution of ε in the respective scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For simplicity, hereafter we will solely focus on the aforementioned (RN BH) case with a constant ε value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' With respect to the other possibilities, this choice is conservative in the sense that, for the same initial value of ε, in the other scenarios the degree of extremality would only increase and hence they would be covered by the results obtained for the constant ε case (see [23, 24, 28, 29] for related but rel- atively limited discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Furthermore, we point out that, although based on a slightly less realistic scenario, ours has to be taken as a useful proof of principle to be applied to more specific examples in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Evaporation Given these possible sources of quasi-extremality, it is interesting to consider how one might observe the decay of these quasi-extremal BHs and set constraints on their modified evaporation emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The key quantity that is modified by the quasi- extremality of the BHs is their evaporation temperature T, which now reads (assuming e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', a RN BH, but with no loss of generality) T = 1 4πr+ � 1 − Q2 r2 + � = 1 8πM 22 ε (1 + ε)2 , (6) where ε encapsulates the deviations from the standard Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Once the temperature is defined, it becomes possible to determine the luminosity L of the BH via to the Stefan-Boltzmann black body formula2, L = AσT 4 ∝ r2 +T 4 ∝ 26 ε4 (1 + ε)6M 2 , (7) where A = 4πr2 + is the area of the BH and σ is the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This simple dependence of the luminosity is however strictly speaking only valid as long as the BH evaporates at a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Neverthe- less, since different particles can be emitted at different BH temperatures, it is more convenient to interpret the 2 Near extremality the Stefan-Boltzmann formula is modified by important grey-body factors that tend to suppress emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We do not consider such factors and therefore our formulae should be considered as upper bounds on Hawking emission near ex- tremality in the RN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the higher-dimensional case such factors are not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' luminosity as the energy emission rate with explicit de- pendence on the mass loss rate dM/dt, such that L = −dM dt 26 ε4 (1 + ε)6 , (8) where the ε dependence needs to be introduced for con- sistency with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The mass loss rate of an evaporating BH is commonly defined in terms of the total energy carried away by the emitted particles (due to energy conservation argu- ments), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', following the relation dM dt = − � � j dNj dtdE EdE , (9) where dNj/dtdE is the number of emitted particles j of spin s in the energy interval between (E, E + dE) and is defined as dNj dtdE = 1 2π Γj e(E−µj)/T − (−1)2sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (10) Here, Γj is the dimensionless absorption probability of the given emitted species, which, in full generality, de- pends on both M and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The presence of charge and spin can in fact enhance or reduce the probability of charged particles or particles with spin (mis-)aligned with that of the BH to be emitted from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For stan- dard RN BHs charged under EM, [27] found that the impact of the charge on Γj is of the order of a few per- cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (10), µj refers to the chemical potential of a given emitted particle and generally depends on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For instance, in the case of standard charged BHs, it would take the form µj ∝ qj � (1 − ε)/(1 + ε), where qj is the charge of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In our simplified scenario, how- ever, we assume that the particles carrying the charge of the quasi-extremal BH are not emitted from the BH at all (or at least very slowly) and we can therefore neglect these ε-dependent contributions to Γj and µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Under this simplifying assumption, the only way ε af- fects the mass loss rate is via the exponential dependence of dNj/dtdQ on the BH temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This determines what particles are kinematically available at a given tem- perature T and it therefore makes sense for it to be de- pendent on the temperature of the system only, regard- less of the characteristics of the BH reaching that tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' One can then simply extend the validity of the results found in [30, 31] according to which dM dt = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='34 × 1025 f(T) M 2 g/s , (11) where f(T) defines the number of emitted species and can be expressed as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (9) of [31] (see also [11, 12] for additional details, updated coefficients and contributions 4 beyond the QCD phase transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='3 With this definition of the mass loss rate it is then possible to compute the lifetime of the BH by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Following again [31], one obtains tev = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='24 × 10−27 M 3 f(T) (1 + ε)6 26 ε4 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (12) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' IMPACT ON THE OBSERVABLES Once the mass loss rate due to the PBH evaporation and the related luminosity have been defined, it is pos- sible to analyse how the emission of particles from the PBH affects various cosmological and astrophysical ob- servables such as the CMB anisotropies as well as the cosmic and γ-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Since all of these probes are sensitive to different epochs of the universe, they also con- strain different mass ranges, allowing us to cover a wide region of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the following sections, we describe all of the constraints and explain how they are affected by the presence of evaporating qPBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' BBN The first observable we focus on is Big Bang Nu- cleosynthesis (BBN), which covers the period of light- element formation, such as deuterium and helium, in the early universe [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The predictions of standard BBN are in extremely good agreement with measurements of the corresponding abundances in the first galaxies, where galactic dynamics and star formation have not had the time to affect the primordial abundances yet [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In- ferred quantities such as the baryon energy density, the baryon-to-photon ratio and the primordial helium abun- dance are also consistent with CMB measurements [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Given the success of the standard BBN model, this probe has been often employed to constrain beyond-the- standard-model (BSM) physics, such as annihilating or decaying DM [35, 36] or PBH evaporation [13, 37], typ- ically delivering the most stringent constraints on these types of models prior to recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, BBN can constrain BSM models in a variety of ways, from the impact that they might have on the expansion of the uni- verse (changing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', the number of relativistic degrees of freedom) to the photo-disintegration of the light elements after BBN is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Precisely this richness of constraints, however, prevents us from deriving simple and general limits that can be 3 Concretely, focusing for instance on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 10 of [12] for a graphical representation, the only aspect of the plot that would be modified by the presence of a non-zero ε would be the relation between the two horizontal axis, reporting the PBH mass M and the corresponding T values, with the latter being shifted more and more to the left the higher the value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' recast for any value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This is due to the fact that, for instance, ε affects both the overall and the relative amount of injected species (via the modification to f(T)) as well as the lifetime of the PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Therefore, different aspects of the standard BBN picture might be modified in non-trivial ways, affecting the magnitude and shape of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' BBN bounds on qPBH evaporation would then have to be derived with dedicated analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For this reason, the accurate inclusion of the BBN con- straints in the following discussion goes beyond the proof- of-principle type of study conducted here and will not be considered any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We note, however, that, albeit the scaling of the constraints is not directly proportional to ε as for the probes discussed below, we do expect a significant suppression of the constraints the lower the value is of ε and that the results of this work will not be affected by the non-inclusion of the BBN constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' CMB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' CMB anisotropies CMB anisotropies are very well known to be affected by exotic energy injections during the dark ages (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [10] for a thorough discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, in that period of the thermal history of the universe, the cos- mic medium was almost perfectly neutral, allowing the CMB photons to travel straight from the last scatter- ing surface (at z ≃ 1100) to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Any injection of parti- cles with enough energy to ionize the abundant hydrogen atoms would have increased the amount of free electrons, thereby enhancing the probability of further scattering of the CMB photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This modification of the so-called visibility function would in turn affect the shape of the CMB anisotropy power spectra (both temperature and polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Since the observed spectra are in perfect agreement with the ΛCDM model in the absence of any energy injection [34], the CMB anisotropies can be used to constrain processes such as the evaporation of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In order to estimate the extent to which PBH evapora- tion affects the CMB anisotropies one needs to determine the energy injection rate, which in this case is given by dE dtdV ���� inj = ρcdmfPBH L M , (13) where fPBH is the (primordial) fractional abundance of PBHs with respect to the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This injected energy does not, however, necessarily coincide with the effectively de- posited energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, for instance, part of the injected energy might be in form of non-electromagnetically in- teracting particles and not all of it is spent to ionize the medium (some of this energy would heat up or excite the plasma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' These contributions are commonly taken into account by deposition efficiency feff and deposition fraction per channel χc, respectively [10–12, 38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Ex- 5 plicitly, this implies dE dtdV ���� dep,c = dE dtdV ���� inj feff χc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (14) A graphical representation of the heating rate due to PBH evaporation is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 5 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The con- sequent impact of PBH evaporation on the free electron fraction can be seen in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 6 of [10]4, while that in relation to the CMB power spectra can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 6 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Some important remarks can be drawn from the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' First of all, the majority of the energy injection takes place around the lifetime of the PBH, similarly to the DM decay scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This means that the injection time can be roughly approximated to coincide with the lifetime of the PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Secondly, only PBHs with masses larger than 1013 g evaporate after recombination and can therefore be constrained with CMB anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Based on the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' II B, the energy depo- sition rate has a dependence on the PBH parameters of the form dE dtdV ���� dep ∝ fPBH f(T) 26 ε4 M 3 (1 + ε)6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (15) In the ε = 1 case, f(T) is almost constant for masses above 1013 g (it varies at most by a factor 3, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 10 of [12]) meaning that the energy deposition rate has a simple dependence of the form fPBH/M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This implies a proportionality of the constraints on the PBH abundance as M 3, which is perfectly recovered in the bounds shown in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' On the other hand, the simple proportionality of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (15) also allows us to take into account the contri- bution of ε by simply rescaling the bounds on fPBH by a factor f(Tε=1)(1 + ε)6/(f(T)26ε4), where f(Tε=1) is the value of f(T) in the ε = 1 case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', for a Schwarzschild BH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The overall order of magnitude of the rescaling is given by the chosen value of ε, with the f(T) ratio in- troducing a further enhancement of at most an order of a few (and never more than ten).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This enables us to recast existing constraints, such as the ones derived in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [11–13], for any value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Here we rely on the bounds derived in [13], which are based on Planck 2015 data [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Corresponding con- straints employing Planck 2018 data [34] have been de- rived in [12] and seem to be approximately one order of magnitude more constraining than those of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Never- theless, [12] employed a simplified thermal history and made use of a mock likelihood instead of real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For this reason, and for sake of being conservative, we choose to focus on the results of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Furthermore, compared to other results based on Planck 2015 data such as [11, 42], 4 From the left panel of the figure it becomes clear that heating and ionization rates are rather correlated, so that the heating rate shown in [12] is also indicative of the ionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' the findings of [13] overlap well for PBH masses cor- responding to lifetimes longer than recombination but improve upon them at lower masses, where the PBHs evaporate before recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This is due to a better analysis of the delay between the energy injection and its deposition which extends the constraints down to evap- oration redshifts of the order of z ≃ 5 × 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' CMB spectral distortions In brief, CMB spectral distortions (SDs) are any type of deviation of the CMB energy spectrum from a pure black body [12, 43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' They are typically created by the injection of energy or photons in the thermal bath, although they can also be produced by effects such as the dissipation of acoustic waves and adiabatic cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Com- plementary to the CMB anisotropies, CMB SDs are very sensitive to the thermal history of the universe prior to recombination, up to redshifts of the order of z ≃ 2 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the context of PBH evaporation, as in the case of the CMB anisotropies, their shape is determined by the amount of injected energy defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' A key difference is, however, that for CMB SDs it is the heating rate that needs to be considered and not the ionization rate (which would anyway be zero before recombination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This implies that the same rescaling of existing bounds (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [12, 46–48]) discussed in the previous section can be employed for SDs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Here we follow the results of [47] based on FIRAS data [49], which perfectly overlap with the more recent and exact calculations of [48] at very high evaporation redshifts (or, equivalently, for very low PBH masses), but extend them until recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 21 cm Another cosmological observable that can be employed to constrain the evaporation of PBHs are the 21 cm ab- sorption lines [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' These lines are generated when- ever a neutral hydrogen atom undergoes a spin-flip tran- sition and are therefore a very important tracker of the neutral hydrogen distribution across space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Cosmologically, the probability of this transition to hap- pen is proportional to the relative abundance of the two spin levels, which in turn depends on what is known as the spin temperature TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Since TS is determined by the CMB and gas temperature, any process that affects the latter inevitably modifies also the 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This logic has been applied to constrain several beyond-ΛCDM models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [51] and references therein for a recent overview), and here we focus on the case of PBH evaporation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' As extensively explained in the reference, the relation between energy injection and modified 21 cm signal is dictated by the same equations discussed in the previous section in the context of CMB anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This similarity is qualitatively confirmed, for instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 4 of the reference, where the same 6 fPBH ∝ M 3 proportionality is shown for the 21 cm con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Therefore, in analogy to the previous section also in the 21 cm case we can simply recast the existing bounds of [15] to account for the role of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Diffuse γ-ray background Next we move to constraints of astrophysical origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', focusing on the evaporation of PBHs in the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Firstly, we consider the case of the dif- fuse extra-galactic γ-ray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The idea in this context is then to consider the observed γ-ray fluxes, ob- served by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', Fermi LAT [53] and HAWK [54], and to impose the condition that the flux of photons emitted from the cosmological PBH population does not exceed this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This exercise has been performed in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [37] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 5 therein) including a number of observations and found that this probe is particularly constraining for PBH masses around 1015 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In a subsequent work [55], the authors also computed the constraints on the flux of galactic origin, which however turn out to be subdomi- nant with respect to the extra-galactic counterpart and will therefore be neglected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Since these constraints depend on the flux of photons emitted from the PBHs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', φγ = 1 4π � Lγ nPBH dt ∝ fPBH tev , (16) where Lγ is the emitted luminosity in form of photons and nPBH is the PBH number density, also in this case the constraints have a power-law dependence on ε4/(1 + ε)6, which allows for a straightforward rescaling of the afore- mentioned bounds derived in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Cosmic rays A similar discussion can be also carried out for galac- tic cosmic rays, such as electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The observational difficulty in this case is, however, that low- energy charged particles are significantly affected by the heliosphere of the sun and this limits the amount of infor- mation that can be extracted from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This prob- lem has been overcome with the exit from the heliosphere of the Voyager 1 spacecraft [56] and now it is therefore possible to combine Voyager 1 [57] and AMS-02 [58] data to constrain the cosmic ray flux over an energy range between a few MeV and hundreds of GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' These data sets have been employed by [14] to bound the PBH abundance in the mass range between 5×1014− 3 × 1016 g, where they are also the most constraining to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' As for the γ-rays, also in this case the limits rely on the definition of the flux of particles from the PBHs, so that the same ε rescaling applies here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Diffuse high-energy neutrino background Finally, the observation of the neutrino flux at facilities such as IceCube [59] and Super-Kamiokande [60] would enable us to constrain the PBH abundance in the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' However, so far the current sensitivity of these experiments has not been able to set competitive bounds with respect to the aforementioned ones [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We will therefore not consider these observations in the fol- lowing discussion, but point them out as a promising avenue for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' RESULTS The collection of the cosmological and astrophysical constraints on the PBH abundance discussed in the pre- vious section is summarized in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' There, we also display the BBN constraints derived in [13, 37] for reference (solid green line), although they are not rescaled as the others and are only to be relied upon for the ϵ = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We remark, however, that the con- clusions drawn below do not depend on this limitation of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the figure, the solid lines represent the cases with ε = 1, while dashed, dashed-dotted and dotted lines re- fer respectively to the ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='001 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' As expected, the constraints are significantly suppressed the lower the value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, as argued in the previ- ous section the upper bound on the PBH abundance re- laxes roughly proportionally to ε4/(1+ε)6, which in turns means that the largest mass allowed by evaporation con- strains for fPBH = 1 reduces to approximately 2 × 1016 g for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='1, to 4×1015 g for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='01 and all PBH masses are allowed for ε ≲ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We also show as vertical lines the PBH masses whose lifetime would correspond to the age of the universe (with the same line style as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This sets the threshold above which the PBHs are still present in the universe today (or, alternatively, below which they are already evaporated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' While in the ε = 1 this corresponds to approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='1×1014 g, this value scales as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (12), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', approximately as (ε4/(1 + ε)6)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This means that for ε ∼ 10−3 even PBHs as light as ∼ 1011 g would survive until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The combination of these two conclusions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', that qPBHs with ε ≤ 10−3 can match the correct DM abun- dance and that they would still be present today, opens the door to the interesting possibility that such light qPBHs could be the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Of course, this will need to be developed in the context of more refined qPBH mod- els, but can still act as a useful (conservative) benchmark for such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Interestingly, for relatively small values of ε, the bounds presented in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 1 can be ap- proximated to be on the parameter combination fPBH ϵ4 (neglecting the dependence on f(T) and on the second order ε term, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This allows for a very simple, 7 1012 1014 1016 M [g] 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 fPBH BBN CMB SDs CMB ani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' γ-rays cosmic rays 21 cm tuni = tev ε = 1 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='1 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='01 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='001 1011 1013 1015 1017 M [g] 10−3 10−2 10−1 100 ε fPBH = 1 fPBH = 10−4 fPBH = 10−8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Left panel: Cosmological and astrophysical constraints on the fractional PBH abundance as a function of the PBH mass for different values of the quasi-extremality parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The vertical gray lines represent the PBH masses whose lifetimes correspond to the age of the universe (with the same line styles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The BBN constraints are only shown for reference in the ε = 1 case and are not rescaled for the other values of ε for the reasons explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Right panel: Same as in the left panel but on the ε − M plane for different values of fPBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' order-of-magnitude reinterpretation of the constraints for any value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Furthermore, it also allows us to present the limits in the ε − M plane for fixed values of fPBH, which can be useful for realistic models where the qPBHs are predicted to be a given sub-component of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We perform this exercise (with the exact dependence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (15)) in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 1 for the representa- tive cases of fPBH = 1, 10−4, 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The figure confirms the aforementioned discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' SUMMARY AND DISCUSSION The analysis carried out here focuses on PBHs in the mass range between ∼ 1010 − 1017 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The allowed abun- dance of such PBHs is mostly constrained by the impact of their evaporation on cosmological and astrophysical observables such as the CMB, the 21 cm lines and cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Nevertheless, these stringent limits are derived as- suming non-spinning, neutral (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', Schwarzschild) BHs, a scenario that maximizes the evaporation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' If one assumes instead that the BHs are e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', charged, spin- ning or even living in a higher-dimensional space, their evaporation temperature decreases, and consequently so does their luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This approach can be pushed to the limit where the evaporation stops completely, leading to what are known as extremal BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In this work we consider so-called quasi-extremal PBHs and show that indeed the assumption of quasi- extremality can greatly suppress the aforementioned con- straints on the PBH evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Concretely, we anal- yse the case of a general and conservative scenario where the degree of quasi-extremality is captured by a model- independent parameter ε, which we assume to be con- stant for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In the context of charged BHs, for instance, this quasi-extremality parameter would be de- fined as ε = 1−Q2/M 2, where Q and M represent charge and mass of the PBH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' As a result, we find i) that all constraints vanish for ε ≲ 10−3 and ii) that for these values of ε all PBHs with masses larger than ∼ 1011 g would still be present today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The combination of these conclusions implies that such light qPBHs are a viable DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' However, the question of observability remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, given the dependencies on ε of the constraints dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' III we do not expect that upcoming ex- periments such as CMB-S4 [62, 63] and SKA [64] (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', [12, 65] for related forecasts) would be able to sig- nificantly change the current picture since qPBHs with ε ≲ 10−3 would still largely evade them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Further- more, even if a survey did observe a signature compati- ble with the energy injection following the evaporation of PBHs, it would be impossible to disentangle the case of a Schwarzschild PBH population from that of a more abun- dant population of lighter qPBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Therefore, cosmolog- ical and astrophysical probes testing Hawking evapora- tion are a priori not sensitive enough to uniquely prove the existence of qPBHs and complementary observations would become fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For instance, while microscopic PBHs might make up for most of the DM if they are quasi-extremal, there may also be a high mass tail, which would provide a unique gravitational wave (GW) signature observable by future observatories such as LISA [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In fact, the de- gree of quasi-extremality has an impact on the GW sig- nature in the case of a merger and values of 1 − a (and, similarly, of ε) as small as 10−9 may be detectable in the waveform measurable by LISA for extreme mass ra- tio merger events [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Such BH-BH interactions might also leave unique signatures in the early universe, as 8 would be the case for opposite charge BH encounters, although we leave a more accurate investigation of this possibility for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Another avenue to disentan- gle Schwarzschild and qPBHs in a potential cosmological observation is to determine the PBH mass independently, which can be achieved by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', gravitational direct detec- tion [68] and other direct detection techniques [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In summary, in this work we have shown that light, quasi-extremal PBHs can be the DM and argued that with the help of complementary GW observations, an accurate determination of their characteristics might be within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The results found here are general and conservative, and should be taken as the basis for future, model-specific studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Marco Hufnagel for very useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' ML is supported by an F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='-FNRS fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Appendix A: More details on the properties of four-dimensional and higher-dimensional black holes In this appendix we collect some useful formulae that pertain to the properties of 4D as well as higher- dimensional Schwarzschild BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 4D Schwarzschild black holes The Schwarzschild radius is rs = 2GM c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='95 M M⊙ km , (A1) where the Planck mass is given by MP = � ℏc G ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='21019 c2 GeV ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='2 × 10−8 kg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A2) The Hawking temperature is given by TH = ℏc3 8πk GM ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='5 × 1021 �1 gr M � eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A3) The decay rate due to Hawking radiation to massless constituents is given by the a Stefan-Boltzmann-like for- mula dM dt = −4πσr2 s T 4 H = − ℏc6 30 · 83πG2 1 M 2 , (A4) where σ = π2 60 k4 c2ℏ3 (A5) is the standard Stefan-Boltzmann coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' This for- mula is also valid for massive particles emitted, provided their mass mc2 ≪ TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For the Standard Model (SM) of particle physics plus Gravity, this means photons and gravitons, For M ≫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='5 × 1018 g we can neglect there- fore the emission of other SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For 5 × 1012 g ≪ M ≪ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='5 × 1018 g, one should also include the three SM neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='5 × 1010 g ≪ M ≪ 5 × 1012 g one should include electron emission, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A4) we obtain for the evolution of the mass M(t) = � M 3 0 − ℏc6 10 · 83πG2 t � 1 3 (A6) and therefore the evaporation time tev is given by ctev = 10 · 83 G2M 3 ℏc5 = 640 ℏG r3 s = 640M 2 P ℏ2 r3 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A7) The evaporation formulae above are assuming massless photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Including gravitons doubles the rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Grey- body factors are ignored as they are not important for standard Schwarzschild BHs as absorption cross sections are geometrical in the IR and suppressed in the UV regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' General dimension d ≥ 4 We now move to d spacetime dimensions with d ≥ 4 and we set ℏ = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In this case, the definitions of Schwarzschild radius and Hawking temperature can be generalized as rd−3 s = 2GdM and kTH = d − 3 4πrs , (A8) while the decay rate due to ”massless” Hawking radiation is given by − dM dt = Ωd−2 rd−2 s σd(kTH)d = Sd r2s = Sd (2GdM) 2 d−3 (A9) with Ωd−2 ≡ 2π d−1 2 Γ � d−1 2 � , Sd ≡ Ωd−2σd �d − 3 4π �d (A10) and σd is the analogue of the Stefan-Boltzman coefficient in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A9) we obtain (2Gd) 2 d−1 M(t) = �� 2GdM d−1 2 0 � 2 d−3 − d − 1 d − 3Sdt � d−3 d−1 (A11) and t(d) ev = d − 3 d − 1 � 2GdM d−1 2 0 � 2 d−3 Sd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A12) 9 Consider now the d − 4 ≡ n extra dimensions to be com- pactified on T n with all radii equal to R for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' In that case the 4D, G, and d-dimensional Newton con- stants, Gd, are related as Gd = G (2πR)d−4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A13) We may rewrite the d-dimensional evaporation time in this case as t(d) evap(M) = d − 3 d − 1 � 2G(2πR)d−4M d−1 2 � 2 d−3 Sd (A14) from which it follows that t(d) ev (M) t(4) ev (M) = 3(d − 3) (d − 1) S4 Sd � πR GM �2 (d−4) (d−3) = 3(d − 3) (d − 1) S4 Sd �2πR rs �2(d−4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A15) Simple physical arguments indicate that when the BH is much smaller in size than R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=', with rs ≪ R , (A16) then it behaves as a d-dimensional BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A8) we obtain that kTH ≫ 1 R and therefore the BH can ra- diate all Kaluza-Klein (KK) modes of the graviton and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Its lifetime from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A15) is much longer than a 4D BH of the same mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Moreover, if rs ≪ R then during the evaporation process, the horizon radius becomes smaller and smaller and the whole evaporation process happens in the d-dimensional regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' On the other hand for the BH to be semi-classical, we must have Gdr2−d s ≪ 1 ⇒ G (2πR)2 �2πR rs �d−2 ≪ 1 , (A17) which implies the inequalities 1 ≪ t(d) ev (M) t(4) ev (M) ≪ �(2πR)2 G �2 d−4 d−2 → �2πcMP R ℏ �4 d−4 d−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A18) In the extreme case, R ≃ 1 µm, we obtain c ℏMP R ≃ 1028 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' If on the other hand we have a BH with a horizon radius rs ≫ R, in this case the BH behaves as 4D BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The temperature is much smaller than the KK mass scale and none of the KK states can be emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' While it is evaporating, it will start doing so by using the 4D formula, but as its horizon radius becomes smaller than R, then it starts evaporating as a d-dimensional BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' The transition mass M∗ is given by rs = R, 2G M∗ = R ⇒ M∗ = MP R 2 MP (A19) with MP R ≲ 1028 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A20) In the extreme case of R = 1 µm, we obtain M∗ ≃ 1023 gr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A21) Simplifying,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' we assume that the evaporation process happens as 4D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' until M reduces to M∗ and as higher d when M < M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' and we obtain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' M(t) = � � � � � � � � � � � � � � � � M 3 − 3S4t (2G)2 � 1 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 0 < t ≤ t∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' � M d−1 d−3 ∗ − d − 1 d − 3 Sd(t − t∗) (2Gd) 2 d−3 � d−3 d−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' t∗ < t ≤ T(M) (A22) where t∗ is the time for the BH to reach the transition mass M∗ MP t∗ = 2 S4 � M 3 M 3 P − M 3 ∗ M 3 P � (A23) and T(M) is the total evaporation time T(M) = t∗ + d − 3 d − 1 (2Gd) 2 d−3 M d−1 d−3 ∗ Sd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A24) When M ≫ M∗, then t∗ T − t∗ ≃ (d − 1) 3(d − 3) Sd S4 �2GM R �3 = (d − 1) 3(d − 3) Sd S4 �rs R �3(d−3) ≫ 1 (A25) and essentially, the decay time is given by t∗ which is approximately equal to the 4D evaporation time T(M) ≃ t∗ ≃ tevap ≡ (2G)2M 3 3S4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A26) If on the other hand the mass M < M∗ then the evapo- ration is higher-dimensional and in that case T(M) = t(d) ev ≡ d − 3 d − 1 (2Gd) 2 d−3 M d−1 d−3 Sd (A27) We conclude that “large” BHs M ≫ M∗ have a 4D decay time, while “small” BHs, M ≪ M∗ have a higher dimen- sional decay time given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' For a small BH, taking the ratio of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A27) with the 4D one we obtain t(d) ev t∗ ≃ t(d) ev t(4) ev = 3d − 3 d − 1 S4 Sd �2πR rs �2(d−4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (A28) Therefore small BHs have rs ≪ R and are relatively long- lived compared to 4D Schwarzschild BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' We can define an effective extremality parameter ϵeff for small higher-dimensional BHs as ε−4 eff = 3(d − 3) (d − 1) S4 Sd �2πR rs �2(d−4) (A29) by comparing it with 4D RN BHs (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' (12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 10 [1] Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Zel’dovich and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Novikov, “The Hypothesis of Cores Retarded during Expansion and the Hot Cosmo- logical Model,” Soviet Astronomy 10, 602 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' [2] Stephen Hawking, “Gravitationally collapsed objects of very low mass,” Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf'} +page_content=' 152, 75 (1971).' 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Let q be a prime power and Fq be the finite field with q elements. In this +article we investigate the space of unramified automorphic forms for PGLn over the +rational function field defined over Fq (i.e. for P1 defined over Fq). In particular, we +prove that the space of unramified cusp form is trivial and (for n = 3) that the space +of eigenforms is one dimensional. Moreover, we show that there are no nontrivial un- +ramified toroidal forms for PGL3 over P1 and conjecture that the space of all toroidal +automorphic forms is trivial. +1. Introduction +Automorphic forms are central objects in modern number theory, for instance, they +play a key role in the celebrated Langlands program, see eg. [JL70]. The goal of +this work is to probe the space of unramified automorphic forms for PGLn over the +projective line. +Let F be either a local or a global field, such as the function field attached to a non- +singular projective curve. In the context of understanding the absolute Galois group +Gal(F,F), which is a fundamental problem on number theory, automorphic forms play +the following role. As the theory of representation of groups suggests, we may study +Gal(F,F) through its finite dimension representations. Given a finite dimensional rep- +resentation ρ : Gal(F,F) → GLn(k), where k equals to C or Qℓ for ℓ a prime number +different from the characteristic of F, there exists L(s,ρ) an analytic invariant attached +to ρ, its L-function. Hence we might investigate ρ, and thus understand Gal(F,F), by +means of its L-function. For n = 1, this approach is fundamental for the development +of abelian class field theory and the understanding of Gal(F,F)ab. +On the other side, when F is a global field, we might consider A its adelic ring and +the so-called automorphic representations (forms) π of GLn(A). Here is where emerge +the main object of this paper. Attached to each automorphic representation π, we also +have an analytic invariant L(s,π), its L-function. The celebrated Langlands conjectures +predict the existence of a correspondence between the n-dimensional representations +of Gal(F,F) and the automorphic representation of GLn(A) which preserves in some +sense the respective L-functions. We refer [BCdS+03, Chap. 10 and 12] for a com- +plete discussion about how automorphic forms (representations) plays a role in the +Langlands program. +Cusp forms. When F is the function field of a nonsingular projective geometrically +irreducible curve X, automorphic forms with trivial central character and fixed ramifi- +cation can be identified with continuous functions on the double quotient +GLn(F)Z(A)\GLn(A)/K +that are of moderate growth (see Section 2 for precise definitions). Where K stands +for a compact open subgroup of GLn(A) and Z(A) is the center of GLn(A). Hence +1 + +2 +Roberto Alvarenga and Valdir Pereira Júnior +the domain of a cusp automorphic form (see Definition 3.1) with fixed ramification +is discrete modulo the right-action of a compact open subgroup K of GLn(A), which +allows for explicit descriptions, see Lemma 3.5. In this context, roughly speaking, the +geometric Langlands correspondence(cf. [Dri80] and [Laf02]) states the existence of a +bijective map between +(1) the set isomorphism classes of irreducible continuous representations +ρ : Gal(F,F) → GLn(Qℓ) +which are unramified outside a finite number of places and whose determinant +is of finite order and, +(2) the space of Qℓ-valued automorphic representations which are cuspidal and +whose central character is of finite order. +Moreover, above correspondence is the bijection given by global class field theory for +n = 1 and “preserves” the respective L-functions for n > 1. We prove in section 3 that +the space of cusp forms over the projective line is trivial. See Definition 3.1 for the +definition of cusp forms. +HK-eigenforms. Let OA := ∏Ox where the product is taken over all places x of F and +Ox is the ring of integers of the completion of F with respect to x. When K above is +given by GLn(OA), the standard maximal compact open subgroup of GLn(A), we have +the so-called unramified automorphic forms. For global function fields, the behave +of unramified automorphic forms are particularly nice: every unramified automorphic +representation contains a unique unramified automorphic form, or spherical vector, up +to scalar multiples. Therefore the unramified automorphic representations correspond +to certain 1-dimensional representations of the (commutative) spherical Hecke algebra +HK, see Definition 2.1. In other words, unramified automorphic representations corre- +spond to eigenforms of HK and are therefore determined by their eigenvalues under the +action of Hecke operators. We use the explicit description of the graphs of Hecke oper- +ators introduced by Lorscheid in [Lor13a] to parametrize the spaces of HK-eigenforms +when F is the rational function field. +Toroidal forms. Associated to a degree n separable extension of F there exists a max- +imal torus of GLn. The automorphic forms which vanish the integral along that torus, +for every degree n separable extension of F, are called toroidal automorphic forms, cf. +Definition 5.3. When F = Q, these special class of automorphic forms are closed re- +lated to the classical Riemann hypothesis, cf. [Zag81]. When F is the rational function +field over Fq, we prove in section 5 that there does not exist nontrivial (unramified) +toroidal automorphic forms. Moreover, we conjecture that this is the situation for the +whole space of toroidal automorphic forms for every n. +Acknowledgements. The authors would like to thank Mikhail Kapranov for fruitful +exchange of emails about formula (1). The first author was supported by FAPESP +[grant number 2022/09476-7]. +2. Background +In this first section we set up the notation used throughout the paper. Let F be a global +function field defined over a finite field Fq, where q is a prime power. We might regard +F as the function field of a geometrically irreducible smooth projective curve X defined + +On unramified automorphic forms over the projective line +3 +over Fq. We shall make it clear when we would like to specialize X to be the projective +line P1 defined over Fq. The reason for employ both notations is because some of the +definitions and properties which we write through the article is true for an arbitrary +curve. +In the subsequent section we shall assume that X is the projective line P1 defined +over Fq and F its function field. +Let g stand for the genus of X and |X| be the set of closed points of X or, equivalently, +the set of places in F. For x ∈ |X|, we denote by Fx the completion of F at x, by Ox the +ring of integers of Fx, by πx ∈ Ox (we can assume πx ∈ F) a uniformizer of x and by qx +the cardinality of the residue field κ(x) := Ox/(πx) ∼= Fqx. Moreover, we denote by |x| +the degree of x, which is defined to be the extension fields degree [κ(x) : Fq]. In other +words, qx = q|x|. Let | · |x the absolute value of Fx (resp. F) such that |πx|x = q−1 +x , we +call |·|x the local norm for each x ∈ |X.| +Let A be the adele ring of F and A∗ the idele group. We denote OA := ∏Ox, where +the product is taken over all places x of F. The idele norm is the quasi-character +|·| : A∗ → C∗ that sends an idele (ax) ∈ A∗ to the product ∏|ax|x over all local norms. +By the product formula, this defines a quasi-character on the idele class group A∗/F∗. +We might assume Fx being embedded into the adele ring A by sending an element +a ∈ Fx to the adele (ay)y∈|X| with ax = a and ay = 0 for y ̸= x. Not quite compatible +with this embedding, we think of the unit group F∗ +x as a subgroup of the idele group +A∗ by sending an element b of F∗ +x to the idele (by) with bx = b and by = 1 for y ̸= x. +We will explain in case of ambiguity, which of these embeddings we use. +Let G(A) := GLn(A), Z(A) be the center of G(A), G(F) := GLn(F) and K := +GLn(OA) the standard maximal compact open subgroup of G(A). Note that G(A) +comes together with an adelic topology that turns G(A) into a locally compact group. +Hence G(A) is endowed with a Haar measure. We fix the Haar measure on G(A) for +which vol(K) = 1. The topology of G(A) has a neighborhood basis V of the identity +matrix that is given by all subgroups +K′ = ∏ +x∈|X| +K′ +x < ∏ +x∈|X| +Kx = K +where Kx := GLn(Ox), such that for all x ∈ |X| the subgroup K′ +x of Kx is open and +consequently of finite index and such that K +′ +x differs from Kx only for a finite number +of places. +Hecke algebra. Consider the space C0(G(A)) of continuous functions f : G(A) → C. +Such a function is called smooth if it is locally constant. The group G(A) acts on +C0(G(A)) through the right regular representation +ρ : G(A) → Aut(C0(G(A))), +which is defined by right translation of the argument: (g. f)(h) := (ρ(g) f)(h) := f(hg) +for g,h ∈ G(A) and f ∈ C0(G(A)). +Let H be a subgroup of G(A). We say that f ∈C0(G(A)) is left or right H-invariant +if for all h ∈ H and g ∈ G(A), f(hg) = f(g) or f(gh) = f(g), respectively. If f is right +and left H-invariant, it is called bi-H-invariant. + +4 +Roberto Alvarenga and Valdir Pereira Júnior +Definition 2.1. The complex vector space H of all smooth compactly supported func- +tions Φ : G(A) → C together with the convolution product +Φ1 ∗Φ2 : g �−→ +� +G(A) Φ1(gh−1)Φ2(h)dh +for Φ1,Φ2 ∈ H is called the Hecke algebra for G(A). Its elements are called Hecke +operators. +The Hecke algebra H acts on C0(G(A)) by +Φ( f) : g �−→ +� +G(A) Φ(h) f(gh)dh. +We say that a function f ∈C0(G(A)) is H-finite if the space H· f is finite dimensional. +The zero element of H is the zero function, but there is no multiplicative unit. For +K′ ∈ V, we define HK′ to be the subalgebra of all bi-K′-invariant elements. These +subalgebras have multiplicative units. Namely, the normalized characteristic function +ǫK′ := (volK′)−1charK′ acts as the identity on HK′ by convolution. +Definition 2.2. When K′ = K above, we call HK the spherical (unramified) part of H +and its elements are called spherical (unramified) Hecke operators. +Every Φ ∈ H is bi-K′-invariant for some K′ ∈ V, see [Lor08, Lemma 1.4.3 and Prop. +1.4.4]. In particular, H = � +K′∈V HK′. Furthermore, the spherical Hecke algebra HK +can be explicit described as follows. We fix 1 ⩽ r ⩽ n an integer and let x be a place of +F. Let Φx,r stand for the characteristic function of +K +� +πxIr +In−r +� +K +where Ir (resp. In−r) stands for the r×r (resp. (n−r)×(n−r)) identity matrix and the +empty entry in the matrix means a zero entry. Observe that Φx,n is invertible and its +inverse is given by the characteristic function of K(πxIn)−1K. The following theorem, +due to Satake, describes HK as a commutative (almost) polynomial algebra. +Theorem 2.3. Identifying ǫK with 1 ∈ C yields +HK ∼= C[Φx,1,...,Φx,n,Φ−1 +x,n]x∈|X|. +an isomorphism of C-algebras. In particular, HK is commutative. +Proof. See [BCdS+03, Chapter 12, 1.6]. +□ +Automorphic forms. A function f ∈C0(G(A)) is called K-finite if the complex vector +space that is generated by {ρ(k) f}k∈K is finite dimensional. +We embed G(A) ֒→ An2+1 via g �→ (g,det(g)−1). We define a local height |gx|x on +G(Fx) := GLn(Fx) by restricting the height function +(v1,...,vn2+1) �→ max{|v1|x,...,|vn2+1|x} +on Fn2+1 +x +. We note that |gx|x ⩾ 1 and that |gx|x = 1 if gx ∈ Kx. We define the global +height |g| to be the product of the local heights. We say that f ∈ C0(G(A)) is of +moderate growth if there exists constants C and N such that +| f(g)|C ⩽ C|g|N + +On unramified automorphic forms over the projective line +5 +for all g ∈ G(A). +Let V ⊂ C0(G(A)) be a sub-representation of G(A) on C0(G(A)). For K′ ∈ V, let +V K′ be the subspace of all f ∈ V that are right K′-invariant i.e. such that ρ(k′) f = f for +all k′ ∈ K′. We say that the representation V is admissible if V K′ is finite dimensional +for every K′ ∈ V. In the following we take V = ρ(G(A)) f for some f ∈ C0(G(A)). +Definition 2.4. The space of automorphic forms A (with trivial central character) is +the complex vector space of all functions f ∈ C0(G(A)) which are smooth, K-finite, +of moderate growth, left G(F)Z(A)-invariant and such that the smooth representation +ρ(G(A)) f is admissible. Its elements are called automorphic forms. +Remark 2.5. We are actually considering automorphic forms for PGLn. This is equiv- +alent to consider in previous definition G = PGLn and remove the left Z(A)-invariance. +However, for technical reasons, we maintain above definition and consider the Hecke +algebra over GLn. +Lemma 2.6. A function f ∈ C0(G(A)) is smooth and K-finite if and only if there is a +K′ ∈ V such that f is right K′-invariant. In particular, +V = +� +K′∈V +V K′ +for every subspace V ⊆ A. +Proof. See [Lor08, Lemma 1.3.2]. +□ +Hence, from previous lemma, functions in AK′ can be identified with functions on +the double quotient +G(F)Z(A)\G(A)/K′ +that are of moderate growth. We call AK by the space of unramified automorphic +forms. +Remark 2.7. Although the subspace AK′ of A is not stable under the action of G(A), it +carries the induced action of the Hecke subalgebra HK′. The Proposition 4.4 re-writes +the action of Hecke operators on automorphic forms given by previous integral as a +finite sum. This is fundamental to define the graphs of Hecke operators which shall be +applied to parametrize the space of HKx-eigenforms, see section 4. +3. Cusp forms +In this section we investigate the space of unramified cusp forms for PGLn over the +projective line. +Definition 3.1. An automorphic form f ∈ A is a cusp form (or cuspidal) if +� +U(F)\U(A) f(ug)du = 0 +for all g ∈ G(A) and all unipotent radicals subgroups U of all standard parabolic sub- +groups P of G(A). We denote the whole space of cusp forms by A0. If f ∈ AK is a +cusp form we call it an unramified cusp form and denote the whole space of unramified +cusp forms by AK +0 . + +6 +Roberto Alvarenga and Valdir Pereira Júnior +Geometric interpretation. Let BunnX be the set of isomorphism classes of rank n +vector bundles on X. Let PBunn X stands for the set of isomorphism classes of rank n +projective vector bundles. The vector bundles E,E′ ∈ Bunn X are in the same class in +PBunn X if there exists a line bundle L ∈ PicX such that E ∼= E′ ⊗L. In this case we +denote E = E′ in PBunnX. +It is well known that the double quotient GL1(F)\GL1(A)/GL1(OA) is in bijection +with the set of classes of divisors on X. Hence, it yields the following bijection +GL1(F)\GL1(A)/GL1(OA) ←→ Bun1X +since the set of classes of divisors on X are in correspondence with the line bundles +over X. The following theorem, due to Weil, extends above bijection to higher rank +vector bundles on X. +Theorem 3.2 (Weil). For every n ⩾ 1, there exists a bijection +GLn(F)\GLn(A)/GLn(OA) ←→ Bunn(X) +g �−→ Eg +such that Eg ⊗La = Eag for a ∈ A× and La the correspondent line bundle. Moreover, +degEg = deg(detg). +Proof. Its well know that vector bundles are completely determined by its transition +maps. The bijection follows, essentially, by associate to a vector bundle the adelic +matrix given by the stalks of its transition maps. See either [Lor08, Lemma 5.1.6] or +[Fre04, Lemma 3.1] for a complete proof. +□ +Corollary 3.3. For every n ⩾ 1, there exists a bijection +GLn(F)Z(A)\GLn(A)/GLn(OA) ←→ PBunn(X) +where Z(A) is the center of GLn(A) +Remark 3.4. Therefore, due to Weil’s theorem, we can interpret unramified automor- +phic forms AK as the space of complex valued functions on PBunn(X) with some +moderate growth condition. +The next lemma reinterprets the cuspidal condition in geometric terms. Despite it +is already known and largely used in the literature, see for example [Kap97, Sec. 2] +and [Bum97, pag. 296], we could not find a proof. Thus we sketch its proof in the +following for sake of completeness. We first observe, following [HC68, Lemma 3], +that f ∈ AK is cuspidal if +� +U(F)\U(A) f(ug)du = 0 +for all g ∈ G(A) and all unipotent radicals subgroups U of all maximal parabolic sub- +groups P of G(A). +We fix P a maximal parabolic subgroup of G. It is well known that there exists +integers r,s > 0 with r + s = n such that P = UM where U is the unipotent of P and +radical M ∼= GLr ×GLs is its Levi subgroup, cf. [Bor91, Cor. 14.19]. One says that P +is a parabolic maximal subgroup of type (r,s). Observe furthermore that an element of +U has the following form +� +Ir×r +hs×r +0 +Is×s +� + +On unramified automorphic forms over the projective line +7 +where Ir×r (resp. Is×s) stands for the r × r (resp. s × s) identity matrix and hs×r is a +matrix with r rows and s columns. That is, U ∼= Mr,s where Mr,s stands for the additive +group of matrices with r rows and s columns. +From Iwasawa decomposition, G(A) = P(A)K where P is the above fixed maximal +parabolic subgroup. Given g ∈ G(A), we write g = xmk where x ∈ U(A),m ∈ M(A) +and k ∈ K. Hence, f ∈ AK is an unramified cusp form if +� +U(F)\U(A) f(ug)du = +� +U(F)\U(A) f(uxmk)du = +� +U(F)\U(A) f(ym)dy = 0. +for all m ∈ M(A) and where y = ux ∈ U. We write +y = +� +Ir×r +hs×r +0 +Is×s +� +and +m = +� +hr +0 +0 +hs +� +where hr ∈ GLr(A), hs ∈ GLs(A) and hs×r ∈ Mr,s(A). Let F ∈ Bunr X (resp. G ∈ +BunsX) be the vector bundle which corresponds to hr (resp. hs) via Theorem 3.2. +Applying once again Weil’s theorem, ym ∈ G(A) corresponds to an extension of G by +F, see [LP97, Sec. 7.3] and [Sha13, Ex. 6.6]. Therefore, considering an unramified +automorphic form as a complex valuated map from PBunn X as observed in 3.4, above +discussion implies the following lemma. +Lemma 3.5. An unramified automorphic form f ∈ AK is a cusp form if for any integers +r,s > 0 with r +s = n and any vector bundles F ∈ Bunr X, G ∈ BunsX, +∑ +E∈Ext(F,G) +f(E) = 0, +(1) +where we abuse the notation and write E meant the middle term of the correspondent +exact sequence. +Theorem 3.6. Let F be the field of rational functions over Fq i.e. the function field of +P1 the projective line defined over Fq. Then AK +0 is trivial for every n ⩾ 2. +Proof. We denote the structural sheaf of P1 simply by O, i.e. O := OP1. Hence the +canonical sheaf ω := ωP1 of P1 is isomorphic to O(−2). +Let F ∈ Bunr X and G ∈ BunsX, for some integers r,s > 0. The Serre duality (see +[Har77, Sec. III.7]) yields +Ext1(F,G)∨ ∼= Hom(G,F ⊗O(−2)) = H0(P1,F ⊗O(−2)⊗G∨). +From the Grothendieck classification (actually this was already known by Dedekind +and Weber, see [DW12]) of vector bundles on P1, see [GW10, Thm. 11.51], every +rank n vector bundle on the projective line is isomorphic to O(d1) ⊕ ··· ⊕ O(dn) for +some integers d1 ⩽ ··· ⩽ dn. Hence we might write F := O(k1)⊕···⊕O(kr) and G := +O(ℓ1)⊕···⊕O(ℓs), for some integers k1 ⩽ ··· ⩽ kr and ℓ1 ⩽ ··· ⩽ ℓs. Thus +F ⊗ω ⊗G∨ = +� +i, j +O(ki −ℓ j −2). +Let k := max{k1,...,kr} and ℓ := min{ℓ1,...,ℓs}. If d −ℓ < 2, then Ext1(F,G) = {0} +since Ext1 commutes with direct sum and Hom(L,L′) = {0} if deg(L′) > deg(L) for +L,L′ line bundles. +Let E := O(d1)⊕···⊕O(dn) be any rank n vector bundle on P1 with d1 ⩽ ··· ⩽ dn. +If we take above F = O(d1) and G = O(d2) ⊕ ··· ⊕ O(dn), thus Ext1(F,G) = {0}, + +8 +Roberto Alvarenga and Valdir Pereira Júnior +i.e. Ext1(F,G) consists only by the extension given by E. Therefore, for f ∈ AK +0 a +cusp form, follows from above discussion and the geometric interpretation of cuspidal +condition, Lemma 3.5, that +f(E) = 0, +for all E ∈ BunnP1 +and hence AK +0 is trivial. +□ +Corollary 3.7. There are also no Eisenstein series other than those induced from the +Borel subgroup. In conclusion, AK consists only of Eisenstein series on the Borel +subgroup. +Proof. Follows from previous theorem and [PJ20, Thm. 1.7.4]. +□ +4. The Φx,r-eigenforms +The aim of this section is to parametrize the space of unramified automorphic +Φx,r-eigenforms (r = 1,2) for PGL3 over the rational function field. In order to achieve +this goal, we need the graphs of Hecke operators introduced by Lorscheid in [Lor13a], +see also [Lor13b] for applications of these graphs on the theory of automorphic forms. +Definition 4.1. We call f ∈ A a HK-eigenform with eigencharacter λf if f is an eigen- +vector for every Φ ∈ HK with eigenvalue λf (Φ). +Note that λf in the above definition defines a homomorphism of C-algebras from +HK to C. Hence λf indeed defines an additive character on HK. +Definition 4.2. Let x ∈ |X| be a closed point and λ := (λ1,...,λn−1) ∈ Cn−1. The +space of Φx,r-eigenforms (or HKx-eigenforms), for r = 1,...,n−1, with eigenvalues λ +is +A(x,λ) := +� +f ∈ AK �� Φx,i( f) = λi f for i = 1,...,n−1 +� +where AK is the space of unramified automorphic forms. +Remark 4.3. If f ∈ AK is a HK-eigenform with eigencharacter λf , then f is a HKx- +eigenform and λr in the above definition is given by λf(Φx,r) for r = 1,...,n−1. +The goal of this section is to parametrize, for n = 3, the space A(x,λ1,λ2) of Φx,r- +eigenforms (r = 1,2) for some x ∈ |P1| of degree one. As we said, we shall need to +introduce the graphs of Hecke operators, which are graphs that encodes the action of +Hecke operators on the space of automorphic forms. For that reason, we need to recall +the following well-known proposition. +Proposition 4.4. Let K′ ∈ V and fix Φ ∈ HK′. For all classes of adelic matrices [g] ∈ +G(F)Z(A)\G(A)/K′, there is a unique set of pairwise distinct classes [g1],...,[gr] ∈ +G(F)Z(A)\G(A)/K′ and numbers m1,...,mr ∈ C∗ such that +Φ( f)(g) = +r +∑ +i=1 +mi f(gi). +for all f ∈ AK′. +Proof. See [Alv19, Prop. 1.6]. +□ + +On unramified automorphic forms over the projective line +9 +Definition 4.5. Let K′ ∈ V and fix Φ ∈ HK′. For classes of adelic matrics [g],[g1],...,[gr] +in the double coset G(F)Z(A)\G(A)/K′ as in the last proposition, we denote +VΦ,K′([g]) := +� +([g],[gi],mi) +� +i=1,...,r. +We define the graph G Φ,K′ of the Hecke operator Φ (relative to K′) whose vertices are +Vert G Φ,K′ = G(F)Z(A)\G(A)/K′ +and the oriented weighted edges +Edge G Φ,K′ = +� +[g]∈VertG Φ,K′ +VΦ,K′([g]). +The classes [gi] are called the Φ-neighbors of [g] (relative to K′). +Remark 4.6. For Φx,r ∈ HK we adopt the shorthand notation G +(n) +x,r for the graph +G Φx,r,K and Vx,r([g]) for the Φx,r-neighborhood VΦx,r,K([g]) of [g], where x ∈ |X| and +r = 1,...,n. +As we have seen in section 3, we can see f ∈ AK as a function on PBunnX. Hence +we can give an algebraic geometry description of the graphs G +(n) +x,r as follows. +The Weil Theorem 3.2 identifies the set of vertices of G +(n) +x,r with the geometric objects +in PBunn X. Next we describe the edges of G +(n) +x,r in geometric terms. We say that two +exact sequences of coherent sheaves on X +0 −→ F1 −→ F −→ F2 −→ 0 and 0 −→ F′ +1 −→ F −→ F′ +2 −→ 0 +are isomorphic with fixed F if there are isomorphism F1 → F′ +1 and F2 → F′ +2 such that +0 +� F1 +� +∼= +� +F +� F2 +� +∼= +� +0 +0 +� F′ +1 +� F +� F′ +2 +� 0 +commutes. Let Kx be the torsion sheaf that is supported at x and has stalk κ(x) at x, i.e. +the skyscraper torsion sheaf at x. Fix E ∈ BunnX. For r ∈ {1,...,n}, and E′ ∈ BunnX +we define mx,r(E,E′) as the number of isomorphism classes of exact sequences +0 −→ E′′ −→ E −→ K⊕r +x +−→ 0 +with fixed E ∈ PBunn X and where E′′ ∼= E′ in PBunnX. Similarly we can consider +mx,r(E,E′) by considering the isomorphisms in Bunn X instead PBunnX. +Definition 4.7. Let x ∈ |X|. For a projective vector bundle E ∈ PBunnX we define +Vx,r(E) := +� +(E,E′,m)|m = mx,r(E,E′) ̸= 0 +� +, +and we call E′ a Φx,r-neighbor of E if mx,r(E,E′) ̸= 0. In this case, mx,r(E,E′) is the +multiplicity of E′ as a Φx,r-neighbor of E. +The geometric interpretation of G +(n) +x,r the graph of the Hecke operator Φx,r ∈ HK is +given by the theorem below. + +10 +Roberto Alvarenga and Valdir Pereira Júnior +Theorem 4.8. Let x ∈ |X|. The graph G +(n) +x,r of Φx,r is described in geometric terms as +Vert G +(n) +x,r = PBunnX +and +Edge G +(n) +x,r = +� +E∈PBunnX +Vx,r(E). +Proof. See [Alv19, Thm. 3.4]. +□ +Remark 4.9. The entire above discussion holds if we consider automorphic forms as +functions on G(F)\G(A)/K instead Z(A)G(F)\G(A)/K i.e. removing the action of +Z(A). In this case we should replace PBunn X by BunnX and denote G Φ,K′ (resp. +G +(n) +x,r ) simply by GΦ,K′ (resp. G (n) +x,r ). We refer [Alv19, Sec. 3] and [ALJ21, Sec. 1] for +further details. +Rank 3 projective vector bundles on P1. For better readability, we adopt the fol- +lowing notation for the rank 3 projective vector bundles on P1. By the classification +of vector bundles on the projective line, every rank 3 projective vector bundle can be +written as +O⊕O(d1)⊕O(d2) +for some integers d1 ⩾ d2 ⩾ 0. Thus we can assume that all the elements in PBun3P1 +are of some of types below: +ε0 := O⊕O⊕O, +εd := O⊕O⊕O(d), +εd1,d2 := O⊕O(d1)⊕O(d2) +for some integers d > 0 and d2 ⩾ d1 > 0. +Proposition 4.10. Let P1 be the projective line defined over the finite field Fq and x be +a degree one closed point at P1. Then G +(3) +x,2 is given as follows: +(i) +Vx,2(ε0) = +�� +ε0,ε1,q2 +q+1 +�� +. +(ii) Let d be a positive integer, then +Vx,2(εd) = +�� +εd,εd+1,q2� +, +� +εd,ε1,d,q+1 +�� +. +(iii) Let d be a positive integer, then +Vx,2(εd,d) = +�� +εd,d,εd,d+1,q2 +q +� +, +� +εd,d,εd−1,d−1,1 +�� +. +(iv) Let d1,d2 be positive integers with d1 < d2, then +Vx,2(εd1,d2) = +�� +εd1,d2,εd1+1,d2,q +� +, +� +εd1,d2,εd1,d2+1,q2� +, +� +εd1,d2,εd1−1,d2−1,1 +�� +. +Proof. By [Alv19, Thm. 2.6] the sum of multiplicities of edges origin in a fixed vertex +must sum up q2 +q+1. The proposition follows from [Alv19, Ex. 2.3]. +□ +Proposition 4.11. Let P1 be the projective line defined over the finite field Fq and x be +a degree one closed point at P1. Then G +(3) +x,1 is given as follows: +(i) +Vx,1(ε0) = +�� +ε0,ε1,1,q2 +q+1 +�� +. +(ii) Let d be a positive integer, then +Vx,1(εd) = +�� +εd,ε1,d+1,q2 +q +� +, +� +εd,εd−1,1 +�� +. + +On unramified automorphic forms over the projective line +11 +(iii) Let d be a positive integer, then +Vx,1(εd,d) = +�� +εd,d,εd+1,d+1,q2� +, +� +εd,d,εd−1,d,q+1 +�� +. +(iv) Let d1,d2 be positive integers with d1 < d2, then +Vx,1(εd1,d2) = +�� +εd1,d2,εd1+1,d2+1,q2� +, +� +εd1,d2,εd1,d2−1,1 +� +, +� +εd1,d2,εd1−1,d2,q +�� +. +Proof. This follows from last proposition coupled with [ALJ21, Cor. 2.5]. +□ +Above description of the graphs G +(3) +x,1 and G +(3) +x,2 allows us to state and prove the main +theorem of this section. +Theorem 4.12. We fix x ∈ |P1| of degree one and λ1,λ2 ∈ C. Let E ∈ PBun3P1 and +f ∈ AK(x,λ1,λ2). Then f(E) is determined by the values of λ1,λ2 and f(ε0). In +particular, +dimAK(x,λ1,λ2) = 1. +Proof. Since f ∈ AK(x,λ1,λ2), by definition Φx,1( f) = λ1 f and Φx,2( f) = λ2 f. +Let d,d1,d2 be positive integers with d1 < d2. From Proposition 4.11 yields +λ1 f(ε0) = (q2 +q+1) f(ε1,1) +(1.1) +λ1 f(εd) = (q2 +q) f(ε1,d+1)+ f(εd−1) +(1.2) +λ1 f(εd,d) = q2 f(εd+1,d+1)+(q+1) f(εd−1,d) +(1.3) +λ1 f(εd1,d2) = q2 f(εd1+1,d2+1)+ f(εd1,d2−1)+q f(εd1−1,d2). +(1.4) +From Proposition 4.10 yields +λ2 f(ε0) = (q2 +q+1) f(ε1) +(2.1) +λ2 f(εd) = q2 f(εd+1)+(q+1) f(ε1,d) +(2.2) +λ2 f(εd,d) = (q2 +q) f(εd,d+1)+ f(εd−1,d−1) +(2.3) +λ2 f(εd1,d2) = q f(εd1+1,d2)+q2 f(εd1,d2+1)+ f(εd1−1,d2−1). +(2.4) +From (1.1) (resp. (2.1)) we can write f(ε1,1) and (resp. f(ε1)) in terms of λ1 (resp. +λ2) and f(ε0). Suppose by induction hypothesis that f(E) is determined by λ1,λ2 +and f(ε0) for E equals to εd′,εd′,d′ and εd′ +1,d′ +2 for all d′ ⩽ d, d′ +1 ⩽ d1 and d′ +2 ⩽ d2 with +d′ +1 ⩽ d′ +2. +The equations (2.2) and (1.2) (in this order) yields +f(εd+1) = 1 +q2 +� +λ2 f(εd)− q+1 +q2+q +� +λ1 f(εd)− f(εd−1) +�� +and thus by induction hypothesis f(εd) is determined by λ1,λ2 and f(ε0) for all posi- +tive integer d. +The equations (1.3) and (2.3) yields +f(εd+1,d+1) = 1 +q2 +� +λ1 f(εd,d)− q+1 +q2+q +� +λ2 f(εd−1,d−1)− f(εd−2,d−2) +�� + +12 +Roberto Alvarenga and Valdir Pereira Júnior +and thus by induction hypothesis f(εd,d) is determined by λ1,λ2 and f(ε0) for all +positive integer d. +The equation (2.4) yields +f(εd1,d2+1) = 1 +q2 +� +λ2 f(εd1,d2)−q f(εd1+1,d2)− f(εd1−1,d2−1) +� +and by induction hypothesis we are left to show that f(εd1+1,d2) is determined by λ1,λ2 +and f(ε0). If d1 + 1 = d2, then we are done by the case f(εd,d) above. Otherwise, if +d2 > d1 +1, we apply (1.4) replacing d2 by d2 −1 and obtain that +f(εd1+1,d2) = 1 +q2 +� +λ1 f(εd1,d2−1)− f(εd1,d2−2)−q f(εd1−1,d2−1) +� +. +Thus both f(εd1,d2+1) and f(εd1+1,d2) are determined by λ1,λ2 and f(ε0). Finally, +identity (1.4) yields that +f(εd1+1,d2+1) = 1 +q2 +� +λ1 f(εd1,d2)− f(εd1,d2−1)−q f(εd1−1,d2) +� +i.e. f(εd1+1,d2+1) is determined by λ1,λ2 and f(ε0). By induction hypothesis we con- +clude that f(εd1,d2) is determined by λ1,λ2 and f(ε0) for all positive integers d1,d2 +with d1 < d2. +□ +As a byproduct of previous theorem we have Theorem 3.6 for n = 3. +Corollary 4.13. There are no unramified cusp forms for PGL3 over the projective line. +Proof. Let T be diagonal torus of GL3. Given λ1,λ2 ∈ C and x ∈ P1 a closed point +of degree one, there exists a nontrivial Eisenstein series induced from an unramified +character of T which is an eigenfunction for Φx,r (r = 1,2) with eigenvalues λ1,λ2, see +[PJ20, Thm. 1.7.7]. It follows from [PJ20, Thm. 1.7.9] and above theorem that +AK +0 ∩AK(x,λ1,λ2) = {0}. +(2) +Furthermore, A0 splits as a direct sum of irreducible representations. Therefore, we +can write every f ∈ AK +0 as a sum of eigenforms and thus f = 0 by above (2). +□ +5. Toroidal autormophic forms +We define the space of toroidal automorphic forms for any global function field F and, +using the results from previous sections, we derive some results when F is the function +field of P1. +Let F be any global function field over Fq and E/F be a separable field extension +of degree n. Choosing a basis for E over F gives an embedding of E∗ in G(F) and +a non-split maximal torus T ⊆ G with T(F) = E∗ and T(AF) = A∗ +E. In this case we +say that T is associates to E/F. We refer [Lor08, Def. 1.5.1] for the definitions of +non-split and maximal torus. +Definition 5.1. Let T be a maximal torus of GLn over F associated with a separable +extension E/F of degree n. Let A := AF. Endow T(A) and T(F)Z(A) with the Haar +measures and T(F)Z(A)\T(A) with the quotient measure. For f ∈ A we define +fT(g) := +� +T(F)Z(A)\T(A) f(tg)dt + +On unramified automorphic forms over the projective line +13 +the toroidal integral of f along T. +Remark 5.2. The quotient T(F)Z(A)\T(A) is compact, see [PJ20, Pag. 42]. +Definition 5.3. Let T be a maximal torus of GLn over F associated with a separable +extension E/F of degree n. We define +Ator(E) := +� +f ∈ A | fT(g) = 0,∀g ∈ G(A) +� +the space of E-toroidal automorphic forms, and +Ator = +� +E/F +Ator(E) +the space of toroidal automorphic forms, where E/F runs over the separable exten- +sions of degree n. +Remark 5.4. The spaces Ator(E) do not depend on the choice of the basis for E over +F, see [PJ20, Rem. 2.1.3]. +Theorem 5.5. Let F be the function field of P1 defined over Fq and E be the constant +field extension of F of degree 3, i.e. E = Fq3F. If x a degree one place of F and λ1,λ2 ∈ +C, then +Ator(E)∩AK(x,λ1,λ2) = {0} +there does not exist nontrivial Φx,r-toroidal eigenforms for n = 3 and r = 1,2. +Proof. Let T be the 3-dimensional torus associated to E/F, where E = Fq3F. Thus E +is the function field of P1 +3 := P1 ⊗SpecFq SpecFq3, the extension of scalars of P1. The +extension of scalars yields +p : P1 +3 → P1 +the projection map. Moreover p induces the inverse image p∗ : Bun3 X → Bun3P1 +3 and +the direct image (or trace) p∗ : Bun1 P1 +3 → Bun3P1. +From [PJ20, Thm. 2.8.1] +� +T(F)Z(A)\T(A) f(tg)µ(t) = cT · +∑ +[L]∈PicP1 +3/p∗PicP1 +f(p∗L) +where and +cT = vol(T(F)Z(A)\T(A)) +#(PicP1 +3/p∗ PicP1) +Hence f ∈ A(x,λ1,λ2) is E-toroidal, thus f(ε0) = 0. Therefore Theorem 4.12 im- +plies that if f(ε0) = 0, then f is trivial. +□ +Since the zeta function of P1 has no zeros, a possible connection with the space of +toroidal automorphic forms lead us to the following conjecture. +Conjecture 5.6. For all n ⩾ 0, Ator = {0}. +We end this article with the following partial answer for previous conjecture. +Theorem 5.7. Let F be the function field of P1 defined over Fq and E be the constant +field extension of F of degree 3, i.e. E = Fq3F. Then, Ator(E)∩AK = {0} and therefore +Ator ∩AK = {0} for n = 3. + +14 +Roberto Alvarenga and Valdir Pereira Júnior +Proof. We suppose by contradiction that Ator(E)∩AK ̸= {0}. Hence, let f ∈ Ator(E)∩ +AK such that f ̸= 0, By admissibility condition, V = HK · f is a finite dimensional +vector space, Moreover, V is invariant by the action of Φx,1,Φx,2 ∈ HK for some x ∈ +|P1| of degree one. Thus, there exists a Φx,r-eigenform (for r = 1,2) in V, which +disagree with Theorem 5.5. +□ +References +[ALJ21] +R. Alvarenga, O. Lorscheid, and V. Pereira. Júnior. Automorphic forms for PGL(3) +over elliptic function fields - Part 1: Graphs of Hecke operators. Online available at +https://arxiv.org/pdf/2107.08375.pdf, 2021. +[Alv19] +R. Alvarenga. On graphs of Hecke operators. J. Number Theory, 199:192–228, 2019. +[BCdS+03] D. Bump, J. W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, and S. S. Kudla. An intro- +duction to the Langlands program. Birkhäuser Boston, Inc., Boston, MA, 2003. Lectures +presented at the Hebrew University of Jerusalem, Jerusalem, March 12–16, 2001, Edited +by Joseph Bernstein and Stephen Gelbart. +[Bor91] +Armand Borel. Linear algebraic groups., volume 126 of Grad. Texts Math. New York etc.: +Springer-Verlag, 2nd enlarged ed. edition, 1991. +[Bum97] +D. Bump. Automorphic forms and representations, volume 55 of Cambridge Studies in +Advanced Mathematics. Cambridge University Press, Cambridge, 1997. +[Dri80] +V. G. Drinfeld. Langlands’ conjecture for GL(2) over functional fields. In Proceedings of +the International Congress of Mathematicians (Helsinki, 1978), pages 565–574. Acad. Sci. +Fennica, Helsinki, 1980. +[DW12] +Richard Dedekind and Heinrich Weber. Theory of algebraic functions of one variable, vol- +ume 39 of History of Mathematics. American Mathematical Society, Providence, RI; Lon- +don Mathematical Society, London, 2012. Translated from the 1882 German original and +with an introduction, bibliography and index by John Stillwell. +[Fre04] +E. Frenkel. Recent advances in the Langlands program. Bull. Amer. Math. Soc. (N.S.), +41(2):151–184, 2004. +[GW10] +U. Görtz and T. Wedhorn. Algebraic geometry I. Advanced Lectures in Mathematics. +Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. +[Har77] +R. Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Gradu- +ate Texts in Mathematics, No. 52. +[HC68] +Harish-Chandra. Automorphic forms on semisimple Lie groups. Notes by J. G. M. Mars., +volume 62 of Lect. Notes Math. Springer, Cham, 1968. +[JL70] +H. Jacquet and R. P. Langlands. Automorphic forms on GL(2). Lecture Notes in Mathemat- +ics, Vol. 114. Springer-Verlag, Berlin-New York, 1970. +[Kap97] +M. M. Kapranov. Eisenstein series and quantum affine algebras. J. Math. Sci. (New York), +84(5):1311–1360, 1997. Algebraic geometry, 7. +[Laf02] +L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math., +147(1):1–241, 2002. +[Lor08] +O. +Lorscheid. +Toroidal +Automorphic +Forms +for +Function +Fields. +http://w3.impa.br/~lorschei/thesis.pdf. 2008. +[Lor13a] +O. Lorscheid. Graphs of Hecke operators. Algebra Number Theory, 7(1):19–61, 2013. +[Lor13b] +O. Lorscheid. Toroidal automorphic forms for function fields. Israel J. Math., 194(2):555– +596, 2013. +[LP97] +Joseph Le Potier. Lectures on vector bundles, volume 54 of Camb. Stud. Adv. Math. Cam- +bridge: Cambridge University Press, 1997. +[PJ20] +V. +Pereira +Junior. +Graphs +of +Hecke +Operators, +Orthog- +onal +Periods, +and +Prime +Numbers +in +Short +Intervals. +https://impa.br/wp-content/uploads/2021/03/tese_dout_Valdir-Pereira-Junior.pdf. +2020. +[Sha13] +Igor R. Shafarevich. Basic algebraic geometry. 2: Schemes and complex manifolds. Transl. +from the Russian by Miles Reid. Berlin: Springer, 3rd ed. edition, 2013. + +On unramified automorphic forms over the projective line +15 +[Zag81] +D. Zagier. Eisenstein series and the Riemann zeta function. In Automorphic forms, repre- +sentation theory and arithmetic (Bombay, 1979), volume 10 of Tata Inst. Fund. Res. Studies +in Math., pages 275–301. Tata Inst. Fundamental Res., Bombay, 1981. +Roberto Alvarenga, Instituto de Ciências Matemáticas e de Computação - USP, São Carlos, Brazil +Email address: alvarenga@icmc.usp.br +Valdir Pereira Júnior, Brazil +Email address: valdirjosepereirajunior@gmail.com + diff --git a/x9FIT4oBgHgl3EQf0ism/content/tmp_files/load_file.txt b/x9FIT4oBgHgl3EQf0ism/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c09055cb90ad434ac8d26c1663a06c45709f645 --- /dev/null +++ b/x9FIT4oBgHgl3EQf0ism/content/tmp_files/load_file.txt @@ -0,0 +1,685 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf,len=684 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='11369v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='NT] 26 Jan 2023 On unramified automorphic forms over the projective line Roberto Alvarenga and Valdir Pereira Júnior Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let q be a prime power and Fq be the finite field with q elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this article we investigate the space of unramified automorphic forms for PGLn over the rational function field defined over Fq (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' for P1 defined over Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In particular, we prove that the space of unramified cusp form is trivial and (for n = 3) that the space of eigenforms is one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover, we show that there are no nontrivial un- ramified toroidal forms for PGL3 over P1 and conjecture that the space of all toroidal automorphic forms is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Introduction Automorphic forms are central objects in modern number theory, for instance, they play a key role in the celebrated Langlands program, see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [JL70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The goal of this work is to probe the space of unramified automorphic forms for PGLn over the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be either a local or a global field, such as the function field attached to a non- singular projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In the context of understanding the absolute Galois group Gal(F,F), which is a fundamental problem on number theory, automorphic forms play the following role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' As the theory of representation of groups suggests, we may study Gal(F,F) through its finite dimension representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Given a finite dimensional rep- resentation ρ : Gal(F,F) → GLn(k), where k equals to C or Qℓ for ℓ a prime number different from the characteristic of F, there exists L(s,ρ) an analytic invariant attached to ρ, its L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence we might investigate ρ, and thus understand Gal(F,F), by means of its L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For n = 1, this approach is fundamental for the development of abelian class field theory and the understanding of Gal(F,F)ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' On the other side, when F is a global field, we might consider A its adelic ring and the so-called automorphic representations (forms) π of GLn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Here is where emerge the main object of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Attached to each automorphic representation π, we also have an analytic invariant L(s,π), its L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The celebrated Langlands conjectures predict the existence of a correspondence between the n-dimensional representations of Gal(F,F) and the automorphic representation of GLn(A) which preserves in some sense the respective L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We refer [BCdS+03, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 10 and 12] for a com- plete discussion about how automorphic forms (representations) plays a role in the Langlands program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' When F is the function field of a nonsingular projective geometrically irreducible curve X, automorphic forms with trivial central character and fixed ramifi- cation can be identified with continuous functions on the double quotient GLn(F)Z(A)\\GLn(A)/K that are of moderate growth (see Section 2 for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Where K stands for a compact open subgroup of GLn(A) and Z(A) is the center of GLn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence 1 2 Roberto Alvarenga and Valdir Pereira Júnior the domain of a cusp automorphic form (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1) with fixed ramification is discrete modulo the right-action of a compact open subgroup K of GLn(A), which allows for explicit descriptions, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this context, roughly speaking, the geometric Langlands correspondence(cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [Dri80] and [Laf02]) states the existence of a bijective map between (1) the set isomorphism classes of irreducible continuous representations ρ : Gal(F,F) → GLn(Qℓ) which are unramified outside a finite number of places and whose determinant is of finite order and, (2) the space of Qℓ-valued automorphic representations which are cuspidal and whose central character is of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover, above correspondence is the bijection given by global class field theory for n = 1 and “preserves” the respective L-functions for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We prove in section 3 that the space of cusp forms over the projective line is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1 for the definition of cusp forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' HK-eigenforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let OA := ∏Ox where the product is taken over all places x of F and Ox is the ring of integers of the completion of F with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' When K above is given by GLn(OA), the standard maximal compact open subgroup of GLn(A), we have the so-called unramified automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For global function fields, the behave of unramified automorphic forms are particularly nice: every unramified automorphic representation contains a unique unramified automorphic form, or spherical vector, up to scalar multiples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore the unramified automorphic representations correspond to certain 1-dimensional representations of the (commutative) spherical Hecke algebra HK, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In other words, unramified automorphic representations corre- spond to eigenforms of HK and are therefore determined by their eigenvalues under the action of Hecke operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We use the explicit description of the graphs of Hecke oper- ators introduced by Lorscheid in [Lor13a] to parametrize the spaces of HK-eigenforms when F is the rational function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Toroidal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Associated to a degree n separable extension of F there exists a max- imal torus of GLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The automorphic forms which vanish the integral along that torus, for every degree n separable extension of F, are called toroidal automorphic forms, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' When F = Q, these special class of automorphic forms are closed re- lated to the classical Riemann hypothesis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [Zag81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' When F is the rational function field over Fq, we prove in section 5 that there does not exist nontrivial (unramified) toroidal automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover, we conjecture that this is the situation for the whole space of toroidal automorphic forms for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The authors would like to thank Mikhail Kapranov for fruitful exchange of emails about formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The first author was supported by FAPESP [grant number 2022/09476-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Background In this first section we set up the notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be a global function field defined over a finite field Fq, where q is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We might regard F as the function field of a geometrically irreducible smooth projective curve X defined On unramified automorphic forms over the projective line 3 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We shall make it clear when we would like to specialize X to be the projective line P1 defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The reason for employ both notations is because some of the definitions and properties which we write through the article is true for an arbitrary curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In the subsequent section we shall assume that X is the projective line P1 defined over Fq and F its function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let g stand for the genus of X and |X| be the set of closed points of X or, equivalently, the set of places in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For x ∈ |X|, we denote by Fx the completion of F at x, by Ox the ring of integers of Fx, by πx ∈ Ox (we can assume πx ∈ F) a uniformizer of x and by qx the cardinality of the residue field κ(x) := Ox/(πx) ∼= Fqx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover, we denote by |x| the degree of x, which is defined to be the extension fields degree [κ(x) : Fq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In other words, qx = q|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let | · |x the absolute value of Fx (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' F) such that |πx|x = q−1 x , we call |·|x the local norm for each x ∈ |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='| Let A be the adele ring of F and A∗ the idele group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We denote OA := ∏Ox, where the product is taken over all places x of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The idele norm is the quasi-character |·| : A∗ → C∗ that sends an idele (ax) ∈ A∗ to the product ∏|ax|x over all local norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' By the product formula, this defines a quasi-character on the idele class group A∗/F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We might assume Fx being embedded into the adele ring A by sending an element a ∈ Fx to the adele (ay)y∈|X| with ax = a and ay = 0 for y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Not quite compatible with this embedding, we think of the unit group F∗ x as a subgroup of the idele group A∗ by sending an element b of F∗ x to the idele (by) with bx = b and by = 1 for y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We will explain in case of ambiguity, which of these embeddings we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let G(A) := GLn(A), Z(A) be the center of G(A), G(F) := GLn(F) and K := GLn(OA) the standard maximal compact open subgroup of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Note that G(A) comes together with an adelic topology that turns G(A) into a locally compact group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence G(A) is endowed with a Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We fix the Haar measure on G(A) for which vol(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The topology of G(A) has a neighborhood basis V of the identity matrix that is given by all subgroups K′ = ∏ x∈|X| K′ x < ∏ x∈|X| Kx = K where Kx := GLn(Ox), such that for all x ∈ |X| the subgroup K′ x of Kx is open and consequently of finite index and such that K ′ x differs from Kx only for a finite number of places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hecke algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Consider the space C0(G(A)) of continuous functions f : G(A) → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Such a function is called smooth if it is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The group G(A) acts on C0(G(A)) through the right regular representation ρ : G(A) → Aut(C0(G(A))), which is defined by right translation of the argument: (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' f)(h) := (ρ(g) f)(h) := f(hg) for g,h ∈ G(A) and f ∈ C0(G(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let H be a subgroup of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We say that f ∈C0(G(A)) is left or right H-invariant if for all h ∈ H and g ∈ G(A), f(hg) = f(g) or f(gh) = f(g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If f is right and left H-invariant, it is called bi-H-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 4 Roberto Alvarenga and Valdir Pereira Júnior Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The complex vector space H of all smooth compactly supported func- tions Φ : G(A) → C together with the convolution product Φ1 ∗Φ2 : g �−→ � G(A) Φ1(gh−1)Φ2(h)dh for Φ1,Φ2 ∈ H is called the Hecke algebra for G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Its elements are called Hecke operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The Hecke algebra H acts on C0(G(A)) by Φ( f) : g �−→ � G(A) Φ(h) f(gh)dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We say that a function f ∈C0(G(A)) is H-finite if the space H· f is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The zero element of H is the zero function, but there is no multiplicative unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For K′ ∈ V, we define HK′ to be the subalgebra of all bi-K′-invariant elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' These subalgebras have multiplicative units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Namely, the normalized characteristic function ǫK′ := (volK′)−1charK′ acts as the identity on HK′ by convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' When K′ = K above, we call HK the spherical (unramified) part of H and its elements are called spherical (unramified) Hecke operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Every Φ ∈ H is bi-K′-invariant for some K′ ∈ V, see [Lor08, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In particular, H = � K′∈V HK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Furthermore, the spherical Hecke algebra HK can be explicit described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We fix 1 ⩽ r ⩽ n an integer and let x be a place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let Φx,r stand for the characteristic function of K � πxIr In−r � K where Ir (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In−r) stands for the r×r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (n−r)×(n−r)) identity matrix and the empty entry in the matrix means a zero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Observe that Φx,n is invertible and its inverse is given by the characteristic function of K(πxIn)−1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The following theorem, due to Satake, describes HK as a commutative (almost) polynomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Identifying ǫK with 1 ∈ C yields HK ∼= C[Φx,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',Φx,n,Φ−1 x,n]x∈|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' an isomorphism of C-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In particular, HK is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See [BCdS+03, Chapter 12, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' A function f ∈C0(G(A)) is called K-finite if the complex vector space that is generated by {ρ(k) f}k∈K is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We embed G(A) ֒→ An2+1 via g �→ (g,det(g)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We define a local height |gx|x on G(Fx) := GLn(Fx) by restricting the height function (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',vn2+1) �→ max{|v1|x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',|vn2+1|x} on Fn2+1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We note that |gx|x ⩾ 1 and that |gx|x = 1 if gx ∈ Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We define the global height |g| to be the product of the local heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We say that f ∈ C0(G(A)) is of moderate growth if there exists constants C and N such that | f(g)|C ⩽ C|g|N On unramified automorphic forms over the projective line 5 for all g ∈ G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let V ⊂ C0(G(A)) be a sub-representation of G(A) on C0(G(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For K′ ∈ V, let V K′ be the subspace of all f ∈ V that are right K′-invariant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' such that ρ(k′) f = f for all k′ ∈ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We say that the representation V is admissible if V K′ is finite dimensional for every K′ ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In the following we take V = ρ(G(A)) f for some f ∈ C0(G(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The space of automorphic forms A (with trivial central character) is the complex vector space of all functions f ∈ C0(G(A)) which are smooth, K-finite, of moderate growth, left G(F)Z(A)-invariant and such that the smooth representation ρ(G(A)) f is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Its elements are called automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We are actually considering automorphic forms for PGLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' This is equiv- alent to consider in previous definition G = PGLn and remove the left Z(A)-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' However, for technical reasons, we maintain above definition and consider the Hecke algebra over GLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' A function f ∈ C0(G(A)) is smooth and K-finite if and only if there is a K′ ∈ V such that f is right K′-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In particular, V = � K′∈V V K′ for every subspace V ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See [Lor08, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Hence, from previous lemma, functions in AK′ can be identified with functions on the double quotient G(F)Z(A)\\G(A)/K′ that are of moderate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We call AK by the space of unramified automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Although the subspace AK′ of A is not stable under the action of G(A), it carries the induced action of the Hecke subalgebra HK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4 re-writes the action of Hecke operators on automorphic forms given by previous integral as a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' This is fundamental to define the graphs of Hecke operators which shall be applied to parametrize the space of HKx-eigenforms, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Cusp forms In this section we investigate the space of unramified cusp forms for PGLn over the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' An automorphic form f ∈ A is a cusp form (or cuspidal) if � U(F)\\U(A) f(ug)du = 0 for all g ∈ G(A) and all unipotent radicals subgroups U of all standard parabolic sub- groups P of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We denote the whole space of cusp forms by A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If f ∈ AK is a cusp form we call it an unramified cusp form and denote the whole space of unramified cusp forms by AK 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 6 Roberto Alvarenga and Valdir Pereira Júnior Geometric interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let BunnX be the set of isomorphism classes of rank n vector bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let PBunn X stands for the set of isomorphism classes of rank n projective vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The vector bundles E,E′ ∈ Bunn X are in the same class in PBunn X if there exists a line bundle L ∈ PicX such that E ∼= E′ ⊗L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this case we denote E = E′ in PBunnX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' It is well known that the double quotient GL1(F)\\GL1(A)/GL1(OA) is in bijection with the set of classes of divisors on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence, it yields the following bijection GL1(F)\\GL1(A)/GL1(OA) ←→ Bun1X since the set of classes of divisors on X are in correspondence with the line bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The following theorem, due to Weil, extends above bijection to higher rank vector bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2 (Weil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For every n ⩾ 1, there exists a bijection GLn(F)\\GLn(A)/GLn(OA) ←→ Bunn(X) g �−→ Eg such that Eg ⊗La = Eag for a ∈ A× and La the correspondent line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover, degEg = deg(detg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Its well know that vector bundles are completely determined by its transition maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The bijection follows, essentially, by associate to a vector bundle the adelic matrix given by the stalks of its transition maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See either [Lor08, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6] or [Fre04, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1] for a complete proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For every n ⩾ 1, there exists a bijection GLn(F)Z(A)\\GLn(A)/GLn(OA) ←→ PBunn(X) where Z(A) is the center of GLn(A) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore, due to Weil’s theorem, we can interpret unramified automor- phic forms AK as the space of complex valued functions on PBunn(X) with some moderate growth condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The next lemma reinterprets the cuspidal condition in geometric terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Despite it is already known and largely used in the literature, see for example [Kap97, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2] and [Bum97, pag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 296], we could not find a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus we sketch its proof in the following for sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We first observe, following [HC68, Lemma 3], that f ∈ AK is cuspidal if � U(F)\\U(A) f(ug)du = 0 for all g ∈ G(A) and all unipotent radicals subgroups U of all maximal parabolic sub- groups P of G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We fix P a maximal parabolic subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' It is well known that there exists integers r,s > 0 with r + s = n such that P = UM where U is the unipotent of P and radical M ∼= GLr ×GLs is its Levi subgroup, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [Bor91, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' One says that P is a parabolic maximal subgroup of type (r,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Observe furthermore that an element of U has the following form � Ir×r hs×r 0 Is×s � On unramified automorphic forms over the projective line 7 where Ir×r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Is×s) stands for the r × r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' s × s) identity matrix and hs×r is a matrix with r rows and s columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' That is, U ∼= Mr,s where Mr,s stands for the additive group of matrices with r rows and s columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' From Iwasawa decomposition, G(A) = P(A)K where P is the above fixed maximal parabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Given g ∈ G(A), we write g = xmk where x ∈ U(A),m ∈ M(A) and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence, f ∈ AK is an unramified cusp form if � U(F)\\U(A) f(ug)du = � U(F)\\U(A) f(uxmk)du = � U(F)\\U(A) f(ym)dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' for all m ∈ M(A) and where y = ux ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We write y = � Ir×r hs×r 0 Is×s � and m = � hr 0 0 hs � where hr ∈ GLr(A), hs ∈ GLs(A) and hs×r ∈ Mr,s(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F ∈ Bunr X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' G ∈ BunsX) be the vector bundle which corresponds to hr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' hs) via Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Applying once again Weil’s theorem, ym ∈ G(A) corresponds to an extension of G by F, see [LP97, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3] and [Sha13, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore, considering an unramified automorphic form as a complex valuated map from PBunn X as observed in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4, above discussion implies the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' An unramified automorphic form f ∈ AK is a cusp form if for any integers r,s > 0 with r +s = n and any vector bundles F ∈ Bunr X, G ∈ BunsX, ∑ E∈Ext(F,G) f(E) = 0, (1) where we abuse the notation and write E meant the middle term of the correspondent exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be the field of rational functions over Fq i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' the function field of P1 the projective line defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Then AK 0 is trivial for every n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We denote the structural sheaf of P1 simply by O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' O := OP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence the canonical sheaf ω := ωP1 of P1 is isomorphic to O(−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F ∈ Bunr X and G ∈ BunsX, for some integers r,s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The Serre duality (see [Har77, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7]) yields Ext1(F,G)∨ ∼= Hom(G,F ⊗O(−2)) = H0(P1,F ⊗O(−2)⊗G∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' From the Grothendieck classification (actually this was already known by Dedekind and Weber, see [DW12]) of vector bundles on P1, see [GW10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='51], every rank n vector bundle on the projective line is isomorphic to O(d1) ⊕ ··· ⊕ O(dn) for some integers d1 ⩽ ··· ⩽ dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence we might write F := O(k1)⊕···⊕O(kr) and G := O(ℓ1)⊕···⊕O(ℓs), for some integers k1 ⩽ ··· ⩽ kr and ℓ1 ⩽ ··· ⩽ ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus F ⊗ω ⊗G∨ = � i, j O(ki −ℓ j −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let k := max{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',kr} and ℓ := min{ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',ℓs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If d −ℓ < 2, then Ext1(F,G) = {0} since Ext1 commutes with direct sum and Hom(L,L′) = {0} if deg(L′) > deg(L) for L,L′ line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let E := O(d1)⊕···⊕O(dn) be any rank n vector bundle on P1 with d1 ⩽ ··· ⩽ dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If we take above F = O(d1) and G = O(d2) ⊕ ··· ⊕ O(dn), thus Ext1(F,G) = {0}, 8 Roberto Alvarenga and Valdir Pereira Júnior i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Ext1(F,G) consists only by the extension given by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore, for f ∈ AK 0 a cusp form, follows from above discussion and the geometric interpretation of cuspidal condition, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5, that f(E) = 0, for all E ∈ BunnP1 and hence AK 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' There are also no Eisenstein series other than those induced from the Borel subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In conclusion, AK consists only of Eisenstein series on the Borel subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Follows from previous theorem and [PJ20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The Φx,r-eigenforms The aim of this section is to parametrize the space of unramified automorphic Φx,r-eigenforms (r = 1,2) for PGL3 over the rational function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In order to achieve this goal, we need the graphs of Hecke operators introduced by Lorscheid in [Lor13a], see also [Lor13b] for applications of these graphs on the theory of automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We call f ∈ A a HK-eigenform with eigencharacter λf if f is an eigen- vector for every Φ ∈ HK with eigenvalue λf (Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Note that λf in the above definition defines a homomorphism of C-algebras from HK to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence λf indeed defines an additive character on HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let x ∈ |X| be a closed point and λ := (λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',λn−1) ∈ Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The space of Φx,r-eigenforms (or HKx-eigenforms), for r = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',n−1, with eigenvalues λ is A(x,λ) := � f ∈ AK �� Φx,i( f) = λi f for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',n−1 � where AK is the space of unramified automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If f ∈ AK is a HK-eigenform with eigencharacter λf , then f is a HKx- eigenform and λr in the above definition is given by λf(Φx,r) for r = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The goal of this section is to parametrize, for n = 3, the space A(x,λ1,λ2) of Φx,r- eigenforms (r = 1,2) for some x ∈ |P1| of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' As we said, we shall need to introduce the graphs of Hecke operators, which are graphs that encodes the action of Hecke operators on the space of automorphic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For that reason, we need to recall the following well-known proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let K′ ∈ V and fix Φ ∈ HK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For all classes of adelic matrices [g] ∈ G(F)Z(A)\\G(A)/K′, there is a unique set of pairwise distinct classes [g1],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',[gr] ∈ G(F)Z(A)\\G(A)/K′ and numbers m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',mr ∈ C∗ such that Φ( f)(g) = r ∑ i=1 mi f(gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' for all f ∈ AK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See [Alv19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ On unramified automorphic forms over the projective line 9 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let K′ ∈ V and fix Φ ∈ HK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For classes of adelic matrics [g],[g1],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',[gr] in the double coset G(F)Z(A)\\G(A)/K′ as in the last proposition, we denote VΦ,K′([g]) := � ([g],[gi],mi) � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We define the graph G Φ,K′ of the Hecke operator Φ (relative to K′) whose vertices are Vert G Φ,K′ = G(F)Z(A)\\G(A)/K′ and the oriented weighted edges Edge G Φ,K′ = � [g]∈VertG Φ,K′ VΦ,K′([g]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The classes [gi] are called the Φ-neighbors of [g] (relative to K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For Φx,r ∈ HK we adopt the shorthand notation G (n) x,r for the graph G Φx,r,K and Vx,r([g]) for the Φx,r-neighborhood VΦx,r,K([g]) of [g], where x ∈ |X| and r = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' As we have seen in section 3, we can see f ∈ AK as a function on PBunnX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence we can give an algebraic geometry description of the graphs G (n) x,r as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The Weil Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2 identifies the set of vertices of G (n) x,r with the geometric objects in PBunn X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Next we describe the edges of G (n) x,r in geometric terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We say that two exact sequences of coherent sheaves on X 0 −→ F1 −→ F −→ F2 −→ 0 and 0 −→ F′ 1 −→ F −→ F′ 2 −→ 0 are isomorphic with fixed F if there are isomorphism F1 → F′ 1 and F2 → F′ 2 such that 0 � F1 � ∼= � F � F2 � ∼= � 0 0 � F′ 1 � F � F′ 2 � 0 commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let Kx be the torsion sheaf that is supported at x and has stalk κ(x) at x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' the skyscraper torsion sheaf at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Fix E ∈ BunnX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For r ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=',n}, and E′ ∈ BunnX we define mx,r(E,E′) as the number of isomorphism classes of exact sequences 0 −→ E′′ −→ E −→ K⊕r x −→ 0 with fixed E ∈ PBunn X and where E′′ ∼= E′ in PBunnX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Similarly we can consider mx,r(E,E′) by considering the isomorphisms in Bunn X instead PBunnX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let x ∈ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For a projective vector bundle E ∈ PBunnX we define Vx,r(E) := � (E,E′,m)|m = mx,r(E,E′) ̸= 0 � , and we call E′ a Φx,r-neighbor of E if mx,r(E,E′) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this case, mx,r(E,E′) is the multiplicity of E′ as a Φx,r-neighbor of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The geometric interpretation of G (n) x,r the graph of the Hecke operator Φx,r ∈ HK is given by the theorem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 10 Roberto Alvarenga and Valdir Pereira Júnior Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let x ∈ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The graph G (n) x,r of Φx,r is described in geometric terms as Vert G (n) x,r = PBunnX and Edge G (n) x,r = � E∈PBunnX Vx,r(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' See [Alv19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The entire above discussion holds if we consider automorphic forms as functions on G(F)\\G(A)/K instead Z(A)G(F)\\G(A)/K i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' removing the action of Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this case we should replace PBunn X by BunnX and denote G Φ,K′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' G (n) x,r ) simply by GΦ,K′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' G (n) x,r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We refer [Alv19, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 3] and [ALJ21, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Rank 3 projective vector bundles on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For better readability, we adopt the fol- lowing notation for the rank 3 projective vector bundles on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' By the classification of vector bundles on the projective line, every rank 3 projective vector bundle can be written as O⊕O(d1)⊕O(d2) for some integers d1 ⩾ d2 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus we can assume that all the elements in PBun3P1 are of some of types below: ε0 := O⊕O⊕O, εd := O⊕O⊕O(d), εd1,d2 := O⊕O(d1)⊕O(d2) for some integers d > 0 and d2 ⩾ d1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let P1 be the projective line defined over the finite field Fq and x be a degree one closed point at P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Then G (3) x,2 is given as follows: (i) Vx,2(ε0) = �� ε0,ε1,q2 +q+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (ii) Let d be a positive integer, then Vx,2(εd) = �� εd,εd+1,q2� , � εd,ε1,d,q+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (iii) Let d be a positive integer, then Vx,2(εd,d) = �� εd,d,εd,d+1,q2 +q � , � εd,d,εd−1,d−1,1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (iv) Let d1,d2 be positive integers with d1 < d2, then Vx,2(εd1,d2) = �� εd1,d2,εd1+1,d2,q � , � εd1,d2,εd1,d2+1,q2� , � εd1,d2,εd1−1,d2−1,1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' By [Alv19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6] the sum of multiplicities of edges origin in a fixed vertex must sum up q2 +q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The proposition follows from [Alv19, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let P1 be the projective line defined over the finite field Fq and x be a degree one closed point at P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Then G (3) x,1 is given as follows: (i) Vx,1(ε0) = �� ε0,ε1,1,q2 +q+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (ii) Let d be a positive integer, then Vx,1(εd) = �� εd,ε1,d+1,q2 +q � , � εd,εd−1,1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' On unramified automorphic forms over the projective line 11 (iii) Let d be a positive integer, then Vx,1(εd,d) = �� εd,d,εd+1,d+1,q2� , � εd,d,εd−1,d,q+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (iv) Let d1,d2 be positive integers with d1 < d2, then Vx,1(εd1,d2) = �� εd1,d2,εd1+1,d2+1,q2� , � εd1,d2,εd1,d2−1,1 � , � εd1,d2,εd1−1,d2,q �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' This follows from last proposition coupled with [ALJ21, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Above description of the graphs G (3) x,1 and G (3) x,2 allows us to state and prove the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We fix x ∈ |P1| of degree one and λ1,λ2 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let E ∈ PBun3P1 and f ∈ AK(x,λ1,λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Then f(E) is determined by the values of λ1,λ2 and f(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In particular, dimAK(x,λ1,λ2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Since f ∈ AK(x,λ1,λ2), by definition Φx,1( f) = λ1 f and Φx,2( f) = λ2 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let d,d1,d2 be positive integers with d1 < d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='11 yields λ1 f(ε0) = (q2 +q+1) f(ε1,1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1) λ1 f(εd) = (q2 +q) f(ε1,d+1)+ f(εd−1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2) λ1 f(εd,d) = q2 f(εd+1,d+1)+(q+1) f(εd−1,d) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3) λ1 f(εd1,d2) = q2 f(εd1+1,d2+1)+ f(εd1,d2−1)+q f(εd1−1,d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4) From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='10 yields λ2 f(ε0) = (q2 +q+1) f(ε1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1) λ2 f(εd) = q2 f(εd+1)+(q+1) f(ε1,d) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2) λ2 f(εd,d) = (q2 +q) f(εd,d+1)+ f(εd−1,d−1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3) λ2 f(εd1,d2) = q f(εd1+1,d2)+q2 f(εd1,d2+1)+ f(εd1−1,d2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4) From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1)) we can write f(ε1,1) and (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' f(ε1)) in terms of λ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' λ2) and f(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Suppose by induction hypothesis that f(E) is determined by λ1,λ2 and f(ε0) for E equals to εd′,εd′,d′ and εd′ 1,d′ 2 for all d′ ⩽ d, d′ 1 ⩽ d1 and d′ 2 ⩽ d2 with d′ 1 ⩽ d′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2) (in this order) yields f(εd+1) = 1 q2 � λ2 f(εd)− q+1 q2+q � λ1 f(εd)− f(εd−1) �� and thus by induction hypothesis f(εd) is determined by λ1,λ2 and f(ε0) for all posi- tive integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3) yields f(εd+1,d+1) = 1 q2 � λ1 f(εd,d)− q+1 q2+q � λ2 f(εd−1,d−1)− f(εd−2,d−2) �� 12 Roberto Alvarenga and Valdir Pereira Júnior and thus by induction hypothesis f(εd,d) is determined by λ1,λ2 and f(ε0) for all positive integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4) yields f(εd1,d2+1) = 1 q2 � λ2 f(εd1,d2)−q f(εd1+1,d2)− f(εd1−1,d2−1) � and by induction hypothesis we are left to show that f(εd1+1,d2) is determined by λ1,λ2 and f(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If d1 + 1 = d2, then we are done by the case f(εd,d) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Otherwise, if d2 > d1 +1, we apply (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4) replacing d2 by d2 −1 and obtain that f(εd1+1,d2) = 1 q2 � λ1 f(εd1,d2−1)− f(εd1,d2−2)−q f(εd1−1,d2−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus both f(εd1,d2+1) and f(εd1+1,d2) are determined by λ1,λ2 and f(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Finally, identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4) yields that f(εd1+1,d2+1) = 1 q2 � λ1 f(εd1,d2)− f(εd1,d2−1)−q f(εd1−1,d2) � i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' f(εd1+1,d2+1) is determined by λ1,λ2 and f(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' By induction hypothesis we con- clude that f(εd1,d2) is determined by λ1,λ2 and f(ε0) for all positive integers d1,d2 with d1 < d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ As a byproduct of previous theorem we have Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6 for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' There are no unramified cusp forms for PGL3 over the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let T be diagonal torus of GL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Given λ1,λ2 ∈ C and x ∈ P1 a closed point of degree one, there exists a nontrivial Eisenstein series induced from an unramified character of T which is an eigenfunction for Φx,r (r = 1,2) with eigenvalues λ1,λ2, see [PJ20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' It follows from [PJ20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='9] and above theorem that AK 0 ∩AK(x,λ1,λ2) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' (2) Furthermore, A0 splits as a direct sum of irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore, we can write every f ∈ AK 0 as a sum of eigenforms and thus f = 0 by above (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Toroidal autormophic forms We define the space of toroidal automorphic forms for any global function field F and, using the results from previous sections, we derive some results when F is the function field of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be any global function field over Fq and E/F be a separable field extension of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Choosing a basis for E over F gives an embedding of E∗ in G(F) and a non-split maximal torus T ⊆ G with T(F) = E∗ and T(AF) = A∗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In this case we say that T is associates to E/F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We refer [Lor08, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1] for the definitions of non-split and maximal torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let T be a maximal torus of GLn over F associated with a separable extension E/F of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let A := AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Endow T(A) and T(F)Z(A) with the Haar measures and T(F)Z(A)\\T(A) with the quotient measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For f ∈ A we define fT(g) := � T(F)Z(A)\\T(A) f(tg)dt On unramified automorphic forms over the projective line 13 the toroidal integral of f along T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The quotient T(F)Z(A)\\T(A) is compact, see [PJ20, Pag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let T be a maximal torus of GLn over F associated with a separable extension E/F of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We define Ator(E) := � f ∈ A | fT(g) = 0,∀g ∈ G(A) � the space of E-toroidal automorphic forms, and Ator = � E/F Ator(E) the space of toroidal automorphic forms, where E/F runs over the separable exten- sions of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The spaces Ator(E) do not depend on the choice of the basis for E over F, see [PJ20, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be the function field of P1 defined over Fq and E be the constant field extension of F of degree 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' E = Fq3F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' If x a degree one place of F and λ1,λ2 ∈ C, then Ator(E)∩AK(x,λ1,λ2) = {0} there does not exist nontrivial Φx,r-toroidal eigenforms for n = 3 and r = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let T be the 3-dimensional torus associated to E/F, where E = Fq3F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus E is the function field of P1 3 := P1 ⊗SpecFq SpecFq3, the extension of scalars of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' The extension of scalars yields p : P1 3 → P1 the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Moreover p induces the inverse image p∗ : Bun3 X → Bun3P1 3 and the direct image (or trace) p∗ : Bun1 P1 3 → Bun3P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' From [PJ20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='1] � T(F)Z(A)\\T(A) f(tg)µ(t) = cT · ∑ [L]∈PicP1 3/p∗PicP1 f(p∗L) where and cT = vol(T(F)Z(A)\\T(A)) #(PicP1 3/p∗ PicP1) Hence f ∈ A(x,λ1,λ2) is E-toroidal, thus f(ε0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Therefore Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='12 im- plies that if f(ε0) = 0, then f is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ Since the zeta function of P1 has no zeros, a possible connection with the space of toroidal automorphic forms lead us to the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' For all n ⩾ 0, Ator = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We end this article with the following partial answer for previous conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Let F be the function field of P1 defined over Fq and E be the constant field extension of F of degree 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' E = Fq3F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Then, Ator(E)∩AK = {0} and therefore Ator ∩AK = {0} for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 14 Roberto Alvarenga and Valdir Pereira Júnior Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' We suppose by contradiction that Ator(E)∩AK ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Hence, let f ∈ Ator(E)∩ AK such that f ̸= 0, By admissibility condition, V = HK · f is a finite dimensional vector space, Moreover, V is invariant by the action of Φx,1,Φx,2 ∈ HK for some x ∈ |P1| of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Thus, there exists a Φx,r-eigenform (for r = 1,2) in V, which disagree with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' □ References [ALJ21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Alvarenga, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Lorscheid, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Pereira.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Wedhorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Algebraic geometry I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Advanced Lectures in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Vieweg + Teubner, Wiesbaden, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Schemes with examples and exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [Har77] R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [PJ20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Pereira Junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Graphs of Hecke Operators, Orthog- onal Periods, and Prime Numbers in Short Intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' https://impa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='br/wp-content/uploads/2021/03/tese_dout_Valdir-Pereira-Junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' [Sha13] Igor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Shafarevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Basic algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' 2: Schemes and complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' from the Russian by Miles Reid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Berlin: Springer, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' edition, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' On unramified automorphic forms over the projective line 15 [Zag81] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Zagier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Eisenstein series and the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' In Automorphic forms, repre- sentation theory and arithmetic (Bombay, 1979), volume 10 of Tata Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Studies in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=', pages 275–301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Tata Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Fundamental Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=', Bombay, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content=' Roberto Alvarenga, Instituto de Ciências Matemáticas e de Computação - USP, São Carlos, Brazil Email address: alvarenga@icmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='br Valdir Pereira Júnior, Brazil Email address: valdirjosepereirajunior@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FIT4oBgHgl3EQf0ism/content/2301.11369v1.pdf'} diff --git a/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/2301.11745v1.pdf.txt b/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/2301.11745v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea9bac831f318949b68360aef252a0b57066ab7b --- /dev/null +++ b/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/2301.11745v1.pdf.txt @@ -0,0 +1,1284 @@ +Side Auth: Synthesizing Virtual Sensors for Authentication +Yan Long +University of Michigan +yanlong@umich.edu +Kevin Fu +Northeastern University +k.fu@northeastern.edu +ABSTRACT +While the embedded security research community aims to protect +systems by reducing analog sensor side channels, our work argues +that sensor side channels can be beneficial to defenders. This work +introduces the general problem of synthesizing virtual sensors from +existing circuits to authenticate physical sensors’ measurands. We +investigate how to apply this approach and present a preliminary +analytical framework and definitions for sensors side channels. +To illustrate the general concept, we provide a proof-of-concept +case study to synthesize a virtual inertial measurement unit from a +camera motion side channel. Our work also provides an example of +applying this technique to protect facial recognition against silicon +mask spoofing attacks. Finally, we discuss downstream problems +of how to ensure that side channels benefit the defender, but not +the adversary, during authentication. +1 +INTRODUCTION +Sensor side channels enable an adversary to violate integrity of +sensor outputs by influencing or controlling the sensor with trans- +duction attacks [15, 39], or to eavesdrop on sensitive information +and compromise confidentiality by exploiting flaws in sensor and +system designs [3, 5, 22, 30]. For example, the eavesdropping exam- +ple PIN Skimmer [30] shows that adversaries can infer smartphone +touchscreen inputs by exploiting side channel motion informa- +tion captured by smartphone cameras. While the security research +community invested significant effort identifying and mitigating +analog sensor side channels, our work argues that it can be benefi- +cial to embrace, understand, and control analog sensor side +channels instead of simply eliminating them. This is moti- +vated by our observation that such side channel information may +also be used for authentication. For example, extensive research +has been conducted on using dedicated motion sensors to capture +smartphone touch dynamics for continuous implicit user authenti- +cation [34]. Relating it to PIN Skimmer, a natural question arises as +to whether cameras support such authentication when dedicated +motion sensors are not available. We thus propose and investigate +the problem of how to utilize sensor side channels for defensive +purposes such as multimodal authentication by synthesizing virtual +sensors from them. +Side channels are inherent to analog sensors’ physics. There exist +a considerable number of potential sensor side channels besides +those revealed by transduction and eavesdropping attacks. However, +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +NSPW’22, October 24–27, 2022, New Hampshire, USA +© 2023 Copyright held by the owner/author(s). +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +Figure 1: Sensor side channels are different from conven- +tional side channels as they measure the measurement pro- +cesses instead of computation processes. Sensor side chan- +nels can measure the byproduct, measurer, and environ- +ment to verify authenticity of intended sensor measurands. +most of these side channels are deliberately “closed” in the design +phase by employing mitigation mechanisms such as calibration and +noise reduction. It is foreseeable that sensor and system designers +will also try to mitigate newly discovered side channels. This work +argues a different perspective and approach to embrace such sensor +side channels. If these side channels can be used in a beneficial way, +we envision future designs allowing mitigation mechanisms to be +strategically disabled or downgraded when needed such as during +authentication sessions. +We provide a preliminary analytical framework for modeling +analog sensor side channels and explaining the origins and charac- +teristics of them. The framework categorizes sensor side channels +according to their separability from intended signals and whether +they have controllable mitigation mechanisms. Based on the frame- +work, we define the problem of virtual sensor synthesis for mul- +timodal measurand authentication and summarize three possible +ways of applying this approach (Figure 1). First, by verifying sig- +natures of signal byproducts and asking the question “What is the +probability that Alice generated both the measurands and byprod- +ucts?” Second, by verifying the person performing the measurement +and asking “What is the probability that Alice generated the measur- +ands if Bob was the measurer?” Third, by verifying the environment +of the measurement process and asking “What is the probability +that Alice generated the measurands if the measurement was taken +in location B?” +A proof-of-concept case study further concretizes the concepts +and related considerations by studying a camera motion side chan- +nel that enables cameras to sense out-of-sight motion. This side +channel is caused by mechanical connections between camera de- +vices and adjacent objects in motion such as a hand holding the +camera. We propose a methodology for synthesizing virtual inertial +measurement units (IMUs) from this side channel that can extract +both inter-frame low-frequency and intra-frame high-frequency +motion information. The case study discusses this side channel’s +potential application in helping facial recognition systems defend +against 3D silicon mask spoofing attacks by verifying postural hand +arXiv:2301.11745v1 [cs.CR] 27 Jan 2023 + +Conventional +Sensor +Side Channel +Side Channel +Memory +f() +Measurands +S +Byproduct +int +m +Output +Input +Sensor +Measurer +CPU +S +Environment +sideNSPW’22, October 24–27, 2022, New Hampshire, USA +Yan Long and Kevin Fu +tremor motion of the person holding the camera device. Prelimi- +nary test with 4 people suggests the camera motion side channel +help reduce false positive rates by up to 87.5%. It also shows that +disabling video stabilization enables higher performance, empha- +sizing the benefits of strategically disabling side channel mitigation +mechanisms. Finally, we discuss the possible issues of temporar- +ily opening sensor side channels during authentication and the +directions future works may take to address the issues. Our main +contributions are summarized as follows: +• A new paradigm to embrace and harness analog sensor side +channels for defensive purposes via active control or influ- +ence of sensor side channels. +• An analytical framework to enable definition and character- +ization of sensor side channels. The framework introduces +the concept of virtual sensor synthesis for multimodal sensor +measurand authentication. +• A case study and methodology of synthesizing virtual IMUs +from camera motion side channels for enhancing perfor- +mance of facial recognition systems. Virtual IMUs decrease +false positive rates when facial recognition is subjected to +emulated 3D silicon mask spoofing attacks. +2 +BACKGROUND & RELATED WORK +Side channels are unintended information output channels. The +idea of synthesizing virtual sensors for authentication can be traced +back to the general concept of using side channel information to +identify people in forensic analysis. For example, forensic hand- +writing recognition allows one to determine the identity of a letter +writer even though the letter was never intended to convey identity +information. In the field of digital forensics [14], the authenticity +of digital evidence can be verified by cross-correlating file data and +metadata with other contextual information. For example, fraud- +ulent documents were reported to be detected by analyzing their +choices of font [6]. Our paper investigates how we can design +mechanisms to actively utilize side channel information in sensor +readings for defending computer systems. We introduce research +works related to this idea in this section. +2.1 +Sensor Side Channel Based Attacks +Sensor side channels have been actively exploited in two lines of +research, namely transduction and eavesdropping attacks. This +section provides some background and examples of these attacks. +Different from these works that try to compromise information in- +tegrity and confidentiality of sensor systems, our work investigates +how designers may defend sensor systems by actively controlling +and utilizing sensor side channels. +Transduction Attacks. Transduction attacks injects analog sig- +nals into sensors where victim sensor circuitry transduces an at- +tacker’s malicious physical signals to untrustworthy sensor mea- +surements. Such malicious physical signals can often be in different +physical modalities (e.g., acoustic vs. optical) or frequency ranges +(e.g., audible vs. ultrasound) than what the sensors are designed to +sense. For example, Light Commands [33] uses lasers to inject false +speech signals into microphones. Works such as Walnut [31, 36] +use acoustic injections to influence and control the output of MEMS +gyroscopes and accelerometers. Ghost Talk [20] uses radio waves to +inject audio signals into microphones. An SoK and a survey [15, 39] +provide a comprehensive review of theses attacks and correspond- +ing mitigation mechanisms. +Eavesdropping Attacks. Sensor side channels are also exploited +for eavesdropping out-of-band information. For example, PIN Skim- +mer [30] used a camera-based side channel to infer PIN inputs by ex- +ploiting correlations between smartphone camera orientations and +tapping locations. Several works such as Gyrophone [3, 5, 22] use +smartphone IMUs which contain accelerometers and gyroscopes +to eavesdrop speech by exploiting side channels enabled by signal +aliasing in analog-to-digital converters (ADCs). +2.2 +Using Conventional Non-sensor Side +Channels For Defensive Purposes +Although we could not find related research that investigates the +concept of utilizing sensor side channels for defending a system, we +found that a few previous works explored using non-sensor side +channels for machine-to-machine authentication. [29] proposes to +use the key-dependent side channel information in wireless commu- +nication channels to enhance existing cryptographic protocols. [10] +presents an extension by analyzing practical and security issues of +the protocol in [29] and providing fixes. Compared to them, this +work focuses on the concept of sensor side channel and authenti- +cation of sensor’s measurands instead of computation-generated +information as the previous works did. Non-sensor side channels +are also utilized in other applications such as code-execution moni- +toring and intrusion detection [4, 9, 26]. +3 +PROBLEM FORMULATION +This section defines the problem of using sensor side channels +for measurand authentication. Our paper proposes the concept of +using sensor side channels for authentication as a new direction +of research for the community. We also fill a gap by suggesting a +mathematical definition of sensor side channels, beginning with +a framework for defining and categorizing sensor side channels. +We then introduce the problem of synthesizing virtual sensors and +using them for multimodal sensor measurand authentication. +3.1 +Sensor Side Channel Analytical Framework +3.1.1 +Sensor. A sensor, or transducer, can be modeled as a function +that maps physical measurands to digital measurements over time. +A measurand is a quantity that a sensor intends to measure [17]. +Different types of sensors are designed to measure different modal- +ities of measurands such as sound, temperature, motion, etc. Users +who are informed of the apparent purpose and specifications of +sensors often see a sensor as the following function over a single +variable of the measurand: +𝑚 = 𝑓 (𝑠𝑖𝑛𝑡 ) +(1) +where 𝑚 and 𝑠𝑖𝑛𝑡 denote the digital measurements and analog mea- +surand respectively and 𝑓 (·) denotes the sensor. +3.1.2 +Sensor Side Channels. Although Equation 1 provides average +sensor users a clean and easy abstraction, actual sensor implemen- +tations are much “dirtier’ and introduce numerous hidden variables +to the equation that result in unintended components in measure- +ment 𝑚. For instance, every conductor wire can be regarded as + +NSPW’22, October 24–27, 2022, New Hampshire, USA +an unintentional antenna, leading to side channels that convert +electromagnetic energy to measurements of non-electromagnetic +sensors [20, 37]. In this case, hidden variables related to electro- +magnetic energy in the environment should be added to Equation 1. +Another example of such variables is temperature. Semiconductors +made of silicon are inherently sensitive to heat due to its ability +to excite electrons. So technically, Equation 1 should also include +temperature as a variable. Electromagnetic energy and temperature +are just examples of hidden variables associated to the underlying +physical characteristics of devices. There are also hidden variables +caused by design flaws and uncontrollable variations in the manu- +facture processes. Thus, Equation 1 should be modified to enable a +side channel-aware modeling of sensors: +𝑚 = 𝑓 (𝑠𝑖𝑛𝑡,𝑠𝑠𝑖𝑑𝑒 ), +𝑠𝑠𝑖𝑑𝑒 = [𝑠𝑣1,𝑠𝑣2, ...] +(2) +where 𝑠𝑠𝑖𝑑𝑒 represent the set of all these hidden variables that can +potentially lead to side channels attacks. +The comparison between Equation 1 and 2 shows that the gap +between users’ understanding and sensors’ actual implementation +gives birth to sensor side channels. On a high level, we believe +the gap can also be attributed to the insufficient specifications of +legitimate and illegitimate sensor behaviors in the existing sys- +tem’s security policies. Note that this differs from conventional +non-sensor side channels where side channels bypass the clearly +specified security policies [16]: there are often no dedicated security +policies for sensors yet in existing sytems. Building upon previous +side channel research [32, 42], we tentatively define sensor side +channels as the following: +• A sensor side channel is a communication channel that allows +someone to recover secret information using unintended sensor +measurement components in a way that violates the associated +system’s security expectations. +Sensor side channels are sometimes more conceptually difficult +to recognize than conventional non-sensor side channels such as +differential power analysis channels. The reason is that non-sensor +side channels are used to mainly measure computation processes +where there exists a clear boundary between computation and +measurement whereas sensor side channels are used to measure +the measurement processes themselves (Figure 1). +A possible way of identifying sensor side channels is to test the +hypothesis that the analog signal of a variable 𝑣𝑖 correlates with 𝑚 +with certain significance, i.e., +|𝐶𝑜𝑟𝑟 (𝑚,𝑠𝑣𝑖 )| > 𝛼, +𝑠𝑣𝑖 ∈ 𝑠𝑠𝑖𝑑𝑒 +(3) +where 𝛼 is a threshold value. Note that this work does not discuss +the actual choice of threshold values and correlation functions since +they can be flexible depending on the actual application scenarios +and security requirements. In cases where it is challenging to project +𝑚 and 𝑠𝑣𝑖 to the same vector space in order to compute correlation +scores, other methods such as supervised classification can also be +used if 𝑠𝑣𝑖 can be converted into data labels. +3.1.3 +Separability and Controllability. The unintended components +in the measurements are caused by the existence of 𝑠𝑠𝑖𝑑𝑒 and can +be either separable or inseparable from the intended components. +The separability between the intended and unintended components +is the key that decides whether a side channel can be mitigated +and controlled or not. Conceptually, separable components can be +defined as the following: there exists at least one function ˜𝑓 (·) that +can break 𝑚 down into intended and unintended components such +that those components only correlate (with significance) with the +measurand and other hidden variables respectively, i.e., +∃ ˜𝑓 (·) +𝑠.𝑡. +˜𝑓 (𝑚) = [𝑚𝑖𝑛𝑡,𝑚𝑠𝑖𝑑𝑒 ], +𝑚𝑠𝑖𝑑𝑒 = [𝑚𝑣1,𝑚𝑣2, ...], +|𝐶𝑜𝑟𝑟 (𝑚𝑖𝑛𝑡,𝑠𝑖𝑛𝑡 )| > 𝛼𝑖1, |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑣𝑖 )| > 𝛼𝑖2, +|𝐶𝑜𝑟𝑟 (𝑚𝑖𝑛𝑡,𝑠𝑣𝑖 )| < 𝛽𝑖1, |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑖𝑛𝑡 )| < 𝛽𝑖2 +(4) +When a sensor side channel has separable components, we say +it is a separable side channel. Separability is decided by sensor im- +plementation 𝑓 (·). Side channels with inseparable components in +existing sensor implementations led to the various unsolvable at- +tacks against sensors (Section 2.1) because designers cannot extract +only the intended components. +Theoretically, those with separable components can be mitigated +by mechanisms referred to as compensation, calibration and noise +reduction. Such mitigation mechanisms can be abstracted as another +function 𝑔(·) that suppresses the unintended components in the +output of ˜𝑓 (·), i.e., 𝑔( ˜𝑓 (𝑚)) = 𝑚𝑖𝑛𝑡 . If the mitigation mechanisms +can be both turned on and off, the user of the sensor system then +have full control of the sensor side channel. We call such a sensor +side channel controllable: +• A controllable sensor side channel is one whose corresponding +unintended measurement component is separable from the +intended component and can be suppressed by a mitigation +mechanism that can be enabled and disabled. +3.1.4 +Examples. We provide some existing examples of each cate- +gory of sensor side channels to shed light on the differences and +possible future evolution. +Inseparable. The Gyrophone eavesdropping attack [22] and its +follow-up works [3, 5] use an aliasing-enabled inseparable acoustic +side channel in smartphone IMUs to recover speech. These IMUs +have intended acceleration and angular velocity measurands mostly +under the frequency range of human speech. However, due to the +lack of effective analog low-pass filtering before the ADC, aliases of +the high-frequency speech signals exist in the output of ADC and +enable adversaries to recover speech information. Furthermore, the +aliases cannot be separated from the intended motion signals since +they are in the same frequency range. Intuitively, adding analog +filters to the sensors make this acoustic side channel separable. In +order to be controllable, the sensor API may further allow CPU to +enable and disable the filters. +Separable But Uncontrollable. Those seemingly intact sen- +sors that have not been reported vulnerable to side channel-based +attacks also have inherent side channels, but just in a suppressed +manner thus these channels are not exposed to attackers. Take +sensors’ heat sensitivity mentioned in Section 3.1.2 as an exam- +ple. MEMS humidity sensors, gyroscopes, accelerometers, etc., are +widely equipped with temperature-compensated designs or online +thermal calibration procedures [7, 13, 40]. It can be anticipated that +if the compensation and calibration mechanisms can be temporarily +disabled, these sensors’ measurements will exhibit significant cor- +relation with the ambient temperature. In this way, the separable +side channel becomes controllable. + +NSPW’22, October 24–27, 2022, New Hampshire, USA +Yan Long and Kevin Fu +Controllable. There already exist sensors with controllable side +channels. A good example is handheld cameras getting equipped +with video stabilization mechanisms. Camera motion is often re- +garded as side effects that degrade the quality of the intended signal, +i.e., the scene in the field of view of the camera [41]. Video stabiliza- +tion mechanisms, including electronic image stabilization (EIS) and +optical image stabilization (OIS), etc., are implemented to mitigate +these side effects by optically or electronically reducing the un- +wanted image scene movements caused by camera motion. Many +operating systems such as Android allow app developers to choose +if these video stabilization mechanisms will be turned on or off +when the underlying camera hardware offers the API to control +it. However, it is worth noting that such existing controllable side +channels are most likely byproducts of OS designers’ conventions of +providing more fine-grained interfaces, especially for open-source +OS like Android which allows users to control EIS and OIS sepa- +rately. In contrast, iOS does not allow explicit and separate control +of EIS and OIS. Such a large degree of control is provided to support +more potential use cases and enhance usability. For example, users +may want to disable smartphone’s built-in optical image stabiliza- +tion when using an external gimbal because the two can interfere +with each other and produce extra image distortions [1]. To the +best of our knowledge, these existing controllable side channels +have not been explored to enhance the security of systems. +3.1.5 +Summary. It is possible to convert existing inseparable or +uncontrollable side channels into controllable side channels by im- +proving sensor designs, as has been suggested by the increasing +popularity of video stabilization in cameras. Thus, it is important +to think from a perspective of technology development when con- +sidering benefits of sensor side channels. Furthermore, protecting +physical sensors from side channel attacks often already means +transforming inseparable side channels to be separable. With some +additional effort of making mitigation mechanisms controllable in- +stead of forever-on, sensor side channels can be used in a beneficial +and controlled manner. The following discussions assume sensors +have controllable side channels. +3.2 +Measurands Authentication Using +Synthesized Virtual Sensors +3.2.1 +Virtual Sensor Synthesis. A virtual sensor is a function that +maps 𝑚 to 𝑚𝑣𝑖 . Ideally, the construction of ˜𝑓 (·) in Equation 4 al- +ready presents such an overarching function that can measure both +the intended and side channel components. Such construction is +apparently challenging since it needs to consider all possible side +channels. Actual implementations can reduce the level of challenge +by focusing on maximizing |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑣𝑖 )| and −|𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑖𝑛𝑡 )| +for only the set of targeted hidden variable {𝑣𝑖}. We denote such a +function specifically crafted for {𝑣𝑖} as ˜𝑓{𝑣𝑖 } and call them virtual +sensor functions. +3.2.2 +Problem Definition. We define the problem as a binary hy- +pothesis test in a comparative manner by first referencing to the +unimodal authentication on the physical sensor’s measurand alone. +Without virtual sensors, objects in Equation 1 including 𝑚, 𝑠𝑖𝑛𝑡 , and +𝑓 are all that the designer of the authentication system can perceive. +Let there be a measurand with a true identity 𝐿 and a claimed iden- +tity ˜𝐿. The 𝐻1 and 𝐻0 hypotheses are ˜𝐿 = 𝐿 and ˜𝐿 ≠ 𝐿 respectively. +Denote the unimodal authentication system as A𝑢 : 𝑚 → {1, 0}, +where it declares 𝐻1 and 𝐻0 when outputting 1 and 0 respectively. +We can then define the total error of the unimodal system 𝐸𝑢 as +𝐸𝑢 = 𝑐1P[declare 𝐻1|𝐻0] + 𝑐2P[declare 𝐻0|𝐻1] += 𝑐1E[A𝑢 (𝑚)|𝐻0] + 𝑐2E[1 − A𝑢 (𝑚)|𝐻1] +(5) +where P[·|·] and E[·|·] denotes conditional probability and expec- +tation respectively, 𝑐1 and 𝑐2 denote the cost coefficients for false +positive and false negatives respectively. +Similarly, a multimodal authentication system with𝑛 synthesized +virtual sensors can be denoted as A𝑚 : [𝑚𝑖𝑛𝑡,𝑚𝑣1, ...,𝑚𝑣𝑛] → +{1, 0}. The total error 𝐸𝑚 is defined as +𝐸𝑚 = 𝑐1E[A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1, ...,𝑚𝑣𝑛])|𝐻0] ++ 𝑐2E[1 − A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1, ...,𝑚𝑣𝑛])|𝐻1] +(6) +As a result, the problem of synthesizing virtual sensors to au- +thenticate the measurand in a multimodal manner can be defined +as: +• Constructing virtual sensor functions ˜𝑓{𝑣𝑖 } and multimodal +authentication system A𝑚 such that better performance is +achieved for measurand authentication, i.e., 𝐸𝑚 − 𝐸𝑢 < 0. +3.2.3 +Security Properties. Although multimodal authentication us- +ing synthesized virtual sensors look similar to that using multiple +physical sensors, it provides two different security properties. +First, it works with existing devices and media that only have +a single physical sensor’s data. Although high-end devices like +smartphones are equipped with multiple physical sensors, there +still exist lower-end devices that only serve a single purpose such +as ultrasonic proximity detectors and humidity monitors. Further- +more, sometimes it is needed to verify the identity of an object +such as a photograph that has already been generated with only a +single sensor. In this case, synthesized virtual sensors can extract +additional information in a retrospective way. +Second, it potentially provide more robustness against spoofing +attacks on individual sensors. The level of attack difficulty depends +on the complexity of ˜𝑓 , i.e., how difficult it is to decouple and then +modify different measurement components. Using multiple indi- +vidual sensors such as cameras, accelerometers, etc., is equivalent +to having a ˜𝑓 that does not need to decouple anything at all since +the inputs already separated. Conceptually, if we regard the mea- +surements corresponding to different virtual or physical sensors +as random variables, we can then regard their variances and co- +variances as the entropy provided for authentication [24]. Virtual +sensors potentially provides more entropy because the coupling +between them adds to the covariances. Such entropy originates +from the intrinsic physics of sensors. +3.2.4 +Application. The general problem definition can be applied +to different sources of side channel variables whose signatures +correlate with the claimed identity of the measurands. Depending +on the sources, we believe synthesized virtual sensors can be applied +in the following three ways to verify authenticity of measurands. + +NSPW’22, October 24–27, 2022, New Hampshire, USA +Byproduct Verification. A physical process generating intended +measurands is likely to generate other forms of energy as byprod- +ucts. Let us explore the example of a loudspeaker that replays a +person Alice’s speech recordings while a nearby microphone is +listening to this replay. Say there is someone claiming the speech +audio collected by the microphone is coming from Alice herself +speaking live and an investigator tries to verify this claim. The in- +vestigator finds out that the loudspeaker also generates unintended, +secondary byproducts in the form of structure-borne vibrations, +electromagnetic emission, heat, etc., which may be sensed by vir- +tual sensors synthesized from the microphone’s side channels. So, +if these byproducts exist, the investigator knows it is not likely a +legitimate recording of Alice’s voice. In this case, the core authenti- +cation question can be summarized as “What is the probability that +Alice generated both the measurands and byproducts?” +Measurer Verification. A Measurer is the person who makes +measurements with a physical sensor. Measurers themselves gen- +erate unintended emissions taking the form of physical signals +containing certain signatures that correlate with the identity of +measurands. For example, say there exists an unmodified photo of a +person who is claimed to be Alice and an investigator tries to verify +this claim. The investigator managed to find out that the camera +operator who took this photo, i.e., the measurer was Bob because +Bob was speaking when he took the photo and his speech induced +identifiable image blurs through a camera motion side channel. If +the investigator also knows that Bob has never been in the vicinity +of Alice, then the investigator knows the person in the photo is +not Alice. Obviously, measurand authentication through measurer +verification may require higher-level contextual information com- +pared to byproduct verification. The core authentication question +is “What is the probability that Alice generated the measurands if +Bob was the measurer?” +Environment Verification. Similar to measurer verification, +verifying the environment surrounding measurands also allows one +to authenticate the measurands. Take the same example above. Say +the photo has a temperature side channel that shows the ambient +temperature was 104°F/40°C at the time of generating the photo, +pointing to a location B. If the investigator knows Alice has never +been in location B, then the investigator knows the person in the +photo is not Alice. The core authentication question is “What is the +probability that Alice generated the measurands if the measurement +was taken in location B?” +4 +CASE STUDY +The case study demonstrates how to use camera motion side chan- +nels (Section 3.1.4) to synthesize virtual IMUs that can collect pos- +tural hand tremor information for measurand authentication in +facial recognition applications. It can be regarded an example of +both byproduct and measurer verification. +4.1 +Primer +4.1.1 +Postural Tremor Information. Tremor is the involuntary rhyth- +mic movement of a human body part caused by reciprocal innerva- +tions of muscles. Such involuntary movements are present in all +people, with those found in healthy people and disease conditions +(e.g., Parkinson disease) classified as physiological and patholog- +ical tremor respectively [35]. Clinical research finds that tremors +measured by accelerometers can effectively predict the category +of tremors. Some works further show that hand tremors measured +by accelerometers and gyroscopes are unique to an individual and +stable over time, suggesting the feasibility of using tremors as a +biometric for personal identification [12, 23]. +4.1.2 +Threat Model. We study a threat model of spoofing attack +against smartphone facial recognition systems where imposters are +assumed to launch a silicone face mask spoofing attack [27]. To +better show the effectiveness of the synthesized IMUs, we further +assume the silicone mask perfectly mimics the face of the victims. +During the attack, the imposter wears the silicone mask and holds +the victim’s smartphone for authentication. Our objective is to +extract camera motion from videos that represents the postural +hand tremor of users to defend against such perfect silicone mask +attacks. +It is worth noting this particular case study’s threat model re- +quires users to hold their phones in their hands during facial recog- +nition as the contact between their phones and hands provides a +propagation path for the vibration information of hand tremor. We +believe this is also the most frequent situation seen in smartphone- +based facial recognition applications. Nevertheless, there do exist +some circumstances where users may want to place their phone +on a table during authentication. Our tremor recognition with syn- +thesized virtual IMUs will not work in this case due to the lack +of camera motion. Similarly, a spoofing attacker cannot authen- +ticate successfully in this case without providing the camera the +correct motion. To enable users to authenticate without holding +their phones, we believe future works may look into other sensor +side channels that acquire a different type of user biometric infor- +mation such as body-radiated electromagnetic/heat energy without +requiring direct contact with the phone. +4.2 +Synthesis Methodology +Different methodologies can be used to synthesize virtual IMUs +from camera motion side channels. For example, a completely +model-based methodology requires understanding 𝑓 (·) and ˜𝑓 (·). +Although the most accurate, it requires thorough understandings of +every targeted camera system and is challenging. Another possible +methodology is to completely rely on neural network to process +the raw videos and let the network figure out ˜𝑓{𝑣𝑖 }, which is sim- +ilar to previous work of inferring sounds from object motions in +videos [25]. This methodology requires intensive computation re- +sources and data collection. This work focuses on the middle ground +by investigating a model-informed methodology that constructs +˜𝑓{𝑣𝑖 } based upon the concepts of image registration. Image registra- +tion is the process of overlaying two or more images of the same +scene that are taken at different times, from different viewpoints, +and/or by different sensors [43]. The methodology aims to extract +both inter-frame motions and intra-frame motions. +4.2.1 +Understand Motion Modulation. To construct ˜𝑓{𝑣𝑖 }, the first +step is to understand how motion signals are modulated onto im- +age streams. We analyze the motion modulation process from two +different perspectives. + +NSPW’22, October 24–27, 2022, New Hampshire, USA +Yan Long and Kevin Fu +Translation +Similarity +Euclidean +Projective + Sensor +(IMU/Camera) +X +Y +Z +Roll +Pitch +Yaw +IMU Output +Sensor Motion +Cam Output +X +Y +Z +Pitch +Yaw +Roll +Accl X Accl Y Accl Z Gyro X Gyro Y Gyro Z +Trans. Trans. Sim. Proj. Proj. Euc. +Figure 2: Types of 2D image transformations corresponding +to the type of camera motion and motion readings measured +by physical IMUs. +Frame Transformation. The frame transformation perspec- +tive considers changes of the frames subjected to camera motions +as 2D image transformations. Figure 2 shows the possible image +transformations corresponding to motion on each one of the six +real-world axes and the measurements of physical IMUs. As a re- +sult, motions that can be measured by IMUs can also be mapped to +inter-frame variations of the camera videos. +Rolling Shutter. Besides inter-frame variations, the rolling shut- +ter property of most cameras on portable devices can generate intra- +frame variations that embed high-frequency motion. Rolling shutter +is the shutter mechanism of commercial CMOS cameras, which +exposes and samples the rows of an image sensor sequentially in- +stead of simultaneously as in a global shutter [21]. If viewing the +possible 2D image transformations as bases, rolling shutter com- +bine multiple transformations into a single frame. It increases the +effective sample rate of the motion signals provided by the camera +side channel. +Based on the knowledge of how camera motion is modulated +onto images, two corresponding categories of virtual IMU synthesis +methods are introduced next to measure low-frequency and high- +frequency information respectively. +4.2.2 +Low-frequency Information Measurement. The frame trans- +formation perspective enables measurements of low-frequency +components. It perceives the difference between two frames as +the result of a single motion vector composed of single-axis mo- +tions (Figure 2) within the period of one frame. The camera imaging +process thus becomes the sampling process of the measurable mo- +tion signals with a sample rate that is the same as the video frame +rate, e.g., 30 Hz in case of 30 fps videos. Theoretically, all image reg- +istration methods are applicable to extract inter-frame variations. +We discuss one possible construction. +Image Transformation Estimation (ITE). A straightforward +way of extracting the frame differences is registering the frames +with respect to a reference frame by estimating the 2D image trans- +formations needed to warp the reference frame to the other frames +as has been explored in [30]. Each 2D transformation estimation +generates a 3-by-3 transformation matrix. By concatenating each en- +try of different transformation matrices chronologically, it produces +9 vectors that represent the output of ˜𝑓{𝑣𝑖 }. Diverse algorithmic +implementations of this method are possible. This works uses an +image registration implementation based on phase correlation [28]. +4.2.3 +High-frequency Information Measurement. The rolling shut- +ter perspective allows for the extraction of intra-frame high-frequency +variations. It perceives the difference between two frames as the +result of multiple sequential motion vectors. The number of motion +vectors is the same as the number of rows of the camera imag- +ing sensor as each row is exposed and sampled sequentially. The +effective sample rate is thus the row-scanning rate of the rolling +shutter, which is higher than 30 kHz for most commercial cameras. +Nevertheless, not all signals within its Nyquist frequency can be +recovered, as the non-zero exposure time causes motion blurs and +attenuate the higher-frequency signals [11]. Similarly, a possible +construction is introduced below. +Rolling Shutter Estimation (RSE). Methods of rolling shut- +ter estimation still compares different frames, but performs such +comparison on the even smaller granularity level of rows or indi- +vidual pixels. Then, the methods concatenate the values generated +by the comparison first across different rows of a single frame, +and then across different frames to form the motion signal vec- +tors. With the proposed methodology, this work converts rolling +shutter estimation into a pixel-level image registration problem. +Algorithms capable of pixel-level registration often generate dis- +placement fields, i.e., matrices of the same size as the registered +images, on the X and Y directions. The produced matrices are appar- +ently high-dimension and difficult to process. We can then group +the matrices column-wise and average the columns in each group +to produce easily understandable signals. This work uses a diffeo- +morphic image registration method [38] to implement RSE. +4.2.4 +Demonstration. Figure 3 shows the motion signals measured +by a physical IMU (408 Hz sample rate) and virtual sensors using +ITE and RSE methods. A Google Pixel 2 smartphone held by a +person recorded the physical IMU readings and camera videos si- +multaneously, where the postural hand tremor of the person caused +the camera motion. The ITE and RSE methods have sample rates +of 30 Hz and 34 kHz respectively. The figure only displays a single +vector of the physical and virtual sensor measurements respectively +that represents the horizontal motion to simplify the visualization. +Figure 3 (a) and (b) shows the measured signals with the video +stabilization functionality being off and on respectively. When +video stabilization is off, the virtual sensor outputs of both the +ITE and RSE method show strong correlation with the physical +IMU measurements. It is also clear that a 30 Hz sample rate is not +sufficient to capture all the motion, as the ITE method’s signal +shows larger distortions than that of the RSE method. When video +stabilization is turned on, the camera motion signals deviate more +from the IMU readings as expected. Although the signal of RSE +method still shows observable correlation with the IMU signal, ITE +produces seemingly uncorrelated signals. +4.3 +Experiment +We conduct preliminary tests with 4 people and a Google Pixel 2 +smartphone. The 4 participants are all healthy males with similar +ages, heights, and weights. As a proof-of-concept instead of an +actual system product, we regard facial recognition and tremor +recognition as two decoupled problems and test them separately. +The tremor recognition mechanism can be regarded as an addi- +tional layer of protection besides the existing facial recognition + +/ +X +1 +Z +Pitch +Yaw +Roll +IMU Output +AcclX +AcclY +AcclZ +GyroX +GyroY +GyroZ +Cam Output +Trans. +Trans. +Sim. +Proj. +Proj. +Euc.SensorMotionNSPW’22, October 24–27, 2022, New Hampshire, USA +(a) +(b) +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +-1 +0 +1 +Amplitude +IMU Accelerometer +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +-1 +0 +1 +Amplitude +Image Transformation Estimation +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time(s) +-1 +0 +1 +Amplitude +Rolling Shutter Estimation +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +-1 +0 +1 +Amplitude +IMU Accelerometer +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +-1 +0 +1 +Amplitude +Image Transformation Estimation +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time(s) +-1 +0 +1 +Amplitude +Rolling Shutter Estimation +Figure 3: Measurements of physical IMU accelerometer (408 Hz) and virtual IMU synthesized with the ITE and RSE methods +from videos (30 fps frame rate, 1080p resolution) in 5 seconds. Amplitudes are normalized to compared different measurement +approaches. (a) Videos stabilization is off. (b) Videos stabilization is turned on. Strategically disabling sensor side channel +mitigation mechanisms boosts up virtual sensors’ capability for measurand authentication. +system. We investigate the impact of disabling and enabling video +stabilization in both of the two tests. +The objective of testing tremor recognition is to verify the ef- +fectiveness of the synthesized IMUs. To that end, we also recorded +the physical IMU readings for comparison. The objective of testing +facial recognition is two-fold. First, it is important to inspect if the +postural hand tremor of different people can already make a differ- +ence in the original facial recognition systems without synthesis of +virtual sensors. This verifies the necessity of constructing dedicated +virtual IMUs. Second, since turning off video stabilization may lead +to better virtual sensor performance, it is also necessary to inspect +if it would degrade the performance of facial recognition given that +the videos are more shaky due to unmitigated camera motion. +4.3.1 +Data Collection. The 4 participants act as the legitimate user +in turn and the remaining 3 participants act as the imposters. During +the legitimate user sessions, each legitimate user holds the phone +and records his own face for 30 times. We refer to these videos as +legitimate videos. During the spoofing attack sessions, each of the +3 imposters holds the phone but records the face of the legitimate +user standing beside the imposter for 6 times to mimic a perfect +silicone mask as assumed in Section 4.1. We refer to these videos as +imposter videos. Each video recording is about 6s in length and the +physical IMU readings are recorded simultaneously. The procedure +is carried out first with video stabilization disabled. At the end, each +participant recorded 48 videos when he held the phone with 30 of +them being legitimate videos and the other 18 being imposter videos. +We then repeat the procedure with video stabilization enabled. The +total 384 videos (192 videos each set) are used for testing facial +recognition and tremor recognition. +4.3.2 +Test Procedure & Result. We generalize the authentication +problem as an identification problem and use classification models +to measure the effectiveness of the two authentication schemes +against the spoofing attack. +Facial recognition Procedure. We tested MobileFaceNets [8] +as the classification model which is a widely used facial recognition +model designed for mobile platforms. 80% of each person’s legit- +imate videos are used to enroll their faces. The remaining 20% of +legitimate videos together with all imposter videos that contain +faces of the legitimate users are used as the authentication test data. +Facial recognition Result. Both the legitimate users and im- +posters’ videos authenticated with 100% success rate no matter +the video stabilization was enabled or disabled. As expected, the +results suggest that existing face authentication systems are mostly +likely not designed to utilize camera motion side channel informa- +tion. Mapping it to Equation 5, it suggests E[A𝑢 (𝑚)|𝐻0] → 1 and +E[1 − A𝑢 (𝑚)|𝐻1] → 0 for the system under this specific spoofing +attack. The results also show that disabling video stabilization to +allow for more capable virtual IMUs did not affect the performance +of the original facial recognition system. +Tremor Recognition Procedure. For each video, we generate +virtual IMU measurements using both the ITE and RSE methods. +We extract common time-domain and frequency-domain features +as the ones used in [5, 23]. As a simple proof-of-concept, we did +not use sophisticated machine learning models but directly utilized +Matlab’s implementation of support vector machine (SVM) with a +quadratic kernel and the default hyper-parameters [2]. 5-fold cross +validation was performed in the training phase along with a one-vs- +one multi-class classification method. Similar to facial recognition, +for each legitimate user we use 80% of the legitimate videos (24 +videos) in the training phase and the remaining 20% legitimate +videos (6 videos) together with all imposter videos (18 videos, 6 +from each of the three imposters) as authentication test data. We +then calculate the true positive and true negative rates on the test + +NSPW’22, October 24–27, 2022, New Hampshire, USA +Yan Long and Kevin Fu +set. To provide comparisons, we repeat the same procedure also for +the physical IMU data. +Tremor Recognition Result. Table 1 shows the results of tremor +recognition. Virtual IMU using RSE had performance approaching +that of the physical IMU. It suggests that under this specific spoofing +attack, E[A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1])|𝐻0] → 0.125 and E[1−A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1])|𝐻1] → +0.083 if using an AND logic to combine facial and tremor recogni- +tion decisions. This results in 𝐸𝑚 −𝐸𝑢 → −0.875𝑐1 +0.083𝑐2, which +is highly likely to be smaller than 0. It is also clear that disabling +video stabilization improves the performance of virtual IMUs. +4.3.3 +Summary & Implication. Our preliminary tests indicate a +high probability that integrating user postural hand tremor infor- +mation from camera motion side channels will help existing facial +recognition systems defend against visual spoofing attacks. Test +results show MobileFaceNets could recognize legitimate users with +100% accuracy but could not detect (with 0% accuracy) a powerful +silicone mask spoofing attack that almost perfectly replicates visual +features of users. This behavior is not a design defect of existing +facial recognition systems, but an anticipated outcome of only using +visual information during an authentication process. On the other +hand, virtual IMUs synthesized from camera motion channel were +able to detect such a visual spoofing attack with over 87.5% accu- +racy at a cost of reducing true positive rate to 91.7%. The simplest +approach of integrating virtual sensor into existing facial recogni- +tion systems is to have a standalone tremor recognition module +that processes camera motion information in the videos, and have +the system declare a legitimate user only when both this tremor +recognition module and the original facial recognition module de- +clare it simultaneously. In this way, the overall system’s security +performance increases in the face of facial spoofing attacks even +with a lower true positive rate. This result also suggests when a +physical sensor system has poor performance on a security task, it +is easy to produce an obvious marginal benefit on the system’s per- +formance by integrating sensor side channel information. Of course, +a more sophisticated decision system can tune its weights on the +facial and tremor recognition modules to strike a better balance +between usability and security. +Beyond camera motion side channels, our tests also provide one +viable data point for the general concept of utilizing sensor side +channels and reveal some common problems it faces. For example, +we expect the same problem of usability-security trade-off in using +virtual sensors synthesized from sensor side channels alongside the +original physical sensors. Essentially, physical sensors and synthe- +sized virtual sensors provide two streams of information, each one +of which is more reliable in one task but also unreliable in another +task. The design trade-off appears when the overall system needs +to complete both tasks to achieve its functionality. +4.3.4 +Limitation & Future Work. With the goal of showing a proof- +of-concept example, our experiment provides empirical statistical +evidence for the benefit of utilizing camera motion side channels +only based on a very limited data distribution. The limitations of +tested data lie in the following 4 main dimensions. +First, the 4 young male participants may not provide a high +enough degree of demographic diversity, especially for evaluating +postural hand tremors which are highly dependent on age, gen- +der, and health conditions [19]. While we based our choice of the +Table 1: Test Accuracy of Tremor Recognition +Physical IMU +Virtual ITE +Virtual RSE +TPR +TNR +TPR +TNR +TPR +TNR +Stab. OFF +95.8% +94.4% +62.5% +65.3% +91.7% +87.5% +Stab. ON +95.8% +93.1% +45.8% +41.7% +70.8% +72.2% +TPR (true positive rate) and TNR (true negative rate) are the percentages of +correctly recognizing a legitimate user and a perfect silicone mask spoofing +attack respectively. In comparison, MobileFaceNets had TPR=100% and +TNR=0% in our test. +4 participants on the hypothesis that more similar participants +produce less distinct tremor patterns and thus help us estimate a +lower bound of tremor recognition performance, we believe study- +ing more diverse groups of people will generate new insights into +recognition performance variability and possible strategies of recog- +nition algorithm design. +Second, we collected 30 samples of legitimate-user videos and +18 spoofing attack videos for each legitimate user’s authentication +session within a single day. We find this initial set of samples pro- +vided evidence to suggest the potential of utilizing hand tremor +information from camera side channels to enhance existing facial +recognition system’s security. It is possible that tremor patterns can +change with time. Although previous research shows hand tremor +remains stable after 78 days [12], a longer duration needs to be +investigated in future complete. The recognition system may need +to periodically update its database if tremor pattern is found to vary +over time. +Third, we emulated perfect silicone masks by using the real faces +of legitimate users. This only provides an estimate of the upper +bound of the overall recognition system’s performance improve- +ment when tremor recognition is used. Specifically, the benefit of +including tremor recognition may get lower when a worse-quality +silicone mask is used because the damage the attack can do to +the original unimodal authentication system is lower while tremor +recognition still causes a decrease in the true positive rate. As a +result, we suggest future works test different qualities of silicone +masks on popular facial recognition systems to better assess the +benefit of including virtual IMUs for tremor recognition. +Fourth, the decoupling of facial recognition and tremor recogni- +tion problems in this proof-of-concept case study prevents us from +utilizing the temporal correlation between the facial and camera +motion signals and investigating the impact of the correlation infor- +mation. Intuitively, systems that inspect such temporal correlation +information require spoofing attackers to further achieve synchro- +nization between the physical and virtual sensors’ data streams and +thus provide additional protection. We envision real-world prod- +ucts building upon the virtual sensors authentication concept to uti- +lize deep-learning approaches for processing temporally-correlated +physical and virtual sensors’ information. +5 +DISCUSSION +Below we discuss the major areas of possible future work and +interesting research questions. +Sensor Side Channel Models. To support future applications +of sensor side channels, we believe more concrete and computable +mathematical models than the framework proposed in Section 3 +are needed as the current framework relies on abstract concepts + +NSPW’22, October 24–27, 2022, New Hampshire, USA +instead of rigorous mathematical derivations. We envision future +models to have the following features. First, they need to enable +exact definitions and determination of different types of sensor +side channels by providing the algorithms for calculating signal +correlations and threshold values. Second, they need to provide +quantitative metrics for measuring the usability-security trade-off +mentioned in Section 4.3.3. Third, they need to delineate mecha- +nisms for measuring the available signal quality and bandwidth of +side channel measurement components. +Security for Sensor Side Channel Authentication. Techni- +cally, inseparable sensor side channels also provide the informa- +tion needed for measurand authentication. We advocate the use of +separable and controllable sensor side channels because they are +protected from adversaries that exploit unmitigated side channels +(Section 2.1). Nevertheless, risks of malicious exploitation still exist +within authentication time. It is thus necessary for future works +to consider how to ensure that side channels benefit the defender, +but not adversaries that attempt eavesdropping and transduction +attacks, during authentication. +We believe an access control and permission system that is simi- +lar to existing systems managing physical sensors on mobile plat- +forms (e.g., Android) can be employed to prevent eavesdropping +attacks. Virtual sensor entries can potentially be created and in- +tegrated into existing permission systems so that knowledge and +methodology of solving physical sensors’ problems can also benefit +virtual sensors. Transduction attacks, on the other hand, are harder +to address. In the context of sensor side channel based measurand +authentication, transduction attacks can be generalized as authen- +tication spoofing that tries to modify perceived characteristics of +the byproducts, measurers, and environments. As a result, existing +methodologies of spoofing detection may be applied. In summary, +we believe there are opportunities to address the problems of virtual +sensors by reflecting on existing methodology for physical sensors. +Side Channels vs. Legitimate Channels. We believe there +will be an interesting phenomenon that sensor side channels are +turned into legitimate communication channels when active con- +trols and dedicated APIs are developed to support as well as regulate +the use of sensor side channels in the future. After all, the key dif- +ference between side channels and legitimate channels is whether +the channels are designed, intended, and allowed by the system’s +security policy or not. When such side channels are regarded as +legitimate channels, however, new side-channel information may +again be discovered to be embedded in such “legitimate” informa- +tion as hardware and computation technologies keep advancing +and extending the boundary of recoverable physical signals. We +thus believe it is necessary for researchers to take a development +perspective and periodically examine the security implications of +sensor side channels. +Fewer Sensors via Sensor Repurposing. In a broader context, +we believe the technique of synthesizing virtual sensors from sensor +side channels aligns with the general idea of repurposing sensors +for different sensing tasks. Essentially, we are trying to shift sensor +hardware functionalities to the software space by understanding the +transformation between different forms of signal energy and car- +rying out additional model-based computations. In contrast to the +current trend of deploying more and more sensors in the Internet +of Things era, we cannot help thinking if such sensor repurposing +ideas would allow us to reduce the number of physical sensors and +achieve more abstract and manageable sensor peripheral systems +that are subjected to smaller attack surfaces. +Besides reducing the number of physical sensors, the technique +could also be applied to enhance existing systems that require new +functionalities but have harsh environmental conditions where a +hardware update is challenging. This idea is revealed in the example +of NASA’s Voyager 1 spacecraft which needed to measure plasma +density in order to determine its location relative to the heliosphere. +Voyager 1’s plasma spectrometer stopped working in 1980, making +a direct plasma density measurement impossible. However, the op- +eration team learned that our sun sometimes emits shock waves +that can cause the plasma surrounding the spacecraft to oscillate. +The team then measured the oscillation using Voyager 1’s onboard +plasma wave sensing system as a proxy of the plasma density [18], +essentially synthesizing a virtual plasma density sensor by under- +standing the energy transformations. +6 +CONCLUSION +This paper argued that analog sensor side channels can benefit +defenders by providing an opportunity to authenticate the sensor +measurands. Future sensor designs can consider actively controlling +sensor side channels after finding ways to mitigate these channels, +instead of simply eliminating sensor side channels. We first in- +troduced a framework for defining and characterizing sensor side +channels, and then formulated the problem of measurand authenti- +cation using virtual sensors synthesized from sensor side channels. +We also introduced three specific ways of applying the model of +measurand authentication by verifying signal byproducts, sensor +measurers, and sensor environments respectively, and provided +examples of each case. +Synthesizing virtual sensors from the side channels of physical +sensors formulates a mechanism for repurposing existing sensor +hardware to harvest extra modalities of information. We believe +the applications of this mechanism can potentially span a much +larger scope than authentication. 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Image +and vision computing 21, 11 (2003), 977–1000. + diff --git a/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/load_file.txt b/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..652a0c08a21ee1ebda2eb4ff6a29ac2fd230e792 --- /dev/null +++ b/xtFKT4oBgHgl3EQfLS1R/content/tmp_files/load_file.txt @@ -0,0 +1,759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf,len=758 +page_content='Side Auth: Synthesizing Virtual Sensors for Authentication Yan Long University of Michigan yanlong@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='edu Kevin Fu Northeastern University k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='fu@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='edu ABSTRACT While the embedded security research community aims to protect systems by reducing analog sensor side channels, our work argues that sensor side channels can be beneficial to defenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This work introduces the general problem of synthesizing virtual sensors from existing circuits to authenticate physical sensors’ measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We investigate how to apply this approach and present a preliminary analytical framework and definitions for sensors side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To illustrate the general concept, we provide a proof-of-concept case study to synthesize a virtual inertial measurement unit from a camera motion side channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our work also provides an example of applying this technique to protect facial recognition against silicon mask spoofing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Finally, we discuss downstream problems of how to ensure that side channels benefit the defender, but not the adversary, during authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 1 INTRODUCTION Sensor side channels enable an adversary to violate integrity of sensor outputs by influencing or controlling the sensor with trans- duction attacks [15, 39], or to eavesdrop on sensitive information and compromise confidentiality by exploiting flaws in sensor and system designs [3, 5, 22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, the eavesdropping exam- ple PIN Skimmer [30] shows that adversaries can infer smartphone touchscreen inputs by exploiting side channel motion informa- tion captured by smartphone cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' While the security research community invested significant effort identifying and mitigating analog sensor side channels, our work argues that it can be benefi- cial to embrace, understand, and control analog sensor side channels instead of simply eliminating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This is moti- vated by our observation that such side channel information may also be used for authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, extensive research has been conducted on using dedicated motion sensors to capture smartphone touch dynamics for continuous implicit user authenti- cation [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Relating it to PIN Skimmer, a natural question arises as to whether cameras support such authentication when dedicated motion sensors are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We thus propose and investigate the problem of how to utilize sensor side channels for defensive purposes such as multimodal authentication by synthesizing virtual sensors from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Side channels are inherent to analog sensors’ physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' There exist a considerable number of potential sensor side channels besides those revealed by transduction and eavesdropping attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' However, Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' NSPW’22, October 24–27, 2022, New Hampshire, USA © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='nnnnnnn Figure 1: Sensor side channels are different from conven- tional side channels as they measure the measurement pro- cesses instead of computation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sensor side chan- nels can measure the byproduct, measurer, and environ- ment to verify authenticity of intended sensor measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' most of these side channels are deliberately “closed” in the design phase by employing mitigation mechanisms such as calibration and noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is foreseeable that sensor and system designers will also try to mitigate newly discovered side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This work argues a different perspective and approach to embrace such sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' If these side channels can be used in a beneficial way, we envision future designs allowing mitigation mechanisms to be strategically disabled or downgraded when needed such as during authentication sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We provide a preliminary analytical framework for modeling analog sensor side channels and explaining the origins and charac- teristics of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The framework categorizes sensor side channels according to their separability from intended signals and whether they have controllable mitigation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Based on the frame- work, we define the problem of virtual sensor synthesis for mul- timodal measurand authentication and summarize three possible ways of applying this approach (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' First, by verifying sig- natures of signal byproducts and asking the question “What is the probability that Alice generated both the measurands and byprod- ucts?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Second, by verifying the person performing the measurement and asking “What is the probability that Alice generated the measur- ands if Bob was the measurer?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Third, by verifying the environment of the measurement process and asking “What is the probability that Alice generated the measurands if the measurement was taken in location B?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A proof-of-concept case study further concretizes the concepts and related considerations by studying a camera motion side chan- nel that enables cameras to sense out-of-sight motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This side channel is caused by mechanical connections between camera de- vices and adjacent objects in motion such as a hand holding the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We propose a methodology for synthesizing virtual inertial measurement units (IMUs) from this side channel that can extract both inter-frame low-frequency and intra-frame high-frequency motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The case study discusses this side channel’s potential application in helping facial recognition systems defend against 3D silicon mask spoofing attacks by verifying postural hand arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='11745v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='CR] 27 Jan 2023 Conventional Sensor Side Channel Side Channel Memory f() Measurands S Byproduct int m Output Input Sensor Measurer CPU S Environment sideNSPW’22, October 24–27, 2022, New Hampshire, USA Yan Long and Kevin Fu tremor motion of the person holding the camera device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Prelimi- nary test with 4 people suggests the camera motion side channel help reduce false positive rates by up to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It also shows that disabling video stabilization enables higher performance, empha- sizing the benefits of strategically disabling side channel mitigation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Finally, we discuss the possible issues of temporar- ily opening sensor side channels during authentication and the directions future works may take to address the issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our main contributions are summarized as follows: A new paradigm to embrace and harness analog sensor side channels for defensive purposes via active control or influ- ence of sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' An analytical framework to enable definition and character- ization of sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The framework introduces the concept of virtual sensor synthesis for multimodal sensor measurand authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A case study and methodology of synthesizing virtual IMUs from camera motion side channels for enhancing perfor- mance of facial recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Virtual IMUs decrease false positive rates when facial recognition is subjected to emulated 3D silicon mask spoofing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 2 BACKGROUND & RELATED WORK Side channels are unintended information output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The idea of synthesizing virtual sensors for authentication can be traced back to the general concept of using side channel information to identify people in forensic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, forensic hand- writing recognition allows one to determine the identity of a letter writer even though the letter was never intended to convey identity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In the field of digital forensics [14], the authenticity of digital evidence can be verified by cross-correlating file data and metadata with other contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, fraud- ulent documents were reported to be detected by analyzing their choices of font [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our paper investigates how we can design mechanisms to actively utilize side channel information in sensor readings for defending computer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We introduce research works related to this idea in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Sensor Side Channel Based Attacks Sensor side channels have been actively exploited in two lines of research, namely transduction and eavesdropping attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This section provides some background and examples of these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Different from these works that try to compromise information in- tegrity and confidentiality of sensor systems, our work investigates how designers may defend sensor systems by actively controlling and utilizing sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Transduction Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Transduction attacks injects analog sig- nals into sensors where victim sensor circuitry transduces an at- tacker’s malicious physical signals to untrustworthy sensor mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such malicious physical signals can often be in different physical modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', acoustic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' optical) or frequency ranges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', audible vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' ultrasound) than what the sensors are designed to sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, Light Commands [33] uses lasers to inject false speech signals into microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Works such as Walnut [31, 36] use acoustic injections to influence and control the output of MEMS gyroscopes and accelerometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Ghost Talk [20] uses radio waves to inject audio signals into microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' An SoK and a survey [15, 39] provide a comprehensive review of theses attacks and correspond- ing mitigation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Eavesdropping Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sensor side channels are also exploited for eavesdropping out-of-band information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, PIN Skim- mer [30] used a camera-based side channel to infer PIN inputs by ex- ploiting correlations between smartphone camera orientations and tapping locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Several works such as Gyrophone [3, 5, 22] use smartphone IMUs which contain accelerometers and gyroscopes to eavesdrop speech by exploiting side channels enabled by signal aliasing in analog-to-digital converters (ADCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Using Conventional Non-sensor Side Channels For Defensive Purposes Although we could not find related research that investigates the concept of utilizing sensor side channels for defending a system, we found that a few previous works explored using non-sensor side channels for machine-to-machine authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' [29] proposes to use the key-dependent side channel information in wireless commu- nication channels to enhance existing cryptographic protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' [10] presents an extension by analyzing practical and security issues of the protocol in [29] and providing fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Compared to them, this work focuses on the concept of sensor side channel and authenti- cation of sensor’s measurands instead of computation-generated information as the previous works did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Non-sensor side channels are also utilized in other applications such as code-execution moni- toring and intrusion detection [4, 9, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3 PROBLEM FORMULATION This section defines the problem of using sensor side channels for measurand authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our paper proposes the concept of using sensor side channels for authentication as a new direction of research for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We also fill a gap by suggesting a mathematical definition of sensor side channels, beginning with a framework for defining and categorizing sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We then introduce the problem of synthesizing virtual sensors and using them for multimodal sensor measurand authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Sensor Side Channel Analytical Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A sensor, or transducer, can be modeled as a function that maps physical measurands to digital measurements over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A measurand is a quantity that a sensor intends to measure [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Different types of sensors are designed to measure different modal- ities of measurands such as sound, temperature, motion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Users who are informed of the apparent purpose and specifications of sensors often see a sensor as the following function over a single variable of the measurand: 𝑚 = 𝑓 (𝑠𝑖𝑛𝑡 ) (1) where 𝑚 and 𝑠𝑖𝑛𝑡 denote the digital measurements and analog mea- surand respectively and 𝑓 (·) denotes the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Sensor Side Channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although Equation 1 provides average sensor users a clean and easy abstraction, actual sensor implemen- tations are much “dirtier’ and introduce numerous hidden variables to the equation that result in unintended components in measure- ment 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For instance, every conductor wire can be regarded as NSPW’22, October 24–27, 2022, New Hampshire, USA an unintentional antenna, leading to side channels that convert electromagnetic energy to measurements of non-electromagnetic sensors [20, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In this case, hidden variables related to electro- magnetic energy in the environment should be added to Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Another example of such variables is temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Semiconductors made of silicon are inherently sensitive to heat due to its ability to excite electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' So technically, Equation 1 should also include temperature as a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Electromagnetic energy and temperature are just examples of hidden variables associated to the underlying physical characteristics of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' There are also hidden variables caused by design flaws and uncontrollable variations in the manu- facture processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Thus, Equation 1 should be modified to enable a side channel-aware modeling of sensors: 𝑚 = 𝑓 (𝑠𝑖𝑛𝑡,𝑠𝑠𝑖𝑑𝑒 ), 𝑠𝑠𝑖𝑑𝑒 = [𝑠𝑣1,𝑠𝑣2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='] (2) where 𝑠𝑠𝑖𝑑𝑒 represent the set of all these hidden variables that can potentially lead to side channels attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The comparison between Equation 1 and 2 shows that the gap between users’ understanding and sensors’ actual implementation gives birth to sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' On a high level, we believe the gap can also be attributed to the insufficient specifications of legitimate and illegitimate sensor behaviors in the existing sys- tem’s security policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Note that this differs from conventional non-sensor side channels where side channels bypass the clearly specified security policies [16]: there are often no dedicated security policies for sensors yet in existing sytems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Building upon previous side channel research [32, 42], we tentatively define sensor side channels as the following: A sensor side channel is a communication channel that allows someone to recover secret information using unintended sensor measurement components in a way that violates the associated system’s security expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sensor side channels are sometimes more conceptually difficult to recognize than conventional non-sensor side channels such as differential power analysis channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The reason is that non-sensor side channels are used to mainly measure computation processes where there exists a clear boundary between computation and measurement whereas sensor side channels are used to measure the measurement processes themselves (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A possible way of identifying sensor side channels is to test the hypothesis that the analog signal of a variable 𝑣𝑖 correlates with 𝑚 with certain significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', |𝐶𝑜𝑟𝑟 (𝑚,𝑠𝑣𝑖 )| > 𝛼, 𝑠𝑣𝑖 ∈ 𝑠𝑠𝑖𝑑𝑒 (3) where 𝛼 is a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Note that this work does not discuss the actual choice of threshold values and correlation functions since they can be flexible depending on the actual application scenarios and security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In cases where it is challenging to project 𝑚 and 𝑠𝑣𝑖 to the same vector space in order to compute correlation scores, other methods such as supervised classification can also be used if 𝑠𝑣𝑖 can be converted into data labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3 Separability and Controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The unintended components in the measurements are caused by the existence of 𝑠𝑠𝑖𝑑𝑒 and can be either separable or inseparable from the intended components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The separability between the intended and unintended components is the key that decides whether a side channel can be mitigated and controlled or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Conceptually, separable components can be defined as the following: there exists at least one function ˜𝑓 (·) that can break 𝑚 down into intended and unintended components such that those components only correlate (with significance) with the measurand and other hidden variables respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', ∃ ˜𝑓 (·) 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' ˜𝑓 (𝑚) = [𝑚𝑖𝑛𝑡,𝑚𝑠𝑖𝑑𝑒 ], 𝑚𝑠𝑖𝑑𝑒 = [𝑚𝑣1,𝑚𝑣2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='], |𝐶𝑜𝑟𝑟 (𝑚𝑖𝑛𝑡,𝑠𝑖𝑛𝑡 )| > 𝛼𝑖1, |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑣𝑖 )| > 𝛼𝑖2, |𝐶𝑜𝑟𝑟 (𝑚𝑖𝑛𝑡,𝑠𝑣𝑖 )| < 𝛽𝑖1, |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑖𝑛𝑡 )| < 𝛽𝑖2 (4) When a sensor side channel has separable components, we say it is a separable side channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Separability is decided by sensor im- plementation 𝑓 (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Side channels with inseparable components in existing sensor implementations led to the various unsolvable at- tacks against sensors (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1) because designers cannot extract only the intended components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Theoretically, those with separable components can be mitigated by mechanisms referred to as compensation, calibration and noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such mitigation mechanisms can be abstracted as another function 𝑔(·) that suppresses the unintended components in the output of ˜𝑓 (·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', 𝑔( ˜𝑓 (𝑚)) = 𝑚𝑖𝑛𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' If the mitigation mechanisms can be both turned on and off, the user of the sensor system then have full control of the sensor side channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We call such a sensor side channel controllable: A controllable sensor side channel is one whose corresponding unintended measurement component is separable from the intended component and can be suppressed by a mitigation mechanism that can be enabled and disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4 Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We provide some existing examples of each cate- gory of sensor side channels to shed light on the differences and possible future evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The Gyrophone eavesdropping attack [22] and its follow-up works [3, 5] use an aliasing-enabled inseparable acoustic side channel in smartphone IMUs to recover speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' These IMUs have intended acceleration and angular velocity measurands mostly under the frequency range of human speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' However, due to the lack of effective analog low-pass filtering before the ADC, aliases of the high-frequency speech signals exist in the output of ADC and enable adversaries to recover speech information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Furthermore, the aliases cannot be separated from the intended motion signals since they are in the same frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Intuitively, adding analog filters to the sensors make this acoustic side channel separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In order to be controllable, the sensor API may further allow CPU to enable and disable the filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Separable But Uncontrollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Those seemingly intact sen- sors that have not been reported vulnerable to side channel-based attacks also have inherent side channels, but just in a suppressed manner thus these channels are not exposed to attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Take sensors’ heat sensitivity mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 as an exam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' MEMS humidity sensors, gyroscopes, accelerometers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', are widely equipped with temperature-compensated designs or online thermal calibration procedures [7, 13, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It can be anticipated that if the compensation and calibration mechanisms can be temporarily disabled, these sensors’ measurements will exhibit significant cor- relation with the ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In this way, the separable side channel becomes controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' NSPW’22, October 24–27, 2022, New Hampshire, USA Yan Long and Kevin Fu Controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' There already exist sensors with controllable side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A good example is handheld cameras getting equipped with video stabilization mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Camera motion is often re- garded as side effects that degrade the quality of the intended signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', the scene in the field of view of the camera [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Video stabiliza- tion mechanisms, including electronic image stabilization (EIS) and optical image stabilization (OIS), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', are implemented to mitigate these side effects by optically or electronically reducing the un- wanted image scene movements caused by camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Many operating systems such as Android allow app developers to choose if these video stabilization mechanisms will be turned on or off when the underlying camera hardware offers the API to control it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' However, it is worth noting that such existing controllable side channels are most likely byproducts of OS designers’ conventions of providing more fine-grained interfaces, especially for open-source OS like Android which allows users to control EIS and OIS sepa- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In contrast, iOS does not allow explicit and separate control of EIS and OIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such a large degree of control is provided to support more potential use cases and enhance usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, users may want to disable smartphone’s built-in optical image stabiliza- tion when using an external gimbal because the two can interfere with each other and produce extra image distortions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To the best of our knowledge, these existing controllable side channels have not been explored to enhance the security of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is possible to convert existing inseparable or uncontrollable side channels into controllable side channels by im- proving sensor designs, as has been suggested by the increasing popularity of video stabilization in cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Thus, it is important to think from a perspective of technology development when con- sidering benefits of sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Furthermore, protecting physical sensors from side channel attacks often already means transforming inseparable side channels to be separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' With some additional effort of making mitigation mechanisms controllable in- stead of forever-on, sensor side channels can be used in a beneficial and controlled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The following discussions assume sensors have controllable side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Measurands Authentication Using Synthesized Virtual Sensors 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Virtual Sensor Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A virtual sensor is a function that maps 𝑚 to 𝑚𝑣𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Ideally, the construction of ˜𝑓 (·) in Equation 4 al- ready presents such an overarching function that can measure both the intended and side channel components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such construction is apparently challenging since it needs to consider all possible side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Actual implementations can reduce the level of challenge by focusing on maximizing |𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑣𝑖 )| and −|𝐶𝑜𝑟𝑟 (𝑚𝑣𝑖,𝑠𝑖𝑛𝑡 )| for only the set of targeted hidden variable {𝑣𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We denote such a function specifically crafted for {𝑣𝑖} as ˜𝑓{𝑣𝑖 } and call them virtual sensor functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Problem Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We define the problem as a binary hy- pothesis test in a comparative manner by first referencing to the unimodal authentication on the physical sensor’s measurand alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Without virtual sensors, objects in Equation 1 including 𝑚, 𝑠𝑖𝑛𝑡 , and 𝑓 are all that the designer of the authentication system can perceive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Let there be a measurand with a true identity 𝐿 and a claimed iden- tity ˜𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The 𝐻1 and 𝐻0 hypotheses are ˜𝐿 = 𝐿 and ˜𝐿 ≠ 𝐿 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Denote the unimodal authentication system as A𝑢 : 𝑚 → {1, 0}, where it declares 𝐻1 and 𝐻0 when outputting 1 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We can then define the total error of the unimodal system 𝐸𝑢 as 𝐸𝑢 = 𝑐1P[declare 𝐻1|𝐻0] + 𝑐2P[declare 𝐻0|𝐻1] = 𝑐1E[A𝑢 (𝑚)|𝐻0] + 𝑐2E[1 − A𝑢 (𝑚)|𝐻1] (5) where P[·|·] and E[·|·] denotes conditional probability and expec- tation respectively, 𝑐1 and 𝑐2 denote the cost coefficients for false positive and false negatives respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Similarly, a multimodal authentication system with𝑛 synthesized virtual sensors can be denoted as A𝑚 : [𝑚𝑖𝑛𝑡,𝑚𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=',𝑚𝑣𝑛] → {1, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The total error 𝐸𝑚 is defined as 𝐸𝑚 = 𝑐1E[A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=',𝑚𝑣𝑛])|𝐻0] + 𝑐2E[1 − A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=',𝑚𝑣𝑛])|𝐻1] (6) As a result, the problem of synthesizing virtual sensors to au- thenticate the measurand in a multimodal manner can be defined as: Constructing virtual sensor functions ˜𝑓{𝑣𝑖 } and multimodal authentication system A𝑚 such that better performance is achieved for measurand authentication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', 𝐸𝑚 − 𝐸𝑢 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3 Security Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although multimodal authentication us- ing synthesized virtual sensors look similar to that using multiple physical sensors, it provides two different security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' First, it works with existing devices and media that only have a single physical sensor’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although high-end devices like smartphones are equipped with multiple physical sensors, there still exist lower-end devices that only serve a single purpose such as ultrasonic proximity detectors and humidity monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Further- more, sometimes it is needed to verify the identity of an object such as a photograph that has already been generated with only a single sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In this case, synthesized virtual sensors can extract additional information in a retrospective way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Second, it potentially provide more robustness against spoofing attacks on individual sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The level of attack difficulty depends on the complexity of ˜𝑓 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', how difficult it is to decouple and then modify different measurement components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Using multiple indi- vidual sensors such as cameras, accelerometers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', is equivalent to having a ˜𝑓 that does not need to decouple anything at all since the inputs already separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Conceptually, if we regard the mea- surements corresponding to different virtual or physical sensors as random variables, we can then regard their variances and co- variances as the entropy provided for authentication [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Virtual sensors potentially provides more entropy because the coupling between them adds to the covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such entropy originates from the intrinsic physics of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4 Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The general problem definition can be applied to different sources of side channel variables whose signatures correlate with the claimed identity of the measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Depending on the sources, we believe synthesized virtual sensors can be applied in the following three ways to verify authenticity of measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' NSPW’22, October 24–27, 2022, New Hampshire, USA Byproduct Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A physical process generating intended measurands is likely to generate other forms of energy as byprod- ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Let us explore the example of a loudspeaker that replays a person Alice’s speech recordings while a nearby microphone is listening to this replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Say there is someone claiming the speech audio collected by the microphone is coming from Alice herself speaking live and an investigator tries to verify this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The in- vestigator finds out that the loudspeaker also generates unintended, secondary byproducts in the form of structure-borne vibrations, electromagnetic emission, heat, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', which may be sensed by vir- tual sensors synthesized from the microphone’s side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' So, if these byproducts exist, the investigator knows it is not likely a legitimate recording of Alice’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In this case, the core authenti- cation question can be summarized as “What is the probability that Alice generated both the measurands and byproducts?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Measurer Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A Measurer is the person who makes measurements with a physical sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Measurers themselves gen- erate unintended emissions taking the form of physical signals containing certain signatures that correlate with the identity of measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, say there exists an unmodified photo of a person who is claimed to be Alice and an investigator tries to verify this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The investigator managed to find out that the camera operator who took this photo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', the measurer was Bob because Bob was speaking when he took the photo and his speech induced identifiable image blurs through a camera motion side channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' If the investigator also knows that Bob has never been in the vicinity of Alice, then the investigator knows the person in the photo is not Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Obviously, measurand authentication through measurer verification may require higher-level contextual information com- pared to byproduct verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The core authentication question is “What is the probability that Alice generated the measurands if Bob was the measurer?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Environment Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Similar to measurer verification, verifying the environment surrounding measurands also allows one to authenticate the measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Take the same example above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Say the photo has a temperature side channel that shows the ambient temperature was 104°F/40°C at the time of generating the photo, pointing to a location B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' If the investigator knows Alice has never been in location B, then the investigator knows the person in the photo is not Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The core authentication question is “What is the probability that Alice generated the measurands if the measurement was taken in location B?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4 CASE STUDY The case study demonstrates how to use camera motion side chan- nels (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4) to synthesize virtual IMUs that can collect pos- tural hand tremor information for measurand authentication in facial recognition applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It can be regarded an example of both byproduct and measurer verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Primer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Postural Tremor Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Tremor is the involuntary rhyth- mic movement of a human body part caused by reciprocal innerva- tions of muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Such involuntary movements are present in all people, with those found in healthy people and disease conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', Parkinson disease) classified as physiological and patholog- ical tremor respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Clinical research finds that tremors measured by accelerometers can effectively predict the category of tremors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Some works further show that hand tremors measured by accelerometers and gyroscopes are unique to an individual and stable over time, suggesting the feasibility of using tremors as a biometric for personal identification [12, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Threat Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We study a threat model of spoofing attack against smartphone facial recognition systems where imposters are assumed to launch a silicone face mask spoofing attack [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To better show the effectiveness of the synthesized IMUs, we further assume the silicone mask perfectly mimics the face of the victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' During the attack, the imposter wears the silicone mask and holds the victim’s smartphone for authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our objective is to extract camera motion from videos that represents the postural hand tremor of users to defend against such perfect silicone mask attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is worth noting this particular case study’s threat model re- quires users to hold their phones in their hands during facial recog- nition as the contact between their phones and hands provides a propagation path for the vibration information of hand tremor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We believe this is also the most frequent situation seen in smartphone- based facial recognition applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Nevertheless, there do exist some circumstances where users may want to place their phone on a table during authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our tremor recognition with syn- thesized virtual IMUs will not work in this case due to the lack of camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Similarly, a spoofing attacker cannot authen- ticate successfully in this case without providing the camera the correct motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To enable users to authenticate without holding their phones, we believe future works may look into other sensor side channels that acquire a different type of user biometric infor- mation such as body-radiated electromagnetic/heat energy without requiring direct contact with the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Synthesis Methodology Different methodologies can be used to synthesize virtual IMUs from camera motion side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, a completely model-based methodology requires understanding 𝑓 (·) and ˜𝑓 (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although the most accurate, it requires thorough understandings of every targeted camera system and is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Another possible methodology is to completely rely on neural network to process the raw videos and let the network figure out ˜𝑓{𝑣𝑖 }, which is sim- ilar to previous work of inferring sounds from object motions in videos [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This methodology requires intensive computation re- sources and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This work focuses on the middle ground by investigating a model-informed methodology that constructs ˜𝑓{𝑣𝑖 } based upon the concepts of image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Image registra- tion is the process of overlaying two or more images of the same scene that are taken at different times, from different viewpoints, and/or by different sensors [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The methodology aims to extract both inter-frame motions and intra-frame motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Understand Motion Modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To construct ˜𝑓{𝑣𝑖 }, the first step is to understand how motion signals are modulated onto im- age streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We analyze the motion modulation process from two different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' NSPW’22, October 24–27, 2022, New Hampshire, USA Yan Long and Kevin Fu Translation Similarity Euclidean Projective Sensor (IMU/Camera) X Y Z Roll Pitch Yaw IMU Output Sensor Motion Cam Output X Y Z Pitch Yaw Roll Accl X Accl Y Accl Z Gyro X Gyro Y Gyro Z Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Euc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Figure 2: Types of 2D image transformations corresponding to the type of camera motion and motion readings measured by physical IMUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Frame Transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The frame transformation perspec- tive considers changes of the frames subjected to camera motions as 2D image transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Figure 2 shows the possible image transformations corresponding to motion on each one of the six real-world axes and the measurements of physical IMUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As a re- sult, motions that can be measured by IMUs can also be mapped to inter-frame variations of the camera videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Rolling Shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Besides inter-frame variations, the rolling shut- ter property of most cameras on portable devices can generate intra- frame variations that embed high-frequency motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Rolling shutter is the shutter mechanism of commercial CMOS cameras, which exposes and samples the rows of an image sensor sequentially in- stead of simultaneously as in a global shutter [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' If viewing the possible 2D image transformations as bases, rolling shutter com- bine multiple transformations into a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It increases the effective sample rate of the motion signals provided by the camera side channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Based on the knowledge of how camera motion is modulated onto images, two corresponding categories of virtual IMU synthesis methods are introduced next to measure low-frequency and high- frequency information respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Low-frequency Information Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The frame trans- formation perspective enables measurements of low-frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It perceives the difference between two frames as the result of a single motion vector composed of single-axis mo- tions (Figure 2) within the period of one frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The camera imaging process thus becomes the sampling process of the measurable mo- tion signals with a sample rate that is the same as the video frame rate, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', 30 Hz in case of 30 fps videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Theoretically, all image reg- istration methods are applicable to extract inter-frame variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We discuss one possible construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Image Transformation Estimation (ITE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A straightforward way of extracting the frame differences is registering the frames with respect to a reference frame by estimating the 2D image trans- formations needed to warp the reference frame to the other frames as has been explored in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Each 2D transformation estimation generates a 3-by-3 transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' By concatenating each en- try of different transformation matrices chronologically, it produces 9 vectors that represent the output of ˜𝑓{𝑣𝑖 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Diverse algorithmic implementations of this method are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This works uses an image registration implementation based on phase correlation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3 High-frequency Information Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The rolling shut- ter perspective allows for the extraction of intra-frame high-frequency variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It perceives the difference between two frames as the result of multiple sequential motion vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The number of motion vectors is the same as the number of rows of the camera imag- ing sensor as each row is exposed and sampled sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The effective sample rate is thus the row-scanning rate of the rolling shutter, which is higher than 30 kHz for most commercial cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Nevertheless, not all signals within its Nyquist frequency can be recovered, as the non-zero exposure time causes motion blurs and attenuate the higher-frequency signals [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Similarly, a possible construction is introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Rolling Shutter Estimation (RSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Methods of rolling shut- ter estimation still compares different frames, but performs such comparison on the even smaller granularity level of rows or indi- vidual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Then, the methods concatenate the values generated by the comparison first across different rows of a single frame, and then across different frames to form the motion signal vec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' With the proposed methodology, this work converts rolling shutter estimation into a pixel-level image registration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Algorithms capable of pixel-level registration often generate dis- placement fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', matrices of the same size as the registered images, on the X and Y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The produced matrices are appar- ently high-dimension and difficult to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We can then group the matrices column-wise and average the columns in each group to produce easily understandable signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This work uses a diffeo- morphic image registration method [38] to implement RSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4 Demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Figure 3 shows the motion signals measured by a physical IMU (408 Hz sample rate) and virtual sensors using ITE and RSE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' A Google Pixel 2 smartphone held by a person recorded the physical IMU readings and camera videos si- multaneously, where the postural hand tremor of the person caused the camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The ITE and RSE methods have sample rates of 30 Hz and 34 kHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The figure only displays a single vector of the physical and virtual sensor measurements respectively that represents the horizontal motion to simplify the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Figure 3 (a) and (b) shows the measured signals with the video stabilization functionality being off and on respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' When video stabilization is off, the virtual sensor outputs of both the ITE and RSE method show strong correlation with the physical IMU measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is also clear that a 30 Hz sample rate is not sufficient to capture all the motion, as the ITE method’s signal shows larger distortions than that of the RSE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' When video stabilization is turned on, the camera motion signals deviate more from the IMU readings as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although the signal of RSE method still shows observable correlation with the IMU signal, ITE produces seemingly uncorrelated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3 Experiment We conduct preliminary tests with 4 people and a Google Pixel 2 smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The 4 participants are all healthy males with similar ages, heights, and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As a proof-of-concept instead of an actual system product, we regard facial recognition and tremor recognition as two decoupled problems and test them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The tremor recognition mechanism can be regarded as an addi- tional layer of protection besides the existing facial recognition / X 1 Z Pitch Yaw Roll IMU Output AcclX AcclY AcclZ GyroX GyroY GyroZ Cam Output Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Euc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='SensorMotionNSPW’22, October 24–27, 2022, New Hampshire, USA (a) (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 1 0 1 Amplitude IMU Accelerometer 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 1 0 1 Amplitude Image Transformation Estimation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 Time(s) 1 0 1 Amplitude Rolling Shutter Estimation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 1 0 1 Amplitude IMU Accelerometer 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 1 0 1 Amplitude Image Transformation Estimation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5 5 Time(s) 1 0 1 Amplitude Rolling Shutter Estimation Figure 3: Measurements of physical IMU accelerometer (408 Hz) and virtual IMU synthesized with the ITE and RSE methods from videos (30 fps frame rate, 1080p resolution) in 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Amplitudes are normalized to compared different measurement approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' (a) Videos stabilization is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' (b) Videos stabilization is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Strategically disabling sensor side channel mitigation mechanisms boosts up virtual sensors’ capability for measurand authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We investigate the impact of disabling and enabling video stabilization in both of the two tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The objective of testing tremor recognition is to verify the ef- fectiveness of the synthesized IMUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To that end, we also recorded the physical IMU readings for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The objective of testing facial recognition is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' First, it is important to inspect if the postural hand tremor of different people can already make a differ- ence in the original facial recognition systems without synthesis of virtual sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This verifies the necessity of constructing dedicated virtual IMUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Second, since turning off video stabilization may lead to better virtual sensor performance, it is also necessary to inspect if it would degrade the performance of facial recognition given that the videos are more shaky due to unmitigated camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1 Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The 4 participants act as the legitimate user in turn and the remaining 3 participants act as the imposters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' During the legitimate user sessions, each legitimate user holds the phone and records his own face for 30 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We refer to these videos as legitimate videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' During the spoofing attack sessions, each of the 3 imposters holds the phone but records the face of the legitimate user standing beside the imposter for 6 times to mimic a perfect silicone mask as assumed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We refer to these videos as imposter videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Each video recording is about 6s in length and the physical IMU readings are recorded simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The procedure is carried out first with video stabilization disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' At the end, each participant recorded 48 videos when he held the phone with 30 of them being legitimate videos and the other 18 being imposter videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We then repeat the procedure with video stabilization enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The total 384 videos (192 videos each set) are used for testing facial recognition and tremor recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2 Test Procedure & Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We generalize the authentication problem as an identification problem and use classification models to measure the effectiveness of the two authentication schemes against the spoofing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Facial recognition Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We tested MobileFaceNets [8] as the classification model which is a widely used facial recognition model designed for mobile platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 80% of each person’s legit- imate videos are used to enroll their faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The remaining 20% of legitimate videos together with all imposter videos that contain faces of the legitimate users are used as the authentication test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Facial recognition Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Both the legitimate users and im- posters’ videos authenticated with 100% success rate no matter the video stabilization was enabled or disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As expected, the results suggest that existing face authentication systems are mostly likely not designed to utilize camera motion side channel informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Mapping it to Equation 5, it suggests E[A𝑢 (𝑚)|𝐻0] → 1 and E[1 − A𝑢 (𝑚)|𝐻1] → 0 for the system under this specific spoofing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The results also show that disabling video stabilization to allow for more capable virtual IMUs did not affect the performance of the original facial recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Tremor Recognition Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For each video, we generate virtual IMU measurements using both the ITE and RSE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We extract common time-domain and frequency-domain features as the ones used in [5, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As a simple proof-of-concept, we did not use sophisticated machine learning models but directly utilized Matlab’s implementation of support vector machine (SVM) with a quadratic kernel and the default hyper-parameters [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 5-fold cross validation was performed in the training phase along with a one-vs- one multi-class classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Similar to facial recognition, for each legitimate user we use 80% of the legitimate videos (24 videos) in the training phase and the remaining 20% legitimate videos (6 videos) together with all imposter videos (18 videos, 6 from each of the three imposters) as authentication test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We then calculate the true positive and true negative rates on the test NSPW’22, October 24–27, 2022, New Hampshire, USA Yan Long and Kevin Fu set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To provide comparisons, we repeat the same procedure also for the physical IMU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Tremor Recognition Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Table 1 shows the results of tremor recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Virtual IMU using RSE had performance approaching that of the physical IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It suggests that under this specific spoofing attack, E[A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1])|𝐻0] → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='125 and E[1−A𝑚([𝑚𝑖𝑛𝑡,𝑚𝑣1])|𝐻1] → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='083 if using an AND logic to combine facial and tremor recogni- tion decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This results in 𝐸𝑚 −𝐸𝑢 → −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='875𝑐1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='083𝑐2, which is highly likely to be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is also clear that disabling video stabilization improves the performance of virtual IMUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3 Summary & Implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Our preliminary tests indicate a high probability that integrating user postural hand tremor infor- mation from camera motion side channels will help existing facial recognition systems defend against visual spoofing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Test results show MobileFaceNets could recognize legitimate users with 100% accuracy but could not detect (with 0% accuracy) a powerful silicone mask spoofing attack that almost perfectly replicates visual features of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This behavior is not a design defect of existing facial recognition systems, but an anticipated outcome of only using visual information during an authentication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' On the other hand, virtual IMUs synthesized from camera motion channel were able to detect such a visual spoofing attack with over 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5% accu- racy at a cost of reducing true positive rate to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The simplest approach of integrating virtual sensor into existing facial recogni- tion systems is to have a standalone tremor recognition module that processes camera motion information in the videos, and have the system declare a legitimate user only when both this tremor recognition module and the original facial recognition module de- clare it simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In this way, the overall system’s security performance increases in the face of facial spoofing attacks even with a lower true positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This result also suggests when a physical sensor system has poor performance on a security task, it is easy to produce an obvious marginal benefit on the system’s per- formance by integrating sensor side channel information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Of course, a more sophisticated decision system can tune its weights on the facial and tremor recognition modules to strike a better balance between usability and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Beyond camera motion side channels, our tests also provide one viable data point for the general concept of utilizing sensor side channels and reveal some common problems it faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' For example, we expect the same problem of usability-security trade-off in using virtual sensors synthesized from sensor side channels alongside the original physical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Essentially, physical sensors and synthe- sized virtual sensors provide two streams of information, each one of which is more reliable in one task but also unreliable in another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The design trade-off appears when the overall system needs to complete both tasks to achieve its functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4 Limitation & Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' With the goal of showing a proof- of-concept example, our experiment provides empirical statistical evidence for the benefit of utilizing camera motion side channels only based on a very limited data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The limitations of tested data lie in the following 4 main dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' First, the 4 young male participants may not provide a high enough degree of demographic diversity, especially for evaluating postural hand tremors which are highly dependent on age, gen- der, and health conditions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' While we based our choice of the Table 1: Test Accuracy of Tremor Recognition Physical IMU Virtual ITE Virtual RSE TPR TNR TPR TNR TPR TNR Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' OFF 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='8% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='4% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='7% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='5% Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' ON 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='8% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='8% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='7% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='8% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='2% TPR (true positive rate) and TNR (true negative rate) are the percentages of correctly recognizing a legitimate user and a perfect silicone mask spoofing attack respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In comparison, MobileFaceNets had TPR=100% and TNR=0% in our test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 4 participants on the hypothesis that more similar participants produce less distinct tremor patterns and thus help us estimate a lower bound of tremor recognition performance, we believe study- ing more diverse groups of people will generate new insights into recognition performance variability and possible strategies of recog- nition algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Second, we collected 30 samples of legitimate-user videos and 18 spoofing attack videos for each legitimate user’s authentication session within a single day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We find this initial set of samples pro- vided evidence to suggest the potential of utilizing hand tremor information from camera side channels to enhance existing facial recognition system’s security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is possible that tremor patterns can change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Although previous research shows hand tremor remains stable after 78 days [12], a longer duration needs to be investigated in future complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The recognition system may need to periodically update its database if tremor pattern is found to vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Third, we emulated perfect silicone masks by using the real faces of legitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This only provides an estimate of the upper bound of the overall recognition system’s performance improve- ment when tremor recognition is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Specifically, the benefit of including tremor recognition may get lower when a worse-quality silicone mask is used because the damage the attack can do to the original unimodal authentication system is lower while tremor recognition still causes a decrease in the true positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As a result, we suggest future works test different qualities of silicone masks on popular facial recognition systems to better assess the benefit of including virtual IMUs for tremor recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Fourth, the decoupling of facial recognition and tremor recogni- tion problems in this proof-of-concept case study prevents us from utilizing the temporal correlation between the facial and camera motion signals and investigating the impact of the correlation infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Intuitively, systems that inspect such temporal correlation information require spoofing attackers to further achieve synchro- nization between the physical and virtual sensors’ data streams and thus provide additional protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We envision real-world prod- ucts building upon the virtual sensors authentication concept to uti- lize deep-learning approaches for processing temporally-correlated physical and virtual sensors’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 5 DISCUSSION Below we discuss the major areas of possible future work and interesting research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Sensor Side Channel Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' To support future applications of sensor side channels, we believe more concrete and computable mathematical models than the framework proposed in Section 3 are needed as the current framework relies on abstract concepts NSPW’22, October 24–27, 2022, New Hampshire, USA instead of rigorous mathematical derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We envision future models to have the following features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' First, they need to enable exact definitions and determination of different types of sensor side channels by providing the algorithms for calculating signal correlations and threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Second, they need to provide quantitative metrics for measuring the usability-security trade-off mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Third, they need to delineate mecha- nisms for measuring the available signal quality and bandwidth of side channel measurement components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Security for Sensor Side Channel Authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Techni- cally, inseparable sensor side channels also provide the informa- tion needed for measurand authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We advocate the use of separable and controllable sensor side channels because they are protected from adversaries that exploit unmitigated side channels (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Nevertheless, risks of malicious exploitation still exist within authentication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' It is thus necessary for future works to consider how to ensure that side channels benefit the defender, but not adversaries that attempt eavesdropping and transduction attacks, during authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We believe an access control and permission system that is simi- lar to existing systems managing physical sensors on mobile plat- forms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=', Android) can be employed to prevent eavesdropping attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Virtual sensor entries can potentially be created and in- tegrated into existing permission systems so that knowledge and methodology of solving physical sensors’ problems can also benefit virtual sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Transduction attacks, on the other hand, are harder to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In the context of sensor side channel based measurand authentication, transduction attacks can be generalized as authen- tication spoofing that tries to modify perceived characteristics of the byproducts, measurers, and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' As a result, existing methodologies of spoofing detection may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In summary, we believe there are opportunities to address the problems of virtual sensors by reflecting on existing methodology for physical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Side Channels vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Legitimate Channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We believe there will be an interesting phenomenon that sensor side channels are turned into legitimate communication channels when active con- trols and dedicated APIs are developed to support as well as regulate the use of sensor side channels in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' After all, the key dif- ference between side channels and legitimate channels is whether the channels are designed, intended, and allowed by the system’s security policy or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' When such side channels are regarded as legitimate channels, however, new side-channel information may again be discovered to be embedded in such “legitimate” informa- tion as hardware and computation technologies keep advancing and extending the boundary of recoverable physical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We thus believe it is necessary for researchers to take a development perspective and periodically examine the security implications of sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Fewer Sensors via Sensor Repurposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In a broader context, we believe the technique of synthesizing virtual sensors from sensor side channels aligns with the general idea of repurposing sensors for different sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Essentially, we are trying to shift sensor hardware functionalities to the software space by understanding the transformation between different forms of signal energy and car- rying out additional model-based computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' In contrast to the current trend of deploying more and more sensors in the Internet of Things era, we cannot help thinking if such sensor repurposing ideas would allow us to reduce the number of physical sensors and achieve more abstract and manageable sensor peripheral systems that are subjected to smaller attack surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Besides reducing the number of physical sensors, the technique could also be applied to enhance existing systems that require new functionalities but have harsh environmental conditions where a hardware update is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' This idea is revealed in the example of NASA’s Voyager 1 spacecraft which needed to measure plasma density in order to determine its location relative to the heliosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Voyager 1’s plasma spectrometer stopped working in 1980, making a direct plasma density measurement impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' However, the op- eration team learned that our sun sometimes emits shock waves that can cause the plasma surrounding the spacecraft to oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The team then measured the oscillation using Voyager 1’s onboard plasma wave sensing system as a proxy of the plasma density [18], essentially synthesizing a virtual plasma density sensor by under- standing the energy transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' 6 CONCLUSION This paper argued that analog sensor side channels can benefit defenders by providing an opportunity to authenticate the sensor measurands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Future sensor designs can consider actively controlling sensor side channels after finding ways to mitigate these channels, instead of simply eliminating sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We first in- troduced a framework for defining and characterizing sensor side channels, and then formulated the problem of measurand authenti- cation using virtual sensors synthesized from sensor side channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We also introduced three specific ways of applying the model of measurand authentication by verifying signal byproducts, sensor measurers, and sensor environments respectively, and provided examples of each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Synthesizing virtual sensors from the side channels of physical sensors formulates a mechanism for repurposing existing sensor hardware to harvest extra modalities of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' We believe the applications of this mechanism can potentially span a much larger scope than authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Going forward, we envision that virtual sensor synthesis could develop into a new research area that actively interacts with the existing research areas of digital forensics, sensor fusion, multimodal deep learning and perception, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' The fundamental research question we will need to explore is how to model the transformations between the energies of different information modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' REFERENCES [1] 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' Trick: Switching off the optical image stabilization of iPhone X, XS, XS Max, XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFKT4oBgHgl3EQfLS1R/content/2301.11745v1.pdf'} +page_content='sir-apfelot.' metadata={'source': 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+https://doi.org/10.5303/JKAS.2022.55.6.207 +55: 207 ∼ 213, 2022 December +pISSN: 1225-4614 · eISSN: 2288-890X +Published under Creative Commons license CC BY-SA 4.0 +http://jkas.kas.org +RENOVATION OF SEOUL RADIO ASTRONOMY OBSERVATORY +AND ITS FIRST MILLIMETER VLBI OBSERVATIONS +Naeun Shin1,2, Yong-Sun Park1,3, Do-Young Byun2,4, Jinguk Seo1, Dongkok Kim1, +Cheulhong Min1, Hyunwoo Kang2, Keiichi Asada5, Wen-Ping Lo5, and Sascha Trippe1,3 +1Department of Physics and Astronomy, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, +Republic of Korea; neshin@kasi.re.kr +2Korea Astronomy and Space science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea +3SNU Astronomy Research Center, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea +4University of Science and Technology, 217, Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea +5Academia Sinica Institute of Astronomy and Astrophysics, Taipei 10617, Taiwan +Received October 16, 2022; accepted November 22, 2022 +Abstract: +The Seoul Radio Astronomy Observatory (SRAO) operates a 6.1-meter radio telescope on the +Gwanak campus of Seoul National University. We present the efforts to reform SRAO to a Very Long +Baseline Interferometry (VLBI) station, motivated by recent achievements by millimeter interferometer +networks such as Event Horizon Telescope, East Asia VLBI Network, and Korean VLBI Network (KVN). +For this goal, we installed a receiver that had been used in the Combined Array for Research in Millimeter- +wave Astronomy and a digital backend, including an H-maser clock. The existing hardware and software +were also revised, which had been dedicated only to single-dish operations. After several years of prepara- +tions and test observations in 1 and 3-millimeter bands, a fringe was successfully detected toward 3C 84 in +86 GHz in June 2022 for a baseline between SRAO and KVN Ulsan station separated by 300 km. Thanks +to the dual frequency operation of the receiver, the VLBI observations will soon be extended to the 1 mm +band and verify the frequency phase referencing technique between 1 and 3-millimeter bands. +Key words: +instrumentation: interferometers — techniques : high angular resolution +1. INTRODUCTION +Since its inauguration in 2001 in a 3 mm band, the +6-meter telescope of the Seoul Radio Astronomy Ob- +servatory (SRAO) has been actively used in research +and education (Koo et al. 2003). The antenna design +is identical to the one used for the Berkeley-Illinois- +Maryland Association array (Hudson 1998). The sur- +face is adjusted by Holographic surface measurements +to an accuracy of around 30 µm, enabling observations +up to 300 GHz (Lee et al. 2003). In 2013, SRAO devel- +oped a 1 mm band receiver with a side band separation +feature and a receiver temperature of 50 K in a single +sideband (SSB) (Lee et al. 2013). After several years of +operations of the receiver, it had suffered from troubles +in cooling systems, suspending regular operation. +Meanwhile, the Event Horizon Telescope (EHT) +presented the first image of the black hole in M87 at +230 GHz, highlighting the crucial role of the Very Long +Baseline Interferometry (VLBI) (The Event Horizon +Telescope collaboration et al. 2019). One of the major +science goals of the next generation EHT is the imag- +ing and monitoring of the jet launching region in M87 +and other active galactic nuclei (AGNs), aiming at a +more detailed understanding of the physics in the ul- +timate vicinity of supermassive black holes. +For this +aim, the EHT collaboration considers the upgrades of +the VLBI network that include a wider observing band- +Corresponding author: +Y.-S. Park +width, higher angular resolution using higher frequen- +cies and possibly longer baselines when extended to +space, and a more dense UV-coverage through the ad- +dition of numerous smaller telescopes. Scientists in east +Asia have also established East Asia VLBI Network op- +erating in the 1 mm band, called EAVN-hi. The Korean +VLBI Network (KVN) tested the performance of its +Yonsei antenna with a 1 mm receiver. The SRAO-KVN +Yonsei pair is separated only by about 10 km, and it +could provide a short baseline for the recovery of ex- +tended features. The fourth telescope of KVN is under +construction, designed to operate up to 230 GHz with +an aperture efficiency of around 30%. +For the possible participation in the next genera- +tion EHT and EAVN-hi and the collaborations with the +extended KVN, SRAO retrofitted the observing system +that had been set to single-dish mode operations. We +describe the new receiver and the revisions of the ob- +serving system in Section 2. The preparation of VLBI +backends is introduced in Section 3. Section 4 presents +the test observation in single-dish mode and the first +fringe detection in the VLBI experiments. The results +are summarised and discussed in Section 5. +2. RETROFIT OF THE RECEIVER SYSTEM +2.1. Import of a CARMA Receiver +Since the trouble with the 4 K cryogenic system men- +tioned above was severe, we had to consider purchasing +a new one. +We finally decided to recycle a receiver +207 +arXiv:2301.11566v1 [astro-ph.IM] 27 Jan 2023 + +208 +Shin et al. +Figure 1. The view of the CARMA receiver. Teflon lenses +and beam combiners for the two bands are shown together. +The LO plates are mounted on both sides of the dewar. +(Photo by Richard Plambeck) +that had once operated in the Combined Array for Re- +search in Millimeter-wave Astronomy (CARMA). The +main reasons for adopting the CARMA receiver are that +it is paired with an economical 15 K cooling system, +and its performance is still good. The only drawback +may be that the CARMA receiver works in double side- +bands (DSB) mode. It features dual-band operation in +the 1 mm and the 3 mm bands, receiver temperatures +of 60–70 K (DSB), and compactness (Hull & Plambeck +2015). +The CARMA receiver is displayed in Figure 1. Mil- +limeter waves from the antenna pass through two Teflon +lenses without complicated beam-guiding optics and go +into the dewar. +The interior of the dewar is divided +mainly into two parts—left for the 1 mm band and right +for the 3 mm band, as shown in Figure 2. The part of +the 1 mm band consists of a feed horn, a circular polar- +izer, an orthomode transducer (OMT), superconductor- +insulator-superconductor (SIS) mixers, and low-noise +amplifiers (LNA) (Hull & Plambeck 2015). This config- +uration allows dual circular polarization observations. +The 3 mm band receiving system is similar but does +not have an OMT, allowing only the right-handed cir- +cular polarization (RCP). Table 1 summarizes the re- +ceiver parameters in the two bands. +Local oscillator +(LO) plates at both sides of the dewar generate the LO +signals for the two bands, as shown in Figure 1. The +mylar beam combiners reflect the LO signals in front of +the dewar. Then the LO signals combined with waves +from celestial sources propagate into the dewar. +After we took the receiver to the laboratory, we +tested its components one by one and measured receiver +temperatures representing its overall performance. Fig- +ure 3 displays the I-V curve of the 1 mm and the 3 mm +mixers without LO power. One can see a steep current +increase at bias voltages of around 10 mV. The mea- +sured receiver temperatures are similar to the ones in +Hull & Plambeck (2015). +2.2. Cryogenic System +The cryogenic system consists of a cold head and a com- +pressor. Within the dewar is a CTI1020 cryogenic cold +Figure 2. The cryogenically cooled dewar contains the 1 mm +band receiver parts on the left side and the 3 mm band parts +on the right side. The copper plates at the bottom are the +first and second cooling stages, and the 3rd is the square +plate at the top of the pillar in the center. +Feed horns, +circular polarizers, an OMT, and mixers are cooled below +4 K, while LNAs are cooled to ∼15 K. +Table 1 +Receiver parameters +1 mm +3 mm +RF frequency +215–270 GHz +84–115 GHz +IF bandwidth +0–1 GHz +Polarization +LCP/RCP +RCP +Receiver temperature +60–70 K (DSB) +System temperature +400 K (DSB) +150 K (DSB) +head that consists of two cooling stages, where the sec- +ond stage cools down to 15 K. It was modified to have +an additional third stage by the University of Califor- +nia, Berkeley, which is shown in Figure 2. The cooling +power of the third stage is only 50 mW, but enough +to cool down mixers and feed horns to around 4 K. +The 15 K cryogenic system is not so expensive but has +gained the 4 K stage after modification, which is one of +the reasons that we adopt the CARMA receiver. + +Feedhorn +Circular +OMT +polarizer +Mixer +4K,3rd stage +LNA +15K,2ndstage +70K, 1st stageBeamcombiner +For3mmband +Beamcombiner +For1mmbandRenovation of SRAO and Its First VLBI Observation +209 +Figure 3. The unpumped I-V curves of the 1 mm and the +3 mm mixers were measured in the laboratory. They exhibit +a nice non-linearity. +Figure 4. The system diagram including the conversion of IF +frequencies. The VLBI instruments and the spectrometer +for single-dish observation are located in the backend room. +The third stage can be further cooled down by low- +ering the pumping frequency. The cold head typically +operates at 72 rpm, which seems the most efficient cy- +cle frequency for the heat transfer of cryogenic regen- +erator (Ogawa et al. 1991). However, it is empirically +found that if the pumping period is increased by a fac- +tor of two, then the temperature of the third stage fur- +ther decreases below 3 K, probably because the cooling +He gas spends more time inside the third stage (Plam- +beck et al. 1993). We usually set the pumping speed +to 30 rpm using a frequency inverter, SV-is7, made by +a local company, LS Electric Co. The typical temper- +atures during the regular operation are 47.4, 12.2, and +2.8 K, respectively, from the first to third stages. As for +the compressor, we use the model M600 made by Tril- +lion. Though a water-cooled compressor has a higher +cooling capacity with a smaller volume, we adopt the +air-cooled one since the air-cooled compressor requires +fewer maintenance efforts. +2.3. Signal Chain +The signal’s intermediate frequency (IF) bandwidth af- +ter the LNAs is rather wide but is narrowed down to +3.5–4.5 GHz through bandpass filters. The IF signal is +Figure 5. The relative output power of the receiver in dB +for various locations of the 80 K absorber. The coordinate +origin of the map refers to the center of the subreflector. +The boresight of the feed horn is slightly shifted to the low +elevation. +then down-converted to 1–2 GHz in the IF processing +box in the cabin. It goes to the observatory building +and is converted to 0–1 GHz before being fed into the +spectrometers and VLBI backends. The spectrometer +covers 1 GHz bandwidth with 214 channels resulting in +a spectral resolution of 61 kHz. The signal flows for +single-dish and VLBI operation modes are summarized +in Figure 4. +2.4. Alignment of the Receiver +Since the configuration changed near the secondary fo- +cus, we need to check whether the direction of the max- +imum gain of the feed horn points to the center of the +subreflector. +First, we point the telescope towards a +mountain, which is seen at low elevations, i.e., the 300 K +background. Then by moving an absorber immersed in +liquid nitrogen at 80 K in front of the subreflector and +by measuring the output power of the receiver, we can +find the direction of the maximum response of the horn. +Figure 5 indicates that the feed horn looks at the sub- +reflector downward by 1◦, while it is well aligned in the +E-W direction. We corrected this misalignment by ad- +justing the heights of the receiver plate in four corners. +The change of the receiver position also affects the +pointing offsets, which must be corrected. We estab- +lished the pointing model by carrying out five-point ob- +servations for the standard stars in the Hipparcos cata- +log (Byun & Yun 2002), using the optical telescope at- +tached to the side of the antenna dish (Koo et al. 2003). +The rms pointing errors are around 15′′ in both direc- +tions right after the model fitting. However, systematic +offsets begin to appear, probably due to the differential +solar heating of the antenna structure, which may affect +the 1 mm band observation because of its smaller beam +size. + +1mm +3mm +200 +200 +175 +175 +150 +150 +(uA) +125 +current (uA) +125 +current +100 +100 +75 +75 +50 +50 +25 +25 +0 +15 +F5 +0 +? +10 +15 +20 +10 +15 +20 +voltage (mv) +voltage (mv)Single dish operation +ab +Spectrometer +Down +converter +TF 0~1 +Down +Mark 6 +R2DBE +口 +converter +VLB operation0.24 +E +0.21 +2 +0.18 +(aap) +1 +0.15 +(8p) +73 +0 +0.12 +relative +-1 +-2 +0.06 +0.03 +-3 +i +2 +E +0.00 +4 +-2 +-1 +0 +4 +relative AZ (degree)210 +Shin et al. +Figure 6. The VLBI backends, R2DBE, Mark 6, and GPS receiver are located in the backend room of SRAO. A spectrometer +and the second IF processing box are shown together on the left. +Figure 7. The first spectrum of CO J = 2−1 at 230.538 GHz +toward the Orion KL was obtained by the SRAO with the +CARMA receiver. The temperature scale and the velocity +of the spectrum are not calibrated accurately. A velocity +resolution is 0.079 km s−1 per channel. +3. INSTALLATION OF VLBI EQUIPMENT +A digital sampler, recorders, and an H-maser clock are +installed to carry out VLBI experiments in SRAO, as +shown in Figure 6. +As for the digital sampler, we +borrowed a ROACH 2 digital backend (R2DBE) from +Academia Sinica Institute of Astronomy and Astro- +physics (ASIAA) (Vertatschitsch et al. 2015). Its max- +imum data rate is 16 Gbps, from 4 Gbps samples per +second in four levels for two data streams. We also bor- +rowed a Mark 6 recorder from ASIAA and four disk +packs from the Korea Astronomy and Space science In- +stitute (KASI) with a total capacity of 256 TBytes. The +H-maser clock, provided by KASI, distributes a refer- +ence frequency of 10 MHz to all the frequency synthe- +sizers in the receiver system and the recorder. An addi- +tional component, the GPS receiver, compares its one +pulse per second (PPS) signal and that of the H-maser +clock for synchronization with other stations. +Several frequency synthesizers in the receiving sys- +tem were of low quality and independent of each other +in the past since it was not so critical for single-dish ob- +servations. For the VLBI experiment, we bought a few +high-quality frequency synthesizers, such as Keysight +E8257D, to reduce the phase noises of the system. They +replaced old synthesizers and are bound to the 10 MHz +reference from the H-maser clock. +4. TEST OBSERVATIONS +4.1. Single-Dish Observations +In February 2019, SRAO detected the first light of the +CO J = 2 − 1 line at 230.538 GHz toward Orion KL +with the CARMA receiver (Figure 7). The best system +temperature for a 1 mm band is measured as 400 K +(DSB). It is found that, contrary to expectations, the +system temperature rises in the winter. The reduction +of cooling capacity due to the hardening of oil in the +compressor may cause this problem. We expect lower +system temperatures by keeping the compressor warm +in the winter. +The 3 mm band receiver was operated in the spring +for recent two years. The lowest system temperature in +the 3 mm band is 150 K (DSB) at 86 GHz. We have +made single-dish observations of several bright spectral + +1ppssignal +comparator +·R2DBE +Spectrometer +Mark 6 +Down/converter +GPS receiverOrionCO2-1atSRAO +60 +'~/DAT/FFT114754.1.Saf +50 +40 +30 +20 +10 +0 +-10 +7000 +7200 +7400 +7600 +7800 +8000 +8200 +channelRenovation of SRAO and Its First VLBI Observation +211 +Table 2 +3 mm VLBI test observation parameters +SRAO +KVN +Equipment +R2DBE / Mark 6 +OCTAD / Mark 6 +Bandwidth +1024 MHz +2048 MHz +Polarization +RCP +LCP/RCP +System temperature +150 K (DSB) +280 K (SSB) +line sources, such as Orion KL and TX-Cam, to inspect +the pointing accuracy and verify the overall system per- +formance before the VLBI observation. +4.2. VLBI Test Observations +4.2.1. Test Observations in the 1 mm Band +As soon as we got the first light at 230 GHz in 2019, +we conducted international VLBI observations. +Dur- +ing UT 11:00 to 18:00 on March 18th and 19th, 2019, +the Greenland Telescope (GLT), built by ASIAA in +Taiwan, and Solar Planetary Atmosphere Research +Telescope, operated by Osaka prefecture university in +Japan, joined the campaign. Targets were three bright +AGNs (M87, NGC 6251, Mrk 501) and four bright cal- +ibrators (3C 371, 1928+738, 3C 345, 1633+382). The +spectral line sources (NGC 7027, IRC+10216, DR21) +are also observed for autocorrelation. No fringes were +detected for baselines that include SRAO. The sus- +pected reason is that one LO accidentally missed the +10 MHz reference signal from the H-maser clock. +The second test observation was run during UT +from 08:00 to 15:00 on February 1st and 5th, 2020, col- +laborating with GLT and James Clerk Maxwell Tele- +scope. +Two bright AGNs (OJ 287, 3C 84) were ob- +served, and data was transferred to the correlation cen- +ter at the Shanghai Astronomical Observatory via the +internet. +The typical data transfer rate was around +500 Mbps. Unfortunatly, the fringes were not detected +in this session too. We concluded that the main reason +for the failure is the substantial system temperature, +probably because of the reason mentioned in Section 4.1 +and cloudy weather during the observation. +4.2.2. VLBI Test Observations with KVN in the 3 mm Band +Since the scheduling is not easy in the international +VLBI observations, and thus the chance of observations +is limited, we cooperated with the domestic VLBI sys- +tem, the KVN. Since the KVN does not have the 1 mm +receivers, the VLBI test observation was made in the +3 mm band. From UT 05:00 June 3rd, 2022, the bright +source, 3C 84 and Orion KL, were observed alternately +in three scans each, with 10 minutes of exposure per +scan. 3C 84, a bright AGN, is selected as the main tar- +get to find a fringe. Orion KL is observed to check the +frequency offset and the pointing accuracy. In SRAO, a +data stream of 1024 MHz bandwidth from 86 to 87 GHz +is Nyquist-sampled and recorded in 4 Gbps. +On the +other hand, KVN stations recorded signals of 2048 MHz +bandwidth from 85 to 87 GHz for two polarization in +16 Gbps using an OCTAD sampler (Oh et al. 2017) and +Mark 6 recorder. The system setup is summarized in +Table 2. Because of the instrumental problems, only the +KVN Ulsan station recorded the data among the three +KVN stations. The recorded data in the Mark 6 was +transferred to the Daejeon correlation center located at +KASI through the internet. The DiFX Software corre- +lator performed the correlation in 65 K channels. +Visibility data is analyzed with the NRAO Astro- +nomical Image Processing System (AIPS). The fringe +fitting solutions are found using the task FRING of +AIPS with a solution interval of 30 seconds. The upper +panel of Figure 8 shows the clear and consistent phase +as a function of frequency after the fringe fitting. The +cross-power spectrum between SRAO and KVN Ulsan +is displayed together. +Figure 9 presents a fringe solution in a delay and +delay rate plane. The visibility amplitude is averaged +over the central 640 MHz of the bandwidth, where +phases remain constant. We can see a strong peak for a +specific delay and delay rate. The obtained delay rate +of 250 mHz between SRAO and KVN Ulsan is mainly +due to the frequency offset of about 200 mHz of the +H-maser clock at KVN Ulsan station. +The observed SNRs are 60 on average for a solution +interval of 30 seconds. We can estimate the expected +SNR using the equation, +SNR = 0.88 F +� +2 ∆ν τ +SEFD1 × SEFD2 +, +where F is a source flux density, τ the integration time, +and ∆ν the observing bandwidth. The SEFDi is the +system equivalent flux density of a station i. We set +∆ν = 640 MHz and τ = 30 seconds. The F is assumed +as 16 Jy based on the single-dish mode observation of +KVN toward 3C 84 in May 2022. +The SEFD of the +KVN is 3200 Jy. The SEFD of the SRAO is not mea- +sured, but it can be accurately inferred as 4.1 × 104 Jy +from the comparison of the antenna temperatures of the +SiO v = 1, J = 2 − 1 transition toward Orion KL ob- +tained by both KVN in single-dish mode and SRAO on +the same day. The resulting SNR is 120, much larger +than the observed one. +The factor of two difference may be originated from +the two reasons. The first one is a longer averaging time +to derive visibility data from raw data streams from the +two antennas. We set one second for it as usual, but the +delay rate of −250 mHz makes the phase rotate by 90◦ +for one second, which results in a factor of 2/π degrada- + +212 +Shin et al. +Figure 8. The visibility phase (top) and the cross power +spectrum (bottom) of 3C 84 for the SRAO to KVN Ulsan +baseline. +tion of visibility amplitudes. The other one is related to +the source size. According to the 86 GHz observation +of 3C 84 with KVN, the core and the jet components +are separated in north-south direction by about 3 mas +(Wajima et al. 2020). The SRAO-KVN Ulsan baseline +length of 300 km results in the minimum fringe spac- +ing of 2.5 mas, and thus the source might be partially +resolved out. Figure 8 shows a flux density of ∼5 Jy, +weaker than that of single-dish observation indeed. The +solution interval comparable to the coherence time of +the atmosphere may also affect the SNR. However, it is +found that data reduction with the solution interval of +10 seconds does not improve the SNR. +5. DISCUSSION AND CONCLUSIONS +The SRAO retrofitted the receiver and cooling systems, +enabling observations in the 1 mm and the 3 mm bands +with reasonable noise temperature. We also installed +instruments such as a digital backend and high-speed +recorder to reform SRAO to a VLBI station. +After +many years of single-dish observation and international +VLBI campaigns, we detected fringes at 86 GHz be- +tween KVN Ulsan and SRAO in June 2022 for the first +time. Since the LO chain of the 1 mm band is identi- +cal to that of the 3 mm band, except for the frequency +tripler, it is expected to find fringes in the 1 mm band +in the near future. +To make routine VLBI observations possible, we +need to improve our system further: A new receiver is +under development, adopting sideband separation mix- +ers and a new cryostat. +It will widen the IF band- +width from 1 GHz to 2 GHz and have a lower noise +temperature than the CARMA receiver. +Moreover, +SRAO will be connected to Korea Research Environ- +ment Open Network (KREONET), and the data trans- +fer rate will be over 10 Gbps. In addition, with the help +of KREONET, a 10 MHz reference signal from KVN +Yonsei may be transmitted to SRAO via dark fibers, +which will replace the H-maser clock operated over its +life span. +−1000−500 0 +500 1000 −500 +−2500250500 +SRAO-KVN Ulsan, June 3rd, 2022, 3C 84 +delay (ns) +rate (mHz) +Figure 9. The visibility amplitude averaged over the central +640 MHz bandwidth as a function of the delay and the delay +rate. The delay and the delay rate at the peak are about +−580 ns and −250 mHz, respectively. +VLBI observations at millimeter wavelengths are +considerably affected by rapid atmospheric phase vari- +ations. +This infection can be minimized by applying +the solutions of the phase variations taken at the lower +frequencies to the higher target frequencies (Rioja et al. +2011, 2014; Algaba et al. 2015; Park et al. 2018). SRAO +can implement the frequency phase referencing with +minimal effort, since the 1 mm and the 3 mm receiving +components are in one dewar, and LOs share a common +frequency reference. The only thing to do is to install +a frequency-selective surface and a reflection mirror in +front of the dewar. +In summary, a series of test observations and +planned future works guarantee that SRAO can be a +member of the international mm VLBI network. +ACKNOWLEDGMENTS +The authors gratefully acknowledge the contribution of +Jongho Park for providing a code of the 3D plot of +fringes and a kind explanation. +We are grateful to +the staff of the KVN who helped to operate the ar- +ray and to correlate the data. The KVN and a high- +performance computing cluster are facilities operated +by the KASI. The KVN observations and correlations +are supported through the high-speed network con- +nections among the KVN sites provided by the KRE- +ONET, which is managed and operated by the KISTI. +This work was supported partially by National Re- +search Foundation of Korea grant funded by the Korean +government (MEST) (No. 2019R1A6A1A10073437 and +2022R1F1A1075115) and partially by KASI under the +R&D program (Project No. 2022-1-860-03) supervised +by the Ministry of Science and ICT. + +Phase (degree) +200 +0 +200 +10 +() +Amplitude +86000 +86200 +86400 +86600 +86800 +87000 +Frequency (MHz)Renovation of SRAO and Its First VLBI Observation +213 +REFERENCES +Algaba, J.-C., Zhao, G.-Y., Lee, S.-S., et al. 2015, Interfero- +metric monitoring of Gamma–Ray bright Active Galactic +Nuclei II: Frequency Phase Transfer, JKAS, 48, 237 +Byun, D., & Yun, Y.-Z. 2002, Development of control soft- +wares for the SRAO 6-meter telescope based on PCs run- +ning Linux, Exp. 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J., Dodson, R., Jung, T., et al. 2014, Verification +of the Astrometric Performance of the Korean VLBI Net- +work, Using Comparative SFPR Studies with the VLBA +at 14/7 mm, AJ, 148, 84 +Vertatschitsch, L., Primiani, R., Young, A., et al. 2015, +R2DBE: A Wideband Digital Backend for the Event Hori- +zon Telescope, PASP, 127, 1226 +Wajima, K., Kino, M., & Kawakatu, N. 2020, Constraints +on the Circumnuclear Disk through Free-Free Absorption +in the Nucleus of 3C 84 with KaVA and KVN at 43 and +86 GHz, ApJ, 895, 35 + diff --git a/yNFJT4oBgHgl3EQfhiyb/content/tmp_files/load_file.txt b/yNFJT4oBgHgl3EQfhiyb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9419c97be3b0e625b488bcaf7d911bca6267cb07 --- /dev/null +++ b/yNFJT4oBgHgl3EQfhiyb/content/tmp_files/load_file.txt @@ -0,0 +1,346 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf,len=345 +page_content='Journal of the Korean Astronomical Society https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='5303/JKAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='207 55: 207 ∼ 213, 2022 December pISSN: 1225-4614 · eISSN: 2288-890X Published under Creative Commons license CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='0 http://jkas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='kas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='org RENOVATION OF SEOUL RADIO ASTRONOMY OBSERVATORY AND ITS FIRST MILLIMETER VLBI OBSERVATIONS Naeun Shin1,2, Yong-Sun Park1,3, Do-Young Byun2,4, Jinguk Seo1, Dongkok Kim1, Cheulhong Min1, Hyunwoo Kang2, Keiichi Asada5, Wen-Ping Lo5, and Sascha Trippe1,3 1Department of Physics and Astronomy, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' neshin@kasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='kr 2Korea Astronomy and Space science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea 3SNU Astronomy Research Center, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea 4University of Science and Technology, 217, Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea 5Academia Sinica Institute of Astronomy and Astrophysics, Taipei 10617, Taiwan Received October 16, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' accepted November 22, 2022 Abstract: The Seoul Radio Astronomy Observatory (SRAO) operates a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1-meter radio telescope on the Gwanak campus of Seoul National University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We present the efforts to reform SRAO to a Very Long Baseline Interferometry (VLBI) station, motivated by recent achievements by millimeter interferometer networks such as Event Horizon Telescope, East Asia VLBI Network, and Korean VLBI Network (KVN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' For this goal, we installed a receiver that had been used in the Combined Array for Research in Millimeter- wave Astronomy and a digital backend, including an H-maser clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The existing hardware and software were also revised, which had been dedicated only to single-dish operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' After several years of prepara- tions and test observations in 1 and 3-millimeter bands, a fringe was successfully detected toward 3C 84 in 86 GHz in June 2022 for a baseline between SRAO and KVN Ulsan station separated by 300 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Thanks to the dual frequency operation of the receiver, the VLBI observations will soon be extended to the 1 mm band and verify the frequency phase referencing technique between 1 and 3-millimeter bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Key words: instrumentation: interferometers — techniques : high angular resolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' INTRODUCTION Since its inauguration in 2001 in a 3 mm band, the 6-meter telescope of the Seoul Radio Astronomy Ob- servatory (SRAO) has been actively used in research and education (Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The antenna design is identical to the one used for the Berkeley-Illinois- Maryland Association array (Hudson 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The sur- face is adjusted by Holographic surface measurements to an accuracy of around 30 µm, enabling observations up to 300 GHz (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' In 2013, SRAO devel- oped a 1 mm band receiver with a side band separation feature and a receiver temperature of 50 K in a single sideband (SSB) (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' After several years of operations of the receiver, it had suffered from troubles in cooling systems, suspending regular operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Meanwhile, the Event Horizon Telescope (EHT) presented the first image of the black hole in M87 at 230 GHz, highlighting the crucial role of the Very Long Baseline Interferometry (VLBI) (The Event Horizon Telescope collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' One of the major science goals of the next generation EHT is the imag- ing and monitoring of the jet launching region in M87 and other active galactic nuclei (AGNs), aiming at a more detailed understanding of the physics in the ul- timate vicinity of supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' For this aim, the EHT collaboration considers the upgrades of the VLBI network that include a wider observing band- Corresponding author: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Park width, higher angular resolution using higher frequen- cies and possibly longer baselines when extended to space, and a more dense UV-coverage through the ad- dition of numerous smaller telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Scientists in east Asia have also established East Asia VLBI Network op- erating in the 1 mm band, called EAVN-hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The Korean VLBI Network (KVN) tested the performance of its Yonsei antenna with a 1 mm receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The SRAO-KVN Yonsei pair is separated only by about 10 km, and it could provide a short baseline for the recovery of ex- tended features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The fourth telescope of KVN is under construction, designed to operate up to 230 GHz with an aperture efficiency of around 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' For the possible participation in the next genera- tion EHT and EAVN-hi and the collaborations with the extended KVN, SRAO retrofitted the observing system that had been set to single-dish mode operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We describe the new receiver and the revisions of the ob- serving system in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The preparation of VLBI backends is introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Section 4 presents the test observation in single-dish mode and the first fringe detection in the VLBI experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The results are summarised and discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' RETROFIT OF THE RECEIVER SYSTEM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Import of a CARMA Receiver Since the trouble with the 4 K cryogenic system men- tioned above was severe, we had to consider purchasing a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We finally decided to recycle a receiver 207 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='11566v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='IM] 27 Jan 2023 208 Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The view of the CARMA receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Teflon lenses and beam combiners for the two bands are shown together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The LO plates are mounted on both sides of the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' (Photo by Richard Plambeck) that had once operated in the Combined Array for Re- search in Millimeter-wave Astronomy (CARMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The main reasons for adopting the CARMA receiver are that it is paired with an economical 15 K cooling system, and its performance is still good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The only drawback may be that the CARMA receiver works in double side- bands (DSB) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' It features dual-band operation in the 1 mm and the 3 mm bands, receiver temperatures of 60–70 K (DSB), and compactness (Hull & Plambeck 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The CARMA receiver is displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Mil- limeter waves from the antenna pass through two Teflon lenses without complicated beam-guiding optics and go into the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The interior of the dewar is divided mainly into two parts—left for the 1 mm band and right for the 3 mm band, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The part of the 1 mm band consists of a feed horn, a circular polar- izer, an orthomode transducer (OMT), superconductor- insulator-superconductor (SIS) mixers, and low-noise amplifiers (LNA) (Hull & Plambeck 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' This config- uration allows dual circular polarization observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The 3 mm band receiving system is similar but does not have an OMT, allowing only the right-handed cir- cular polarization (RCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Table 1 summarizes the re- ceiver parameters in the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Local oscillator (LO) plates at both sides of the dewar generate the LO signals for the two bands, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The mylar beam combiners reflect the LO signals in front of the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Then the LO signals combined with waves from celestial sources propagate into the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' After we took the receiver to the laboratory, we tested its components one by one and measured receiver temperatures representing its overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Fig- ure 3 displays the I-V curve of the 1 mm and the 3 mm mixers without LO power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' One can see a steep current increase at bias voltages of around 10 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The mea- sured receiver temperatures are similar to the ones in Hull & Plambeck (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Cryogenic System The cryogenic system consists of a cold head and a com- pressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Within the dewar is a CTI1020 cryogenic cold Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The cryogenically cooled dewar contains the 1 mm band receiver parts on the left side and the 3 mm band parts on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The copper plates at the bottom are the first and second cooling stages, and the 3rd is the square plate at the top of the pillar in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Feed horns, circular polarizers, an OMT, and mixers are cooled below 4 K, while LNAs are cooled to ∼15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Table 1 Receiver parameters 1 mm 3 mm RF frequency 215–270 GHz 84–115 GHz IF bandwidth 0–1 GHz Polarization LCP/RCP RCP Receiver temperature 60–70 K (DSB) System temperature 400 K (DSB) 150 K (DSB) head that consists of two cooling stages, where the sec- ond stage cools down to 15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' It was modified to have an additional third stage by the University of Califor- nia, Berkeley, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The cooling power of the third stage is only 50 mW, but enough to cool down mixers and feed horns to around 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The 15 K cryogenic system is not so expensive but has gained the 4 K stage after modification, which is one of the reasons that we adopt the CARMA receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Feedhorn Circular OMT polarizer Mixer 4K,3rd stage LNA 15K,2ndstage 70K, 1st stageBeamcombiner For3mmband Beamcombiner For1mmbandRenovation of SRAO and Its First VLBI Observation 209 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The unpumped I-V curves of the 1 mm and the 3 mm mixers were measured in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' They exhibit a nice non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The system diagram including the conversion of IF frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The VLBI instruments and the spectrometer for single-dish observation are located in the backend room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The third stage can be further cooled down by low- ering the pumping frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The cold head typically operates at 72 rpm, which seems the most efficient cy- cle frequency for the heat transfer of cryogenic regen- erator (Ogawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' However, it is empirically found that if the pumping period is increased by a fac- tor of two, then the temperature of the third stage fur- ther decreases below 3 K, probably because the cooling He gas spends more time inside the third stage (Plam- beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We usually set the pumping speed to 30 rpm using a frequency inverter, SV-is7, made by a local company, LS Electric Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The typical temper- atures during the regular operation are 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='4, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='8 K, respectively, from the first to third stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' As for the compressor, we use the model M600 made by Tril- lion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Though a water-cooled compressor has a higher cooling capacity with a smaller volume, we adopt the air-cooled one since the air-cooled compressor requires fewer maintenance efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Signal Chain The signal’s intermediate frequency (IF) bandwidth af- ter the LNAs is rather wide but is narrowed down to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='5 GHz through bandpass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The IF signal is Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The relative output power of the receiver in dB for various locations of the 80 K absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The coordinate origin of the map refers to the center of the subreflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The boresight of the feed horn is slightly shifted to the low elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' then down-converted to 1–2 GHz in the IF processing box in the cabin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' It goes to the observatory building and is converted to 0–1 GHz before being fed into the spectrometers and VLBI backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The spectrometer covers 1 GHz bandwidth with 214 channels resulting in a spectral resolution of 61 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The signal flows for single-dish and VLBI operation modes are summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Alignment of the Receiver Since the configuration changed near the secondary fo- cus, we need to check whether the direction of the max- imum gain of the feed horn points to the center of the subreflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' First, we point the telescope towards a mountain, which is seen at low elevations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=', the 300 K background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Then by moving an absorber immersed in liquid nitrogen at 80 K in front of the subreflector and by measuring the output power of the receiver, we can find the direction of the maximum response of the horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 5 indicates that the feed horn looks at the sub- reflector downward by 1◦, while it is well aligned in the E-W direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We corrected this misalignment by ad- justing the heights of the receiver plate in four corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The change of the receiver position also affects the pointing offsets, which must be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We estab- lished the pointing model by carrying out five-point ob- servations for the standard stars in the Hipparcos cata- log (Byun & Yun 2002), using the optical telescope at- tached to the side of the antenna dish (Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The rms pointing errors are around 15′′ in both direc- tions right after the model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' However, systematic offsets begin to appear, probably due to the differential solar heating of the antenna structure, which may affect the 1 mm band observation because of its smaller beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 1mm 3mm 200 200 175 175 150 150 (uA) 125 current (uA) 125 current 100 100 75 75 50 50 25 25 0 15 F5 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 10 15 20 10 15 20 voltage (mv) voltage (mv)Single dish operation ab Spectrometer Down converter TF 0~1 Down Mark 6 R2DBE 口 converter VLB operation0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='24 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='21 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='18 (aap) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='15 (8p) 73 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='12 relative 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='03 3 i 2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='00 4 2 1 0 4 relative AZ (degree)210 Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The VLBI backends, R2DBE, Mark 6, and GPS receiver are located in the backend room of SRAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' A spectrometer and the second IF processing box are shown together on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The first spectrum of CO J = 2−1 at 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='538 GHz toward the Orion KL was obtained by the SRAO with the CARMA receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The temperature scale and the velocity of the spectrum are not calibrated accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' A velocity resolution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='079 km s−1 per channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' INSTALLATION OF VLBI EQUIPMENT A digital sampler, recorders, and an H-maser clock are installed to carry out VLBI experiments in SRAO, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' As for the digital sampler, we borrowed a ROACH 2 digital backend (R2DBE) from Academia Sinica Institute of Astronomy and Astro- physics (ASIAA) (Vertatschitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Its max- imum data rate is 16 Gbps, from 4 Gbps samples per second in four levels for two data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We also bor- rowed a Mark 6 recorder from ASIAA and four disk packs from the Korea Astronomy and Space science In- stitute (KASI) with a total capacity of 256 TBytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The H-maser clock, provided by KASI, distributes a refer- ence frequency of 10 MHz to all the frequency synthe- sizers in the receiver system and the recorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' An addi- tional component, the GPS receiver, compares its one pulse per second (PPS) signal and that of the H-maser clock for synchronization with other stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Several frequency synthesizers in the receiving sys- tem were of low quality and independent of each other in the past since it was not so critical for single-dish ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' For the VLBI experiment, we bought a few high-quality frequency synthesizers, such as Keysight E8257D, to reduce the phase noises of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' They replaced old synthesizers and are bound to the 10 MHz reference from the H-maser clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' TEST OBSERVATIONS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Single-Dish Observations In February 2019, SRAO detected the first light of the CO J = 2 − 1 line at 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='538 GHz toward Orion KL with the CARMA receiver (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The best system temperature for a 1 mm band is measured as 400 K (DSB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' It is found that, contrary to expectations, the system temperature rises in the winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The reduction of cooling capacity due to the hardening of oil in the compressor may cause this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We expect lower system temperatures by keeping the compressor warm in the winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The 3 mm band receiver was operated in the spring for recent two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The lowest system temperature in the 3 mm band is 150 K (DSB) at 86 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=" We have made single-dish observations of several bright spectral 1ppssignal comparator R2DBE Spectrometer Mark 6 Down/converter GPS receiverOrionCO2-1atSRAO 60 '~/DAT/FFT114754." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='Saf 50 40 30 20 10 0 10 7000 7200 7400 7600 7800 8000 8200 channelRenovation of SRAO and Its First VLBI Observation 211 Table 2 3 mm VLBI test observation parameters SRAO KVN Equipment R2DBE / Mark 6 OCTAD / Mark 6 Bandwidth 1024 MHz 2048 MHz Polarization RCP LCP/RCP System temperature 150 K (DSB) 280 K (SSB) line sources, such as Orion KL and TX-Cam, to inspect the pointing accuracy and verify the overall system per- formance before the VLBI observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' VLBI Test Observations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Test Observations in the 1 mm Band As soon as we got the first light at 230 GHz in 2019, we conducted international VLBI observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Dur- ing UT 11:00 to 18:00 on March 18th and 19th, 2019, the Greenland Telescope (GLT), built by ASIAA in Taiwan, and Solar Planetary Atmosphere Research Telescope, operated by Osaka prefecture university in Japan, joined the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Targets were three bright AGNs (M87, NGC 6251, Mrk 501) and four bright cal- ibrators (3C 371, 1928+738, 3C 345, 1633+382).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The spectral line sources (NGC 7027, IRC+10216, DR21) are also observed for autocorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' No fringes were detected for baselines that include SRAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The sus- pected reason is that one LO accidentally missed the 10 MHz reference signal from the H-maser clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The second test observation was run during UT from 08:00 to 15:00 on February 1st and 5th, 2020, col- laborating with GLT and James Clerk Maxwell Tele- scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Two bright AGNs (OJ 287, 3C 84) were ob- served, and data was transferred to the correlation cen- ter at the Shanghai Astronomical Observatory via the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The typical data transfer rate was around 500 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Unfortunatly, the fringes were not detected in this session too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We concluded that the main reason for the failure is the substantial system temperature, probably because of the reason mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1 and cloudy weather during the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' VLBI Test Observations with KVN in the 3 mm Band Since the scheduling is not easy in the international VLBI observations, and thus the chance of observations is limited, we cooperated with the domestic VLBI sys- tem, the KVN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Since the KVN does not have the 1 mm receivers, the VLBI test observation was made in the 3 mm band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' From UT 05:00 June 3rd, 2022, the bright source, 3C 84 and Orion KL, were observed alternately in three scans each, with 10 minutes of exposure per scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 3C 84, a bright AGN, is selected as the main tar- get to find a fringe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Orion KL is observed to check the frequency offset and the pointing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' In SRAO, a data stream of 1024 MHz bandwidth from 86 to 87 GHz is Nyquist-sampled and recorded in 4 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' On the other hand, KVN stations recorded signals of 2048 MHz bandwidth from 85 to 87 GHz for two polarization in 16 Gbps using an OCTAD sampler (Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2017) and Mark 6 recorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The system setup is summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Because of the instrumental problems, only the KVN Ulsan station recorded the data among the three KVN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The recorded data in the Mark 6 was transferred to the Daejeon correlation center located at KASI through the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The DiFX Software corre- lator performed the correlation in 65 K channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Visibility data is analyzed with the NRAO Astro- nomical Image Processing System (AIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The fringe fitting solutions are found using the task FRING of AIPS with a solution interval of 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The upper panel of Figure 8 shows the clear and consistent phase as a function of frequency after the fringe fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The cross-power spectrum between SRAO and KVN Ulsan is displayed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 9 presents a fringe solution in a delay and delay rate plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The visibility amplitude is averaged over the central 640 MHz of the bandwidth, where phases remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We can see a strong peak for a specific delay and delay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The obtained delay rate of 250 mHz between SRAO and KVN Ulsan is mainly due to the frequency offset of about 200 mHz of the H-maser clock at KVN Ulsan station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The observed SNRs are 60 on average for a solution interval of 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We can estimate the expected SNR using the equation, SNR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='88 F � 2 ∆ν τ SEFD1 × SEFD2 , where F is a source flux density, τ the integration time, and ∆ν the observing bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The SEFDi is the system equivalent flux density of a station i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We set ∆ν = 640 MHz and τ = 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The F is assumed as 16 Jy based on the single-dish mode observation of KVN toward 3C 84 in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The SEFD of the KVN is 3200 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The SEFD of the SRAO is not mea- sured, but it can be accurately inferred as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='1 × 104 Jy from the comparison of the antenna temperatures of the SiO v = 1, J = 2 − 1 transition toward Orion KL ob- tained by both KVN in single-dish mode and SRAO on the same day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The resulting SNR is 120, much larger than the observed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The factor of two difference may be originated from the two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The first one is a longer averaging time to derive visibility data from raw data streams from the two antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We set one second for it as usual, but the delay rate of −250 mHz makes the phase rotate by 90◦ for one second, which results in a factor of 2/π degrada- 212 Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The visibility phase (top) and the cross power spectrum (bottom) of 3C 84 for the SRAO to KVN Ulsan baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' tion of visibility amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The other one is related to the source size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' According to the 86 GHz observation of 3C 84 with KVN, the core and the jet components are separated in north-south direction by about 3 mas (Wajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The SRAO-KVN Ulsan baseline length of 300 km results in the minimum fringe spac- ing of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content='5 mas, and thus the source might be partially resolved out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Figure 8 shows a flux density of ∼5 Jy, weaker than that of single-dish observation indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The solution interval comparable to the coherence time of the atmosphere may also affect the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' However, it is found that data reduction with the solution interval of 10 seconds does not improve the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS The SRAO retrofitted the receiver and cooling systems, enabling observations in the 1 mm and the 3 mm bands with reasonable noise temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We also installed instruments such as a digital backend and high-speed recorder to reform SRAO to a VLBI station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' After many years of single-dish observation and international VLBI campaigns, we detected fringes at 86 GHz be- tween KVN Ulsan and SRAO in June 2022 for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Since the LO chain of the 1 mm band is identi- cal to that of the 3 mm band, except for the frequency tripler, it is expected to find fringes in the 1 mm band in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' To make routine VLBI observations possible, we need to improve our system further: A new receiver is under development, adopting sideband separation mix- ers and a new cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' It will widen the IF band- width from 1 GHz to 2 GHz and have a lower noise temperature than the CARMA receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Moreover, SRAO will be connected to Korea Research Environ- ment Open Network (KREONET), and the data trans- fer rate will be over 10 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' In addition, with the help of KREONET, a 10 MHz reference signal from KVN Yonsei may be transmitted to SRAO via dark fibers, which will replace the H-maser clock operated over its life span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' −1000−500 0 500 1000 −500 −2500250500 SRAO-KVN Ulsan, June 3rd, 2022, 3C 84 delay (ns) rate (mHz) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The visibility amplitude averaged over the central 640 MHz bandwidth as a function of the delay and the delay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The delay and the delay rate at the peak are about −580 ns and −250 mHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' VLBI observations at millimeter wavelengths are considerably affected by rapid atmospheric phase vari- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' This infection can be minimized by applying the solutions of the phase variations taken at the lower frequencies to the higher target frequencies (Rioja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2011, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Algaba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' SRAO can implement the frequency phase referencing with minimal effort, since the 1 mm and the 3 mm receiving components are in one dewar, and LOs share a common frequency reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The only thing to do is to install a frequency-selective surface and a reflection mirror in front of the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' In summary, a series of test observations and planned future works guarantee that SRAO can be a member of the international mm VLBI network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of Jongho Park for providing a code of the 3D plot of fringes and a kind explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' We are grateful to the staff of the KVN who helped to operate the ar- ray and to correlate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The KVN and a high- performance computing cluster are facilities operated by the KASI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' The KVN observations and correlations are supported through the high-speed network con- nections among the KVN sites provided by the KRE- ONET, which is managed and operated by the KISTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' This work was supported partially by National Re- search Foundation of Korea grant funded by the Korean government (MEST) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFJT4oBgHgl3EQfhiyb/content/2301.11566v1.pdf'} +page_content=' 2019R1A6A1A10073437 and 2022R1F1A1075115) and partially by KASI under the R&D 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